Archived NIST Technical Series Publication
The attached publication has been archived (withdrawn), and is provided solely for historical purposes.
It may have been superseded by another publication (indicated below).

Archived Publication
Series/Number:
Title:NIST Special Publication 800-56A Revision 2
Recommendation for Pair-Wise Key Establishment Schemes Using
Discrete Logarithm Cryptography

Publication Date(s):May 2013

Withdrawal Date:

April 16, 2018

Withdrawal Note:

SP 800-56A Rev. 2 is superseded in its entirety by the publication of
SP 800-56A Rev. 3.

Superseding Publication(s)The attached publication has been superseded by the following publication(s):Series/Number:
Title:
Author(s):NIST Special Publication 800-56A Rev. 3
Recommendation for Pair-Wise Key Establishment Schemes Using
Discrete Logarithm Cryptography

Elaine Barker; Lily Chen; Allen Roginsky; Apostol Vassilev; Richard
Davis

Publication Date(s):April 2018

URL/DOI:

https://doi.org/10.6028/NIST.SP.800-56Ar3

Additional Information (if applicable)
Contact:

Computer Security Division (Information Technology Laboratory)

Latest revision of theattached publication:
Related information:
Withdrawal
announcement (link):

https://csrc.nist.gov/Projects/Key-Management/Key-Establishment
https://csrc.nist.gov/publications/detail/sp/800-56A/rev-3/final

N/A

Date updated: Ɖƌŝůϭϲ͕ϮϬϭϴNIST Special Publication 800-56A
Revision 2

Recommendation for Pair-Wise Key
Establishment Schemes Using Discrete
Logarithm Cryptography
Elaine Barker
Lily Chen
Allen Roginsky
Miles Smid
http://dx.doi.org/10.6028/NIST.SP.800-56Ar2

C O M P U T E R

S E C U R I T Y

NIST Special Publication 800-56A
Revision 2

Recommendation for Pair-Wise Key
Establishment Schemes Using Discrete
Logarithm Cryptography
Elaine Barker
Lily Chen
Allen Roginsky
Computer Security Division
Information Technology Laboratory
Miles Smid
Orion Security Solutions, Inc.
McLean, VA
http://dx.doi.org/10.6028/NIST.SP.800-56Ar2

May 2013

U.S. Department of Commerce
Rebecca Blank, Acting Secretary
National Institute of Standards and Technology
Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director

Authority
This publication has been developed by NIST to further its statutory responsibilities under the Federal
Information Security Management Act (FISMA), Public Law (P.L.) 107-347. NIST is responsible fordeveloping information security standards and guidelines, including minimum requirements for Federal
information systems, but such standards and guidelines shall not apply to national security systems
without the express approval of appropriate Federal officials exercising policy authority over such
systems. This guideline is consistent with the requirements of the Office of Management and Budget(OMB) Circular A-130, Section 8b(3), Securing Agency Information Systems, as analyzed in Circular A130, Appendix IV: Analysis of Key Sections. Supplemental information is provided in Circular A-130,
Appendix III, Security of Federal Automated Information Resources.
Nothing in this publication should be taken to contradict the standards and guidelines made mandatory
and binding on Federal agencies by the Secretary of Commerce under statutory authority. Nor should
these guidelines be interpreted as altering or superseding the existing authorities of the Secretary ofCommerce, Director of the OMB, or any other Federal official. This publication may be used bynongovernmental organizations on a voluntary basis and is not subject to copyright in the United States.
Attribution would, however, be appreciated by NIST.
National Institute of Standards and Technology Special Publication 800-56A
Natl. Inst. Stand. Technol. Spec. Publ. 800-56A, Rev2, 138 pages (May 2013)
CODEN: NSPUE2

http://dx.doi.org/10.6028/NIST.SP.800-56Ar2
Certain commercial entities, equipment, or materials may be identified in this document in order todescribe an experimental procedure or concept adequately. Such identification is not intended to imply
recommendation or endorsement by NIST, nor is it intended to imply that the entities, materials, or
equipment are necessarily the best available for the purpose.
There may be references in this publication to other publications currently under development by NIST
in accordance with its assigned statutory responsibilities. The information in this publication, including
concepts and methodologies, may be used by Federal agencies even before the completion of suchcompanion publications. Thus, until each publication is completed, current requirements, guidelines,
and procedures, where they exist, remain operative. For planning and transition purposes, Federal
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Organizations are encouraged to review all draft publications during public comment periods andprovide feedback to NIST. All NIST Computer Security Division publications, other than the ones
noted above, are available at http://csrc.nist.gov/publications.

Comments on this publication may be submitted to:
National Institute of Standards and Technology
Attn: Computer Security Division, Information Technology Laboratory
100 Bureau Drive (Mail Stop 8930) Gaithersburg, MD 20899-8930
Email: 56A2012rev-comments@nist.gov

ii

Reports on Computer Systems Technology
The Information Technology Laboratory (ITL) at the National Institute of Standards and Technology(NIST) promotes the U.S. economy and public welfare by providing technical leadership for the Nation’s
measurement and standards infrastructure. ITL develops tests, test methods, reference data, proof ofconcept implementations, and technical analyses to advance the development and productive use of
information technology. ITL’s responsibilities include the development of management, administrative,
technical, and physical standards and guidelines for the cost-effective security and privacy of other thannational security-related information in Federal information systems. The Special Publication 800-series
reports on ITL’s research, guidelines, and outreach efforts in information system security, and its
collaborative activities with industry, government, and academic organizations.

Abstract
This Recommendation specifies key-establishment schemes based on the discrete logarithm problem overfinite fields and elliptic curves, including several variations of Diffie-Hellman and Menezes-QuVanstone(MQV) key establishment schemes.

Keywords
Diffie-Hellman; elliptic curve cryptography; finite field cryptography; key-agreement; key-confirmation;
key derivation; key establishment; key-transport; MQV.

Acknowledgements
The authors gratefully acknowledge and appreciate the contributions by Rich Davis of the
National Security Agency and Dustin Moody of the National Institute of Standards andTechnology. The authors acknowledge the contributions by Don Johnson made to the previous
versions of this Recommendation. The authors wish to thank their colleagues who reviewed
drafts of this document and provided valuable comments. The authors also appreciate thethoughtful and instructive comments received during the public comment period, which helped
to improve the quality of this publication.
Conformance Testing
Conformance testing for implementations of this Recommendation will be conducted within theframework of the Cryptographic Algorithm Validation Program (CAVP) and the Cryptographic
Module Validation Program (CMVP). The requirements of this Recommendation are indicatedby the word “shall.” Some of these requirements may be out-of-scope for CAVP or CMVP
validation testing, and thus are the responsibility of entities using, implementing, installing or
configuring applications that incorporate this Recommendation.

iii

Table of Contents
1. Introduction .............................................................................................................. 1
2. Scope and Purpose ................................................................................................. 1
3. Definitions, Symbols and Abbreviations ............................................................... 2
3.1

Definitions........................................................................................................................ 2

3.2

Symbols and Abbreviations ............................................................................................. 6

4. Overview of Key-Establishment Schemes ........................................................... 11
4.1

Key Establishment Preparations .................................................................................... 12

4.2

Key Agreement Process ................................................................................................. 14

4.3

DLC-based Key-Transport Process ............................................................................... 16

5. Cryptographic Elements ....................................................................................... 18
5.1

Cryptographic Hash Functions ...................................................................................... 18

5.2

Message Authentication Code (MAC) Algorithm ......................................................... 18
5.2.1

MAC Tag Computation for Key-Confirmation ................................................. 18

5.2.2

MAC Tag Verification for Key-Confirmation................................................... 19

5.3

Random Number Generation ......................................................................................... 19

5.4

Nonces........................................................................................................................... 19

5.5

Domain Parameters ........................................................................................................ 20
5.5.1

5.6

Domain Parameter Generation ........................................................................... 21
5.5.1.1

FFC Domain Parameter Generation.................................................... 21

5.5.1.2

ECC Domain Parameter Generation ................................................... 22

5.5.2

Assurances of Domain Parameter Validity ........................................................ 23

5.5.3

Domain Parameter Management........................................................................ 24

Key-Establishment Key Pairs ........................................................................................ 24
5.6.1

5.6.2

Key-Pair Generation .......................................................................................... 24
5.6.1.1

FFC Key-Pair Generation ................................................................... 24

5.6.1.2

ECC Key-Pair Generation................................................................... 24

Required Assurances .......................................................................................... 25
5.6.2.1

Assurances Required by the Key Pair Owner ..................................... 25iv

5.6.2.1.1 Owner Assurance of Correct Generation ............................. 27
5.6.2.1.2 Owner Assurance of Private-Key Validity .......................... 27
5.6.2.1.3 Owner Assurance of Public-Key Validity ........................... 28
5.6.2.1.4 Owner Assurance of Pair-wise Consistency ........................ 28
5.6.2.1.5 Owner Assurance of Possession of the Private Key ............ 30
5.6.2.2

Assurances Required by a Public Key Recipient ................................ 30
5.6.2.2.1 Recipient Assurance of Static Public-Key Validity ............. 32
5.6.2.2.2 Recipient Assurance of Ephemeral Public-Key Validity..... 32
5.6.2.2.3 Recipient Assurance of the Owner’s Possession of a Static
Private Key........................................................................... 33
5.6.2.2.4 Recipient Assurance of the Owner’s Possession of an
Ephemeral Private Key ........................................................ 34

5.6.2.3

Public Key Validation Routines.......................................................... 36
5.6.2.3.1 FFC Full Public-Key Validation Routine ............................ 36
5.6.2.3.2 ECC Full Public-Key Validation Routine............................ 36
5.6.2.3.3 ECC Partial Public-Key Validation Routine........................ 37

5.6.3

5.7

Key Pair Management........................................................................................ 38
5.6.3.1

Common Requirements on Static and Ephemeral Key Pairs.............. 38

5.6.3.2

Specific Requirements on Static Key Pairs ........................................ 38

5.6.3.3

Specific Requirements on Ephemeral Key Pairs ................................ 39

DLC Primitives .............................................................................................................. 40
5.7.1

5.7.2

Diffie-Hellman Primitives ................................................................................. 40
5.7.1.1

Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive........ 40

5.7.1.2

Elliptic Curve Cryptography Cofactor Diffie-Hellman (ECC CDH)
Primitive.............................................................................................. 41

MQV Primitives ................................................................................................. 41
5.7.2.1

Finite Field Cryptography MQV (FFC MQV) Primitive ................... 41
5.7.2.1.1 MQV2 Form of the FFC MQV Primitive ............................ 42
5.7.2.1.2 MQV1 Form of the FFC MQV Primitive ............................ 42

5.7.2.2

ECC MQV Associate Value Function ................................................ 43

5.7.2.3

Elliptic Curve Cryptography MQV (ECC MQV) Primitive ............... 43v

5.7.2.3.1 Full MQV Form of the ECC MQV Primitive ...................... 44
5.7.2.3.2 One-Pass Form of the ECC MQV Primitive........................ 44
5.8

Key-Derivation Methods for Key-Agreement Schemes ................................................ 45
5.8.1

The Single-step Key-Derivation Function ......................................................... 45
5.8.1.1

The Single-Step KDF Specification.................................................... 46

5.8.1.2

OtherInfo ............................................................................................. 48
5.8.1.2.1 The Concatenation Format for OtherInfo ............................ 49
5.8.1.2.2 The ASN.1 Format for OtherInfo ........................................ 50
5.8.1.2.3 Other Formats for OtherInfo ................................................ 51

5.8.2 The Extraction-then-Expansion Key-Derivation Procedure ........................................ 51
5.8.3 Application-Specific Key-Derivation Methods ........................................................... 51
5.9 Key-Confirmation ........................................................................................................... 51
5.9.1

Unilateral Key-Confirmation for Key-Agreement Schemes ............................. 52
5.9.1.1

5.9.2

Bilateral Key-Confirmation for Key-Agreement Schemes................................ 54
5.9.2.1

5.9.3

Adding Unilateral Key-Confirmation to a Key-Agreement Scheme .. 52

Adding Bilateral Key-Confirmation to a Key-Agreement Scheme .... 54

Security Strength of the Mac Tag ...................................................................... 55

6. Key-Agreement ...................................................................................................... 56
6.1

Schemes Using Two Ephemeral Key Pairs, C(2e) ........................................................ 59
6.1.1

C(2e, 2s) Schemes .............................................................................................. 59
6.1.1.1

dhHybrid1, C(2e, 2s, FFC DH) Scheme ............................................. 61

6.1.1.2(Cofactor) Full Unified Model, C(2e, 2s, ECC CDH) Scheme .......... 63

6.1.1.3

MQV2, C(2e, 2s, FFC MQV) Scheme ............................................... 64

6.1.1.4

Full MQV, C(2e, 2s, ECC MQV) Scheme ......................................... 66

6.1.1.5

Incorporating Key-Confirmation into a C(2e, 2s) Scheme ................. 68
6.1.1.5.1 C(2e, 2s) Scheme with Unilateral Key-Confirmation
Provided by party U to party V ............................................ 68
6.1.1.5.2 C(2e, 2s) Scheme with Unilateral Key-Confirmation
Provided by party V to party U ............................................ 69
6.1.1.5.3 C(2e, 2s) Scheme with Bilateral Key-Confirmation ............ 70vi

6.1.2C(2e, 0s) Schemes .............................................................................................. 71
6.1.2.1

dhEphem, C(2e, 0s, FFC DH) Scheme ............................................... 72

6.1.2.2(Cofactor) Ephemeral Unified Model, C(2e, 0s, ECC CDH) ............. 74

6.1.2.3 Key-Confirmation for C(2e, 0s) ............................................................ 75
6.2

Schemes Using One Ephemeral Key Pair, C(1e)........................................................... 75
6.2.1

C(1e, 2s) Schemes .............................................................................................. 75
6.2.1.1

dhHybridOneFlow, C(1e, 2s, FFC DH) Scheme ................................ 77

6.2.1.2(Cofactor) One-Pass Unified Model, C(1e, 2s, ECC CDH) Scheme.. 79

6.2.1.3

MQV1, C(1e, 2s, FFC MQV) Scheme ............................................... 81

6.2.1.4

One-Pass MQV, C(1e, 2s, ECC MQV) Scheme ................................. 83

6.2.1.5

Incorporating Key-Confirmation into a C(1e, 2s) Scheme ................. 85
6.2.1.5.1 C(1e, 2s) Scheme with Unilateral Key-Confirmation
Provided by party U to party V ............................................ 85
6.2.1.5.2 C(1e, 2s) Scheme with Unilateral Key-Confirmation
Provided by party V to party U ............................................ 87
6.2.1.5.3 C(1e, 2s) Scheme with Bilateral Key-Confirmation ............ 87

6.2.2

C(1e, 1s) Schemes .............................................................................................. 89
6.2.2.1

dhOneFlow, C(1e, 1s, FFC DH) Scheme ........................................... 90

6.2.2.2(Cofactor) One-Pass Diffie-Hellman, C(1e, 1s, ECC CDH) Scheme. 92

6.2.2.3

Incorporating Key-Confirmation into a C(1e, 1s) Scheme ................. 94
6.2.2.3.1 C(1e, 1s) Scheme with Unilateral Key-Confirmation
Provided by party V to party U ............................................ 94

6.3

C(0e, 2s) Schemes .......................................................................................................... 95
6.3.1

dhStatic, C(0e, 2s, FFC DH) Scheme ................................................................ 97

6.3.2(Cofactor) Static Unified Model, C(0e, 2s, ECC CDH) Scheme....................... 98

6.3.3

Incorporating Key-Confirmation into a C(0e, 2s) Scheme .............................. 100
6.3.3.1C(0e, 2s) Scheme with Unilateral Key-Confirmation Provided by party
U to party V ...................................................................................... 100

6.3.3.2C(0e, 2s) Scheme with Unilateral Key-Confirmation Provided by party
V to party U ...................................................................................... 101

6.3.3.3C(0e, 2s) Scheme with Bilateral Key-Confirmation ......................... 102

vii

7. DLC-Based Key-Transport .................................................................................. 103
7.1

Key-Transport Scheme ................................................................................................ 104

7.2

Key-Confirmation for Transported Keying Material ................................................... 105

8. Rationale for Selecting a Specific Scheme ........................................................ 106
9. Key Recovery ....................................................................................................... 117
10. Implementation Validation .................................................................................. 117
Appendix A: References ........................................................................................... 119
A.1 Normative References ................................................................................................... 119
A.2 Informative References ................................................................................................. 120
Appendix B: Rationale for Including Identifiers in the KDF Input ......................... 121
Appendix C: Data Conversions (Normative) ........................................................... 122
C.1 Integer-to-Byte String Conversion............................................................................... 122
C.2 Field-Element-to-Byte String Conversion ................................................................... 122
C.3 Field-Element-to-Integer Conversion .......................................................................... 122
Appendix D: Revisions (Informative)....................................................................... 123

viii

Figures
Figure 1: Owner key establishment preparations...........................................................................14
Figure 2: Key-agreement process ..................................................................................................16
Figure 3: Key-transport process .....................................................................................................17
Figure 4: C(2e, 2s) schemes: each party contributes a static and an ephemeral key pair ..............60
Figure 5: C(2e, 2s) scheme with unilateral key-confirmation from party U to party V ................69
Figure 6: C(2e, 2s) scheme with unilateral key-confirmation from party V to party U ...............70
Figure 7: C(2e, 2s) scheme with bilateral key-confirmation .........................................................71
Figure 8: C(2e, 0s) schemes: each party contributes only an ephemeral key pair .........................72
Figure 9: C(1e, 2s) schemes: party U contributes a static and an ephemeral key pair while party
V contributes only a static key pair ................................................................................76
Figure 10: C(1e, 2s) scheme with unilateral key-confirmation from party U to party V ..............86
Figure 11: C(1e, 2s) scheme with unilateral key-confirmation from party V to party U ..............87
Figure 12: C(1e, 2s) scheme with bilateral key-confirmation .......................................................88
Figure 13: C(1e, 1s) schemes: party U contributes an ephemeral key pair, and party V contributes
a static key pair ...............................................................................................................89
Figure 14: C(1e, 1s) scheme with unilateral key-confirmation from party V to party U ..............95
Figure 15: C(0e, 2s) schemes: each party contributes only a static key pair .................................96
Figure 16: C(0e, 2s) scheme with unilateral key-confirmation from party U to party V ............101
Figure 17: C(0e, 2s) scheme with unilateral key-confirmation from party V to party U ............102
Figure 18: C(0e, 2s) scheme with bilateral key-confirmation .....................................................103ix

Tables
Table 1: FFC parameter-size sets ...................................................................................................21
Table 2: ECC parameter-size sets ..................................................................................................23
Table 3: Initial assurances required by the key-pair owner ...........................................................26
Table 4: Optional renewal of assurances by the key-pair owner ...................................................27
Table 5: Assurances required by a public-key recipient ................................................................31
Table 6: Minimum output length of H for FFC domain parameter-size sets.................................46
Table 7: Minimum output length of H for ECC domain parameter sets .......................................46
Table 8: Minimum MacKey length and MacLen for FFC domain parameter sets ........................55
Table 9: Minimum MacKey length and MacLen for ECC domain parameter sets ........................56
Table 10: Key-agreement scheme categories ................................................................................57
Table 11: Key-agreement scheme subcategories ...........................................................................57
Table 12: Key-agreement schemes ................................................................................................57
Table 13: dhHybrid1 key-agreement scheme summary ................................................................62
Table 14: Full unified model key-agreement scheme summary ....................................................64
Table 15: MQV2 key-agreement scheme summary ......................................................................66
Table 16: Full MQV key-agreement Scheme Summary................................................................67
Table 17: dhEphem key-agreement scheme summary ..................................................................73
Table 18: Ephemeral unified model key-agreement scheme .........................................................74
Table 19: dhHybridOneFlow key-agreement scheme summary ...................................................79
Table 20: One-pass unified model key-agreement scheme summary ...........................................81
Table 21: MQV1 Key-agreement scheme summary .....................................................................83
Table 22: One-pass MQV model key-agreement scheme summary .............................................85
Table 23: dhOneFlow key-agreement scheme summary ...............................................................92
Table 24: One-pass Diffie-Hellman key-agreement scheme summary .........................................94
Table 25: dhStatic key-agreement scheme summary.....................................................................98
Table 26: Static unified model key-agreement scheme summary ...............................................100x

NIST SP 56A: Revision 2, May 2013

1.

Introduction

Many U.S. Government Information Technology (IT) systems need to employ well-establishedcryptographic schemes to protect the integrity and confidentiality of the data that they process.
Algorithms such as the Advanced Encryption Standard (AES) as defined in Federal Information
Processing Standard (FIPS) 197, Triple DES as specified in NIST Special Publication (SP) 80067, and HMAC as defined in FIPS 198 make attractive choices for the provision of these
services. These algorithms have been standardized to facilitate interoperability between systems.
However, the use of these algorithms requires the establishment of keying material between theparticipating entities in advance. Trusted couriers may manually distribute this secret keying
material. However, as the number of entities using a system grows, the work involved in the
distribution of the secret keying material could grow rapidly. Therefore, it is essential to support
the cryptographic algorithms used in modern U.S. Government applications with automated keyestablishment schemes.
A key-establishment scheme can be characterized as either a key-agreement scheme or a keytransport scheme. The asymmetric-key-based key-establishment schemes in thisRecommendation are based on the Diffie-Hellman (DH) and Menezes-Qu-Vanstone (MQV)
algorithms. Asymmetric-key-based key-establishment schemes are also specified in SP 800-56B,
Recommendation for Pair-Wise Key-establishment Schemes Using Integer Factorization
Cryptography. The selection of schemes specified in this Recommendation is based onstandards for key-establishment schemes developed by the Accredited Standards Committee(ASC) X9, Inc.: ANS X9.42, Agreement of Symmetric Keys using Discrete Logarithm
Cryptography, and ANS X9.63, Key Agreement and Key Transport using Elliptic Curve
Cryptography.

2.

Scope and Purpose

This Recommendation provides the specifications for key-establishment schemes that areappropriate for use by the U.S. Federal Government and is intended for use in conjunction with
NIST Special Publication 800-57, Recommendation for Key Management [SP 800-57]. This
Recommendation (i.e., SP 800-56A) and the Recommendation for Key Management [SP 800-57]
are intended to provide sufficient information for a vendor to implement secure key
establishment using asymmetric algorithms in FIPS 140 [FIPS 140] validated modules.
A scheme may be a component of a protocol, which in turn provides additional security
properties not provided by the scheme when considered by itself. Note that protocols, per se, arenot specified in this Recommendation.

1

NIST SP 56A: Revision 2, May 2013

3.

Definitions, Symbols and Abbreviations

3.1

Definitions

Approved

FIPS-approved or NIST-Recommended. An algorithm or technique thatis either 1) specified in a FIPS or NIST Recommendation, or 2) adopted
in a FIPS or NIST Recommendation and specified either (a) in an
appendix to the FIPS or NIST Recommendation, or (b) in a document
referenced by the FIPS or NIST Recommendation.

Assumption

This term is used to indicate the conditions that are required to be true
when an approved key-establishment scheme is executed in accordancewith this Recommendation.

Assurance ofprivate-key
possession

Confidence that an entity possesses a private key corresponding to a
public key.

Assurance ofvalidity

Confidence that either a key or a set of domain parameters isarithmetically correct.

Binding

Assurance of the integrity of an asserted relationship between items of
information that is provided by cryptographic means. Also see Trusted
association.

Bit length

The length in bits of a bit string.

Bit string

An ordered sequence of 0’s and 1’s.

Byte

A bit string consisting of eight bits. A byte is represented by ahexadecimal string of length two. The right-most hexadecimal character
represents the rightmost four bits of the byte, and the left-mosthexadecimal character of the byte represents the left-most four bits of thebyte. For example, 9d represents the bit string 10011101.

Byte string

An ordered sequence of bytes.

Certification
Authority (CA)The entity in a Public-Key Infrastructure (PKI) that is responsible for
issuing public key certificates and exacting compliance to a PKI policy.

Cofactor

The order of the elliptic curve group divided by the (prime) order of thegenerator point (i.e. the base point) specified in the domain parameters.

DestroyIn this Recommendation, to destroy is an action applied to a key or apiece of secret data. After a key or a piece of secret data is destroyed, no
information about its value can be recovered.

Domain parameters

The parameters used with a cryptographic algorithm that are common to adomain of users.

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NIST SP 56A: Revision 2, May 2013
EntityAn individual (person), organization, device, or process. “Party” is asynonym.

Ephemeral key pair

A key pair, consisting of a public key (i.e., an ephemeral public key) anda private key (i.e., an ephemeral private key) that is intended for a very
short period of use. The key pair is ordinarily used in exactly one
transaction of a cryptographic scheme; an exception to this is when theephemeral key pair is used in multiple transactions for a key-transport
broadcast. Contrast with a static key pair.

Fresh

Newly established keying material is considered to be fresh if theprobability of being equal to any previously established keying material isacceptably small. The “acceptably small probability” may be application
specific.

Hash function

A function that maps a bit string of arbitrary length to a fixed-length bit
string. Approved hash functions are expected to satisfy the following
properties:
1. One-way: It is computationally infeasible to find any input thatmaps to any pre-specified output, and
2. Collision resistant: It is computationally infeasible to find any two
distinct inputs that map to the same output.

Identifier

A bit string that is associated with a person, device or organization. It may
be an identifying name or a nickname, or may be something more abstract(for example, a string consisting of an IP address).

Key agreement

A (pair-wise) key-establishment procedure in which the resultant secretkeying material is a function of information contributed by bothparticipants, so that neither party can predetermine the value of the secretkeying material independently from the contributions of the other party.
Contrast with key-transport.

Key-agreementtransaction

An execution of a key-agreement scheme.

Key-confirmation

A procedure to provide assurance to one party (the key-confirmation
recipient) that another party (the key-confirmation provider) actuallypossesses the correct secret keying material and/or shared secret.

Key-confirmation
providerThe party that provides assurance to the other party (the recipient) that thetwo parties have indeed established a shared secret or shared keying
material.

3

NIST SP 56A: Revision 2, May 2013
Key-derivation
functionA function by which keying material is derived from a shared secret (or akey) and other information.

Key-derivation
methodA method by which keying material is derived from a shared secret andother information. A key-derivation method may use a key-derivation
function or a key-derivation procedure.

Key-derivation
procedure

A procedure consisting of multiple steps and using an approved
algorithm (e.g., a MAC algorithm) by which keying material is derived
from a shared secret and other information.

Key establishment

The procedure that results in keying material that is shared among
different parties.

Key-establishment
key pairA private/public key pair used in a key-establishment scheme. It can be astatic key pair or an ephemeral key pair.

Key-establishmenttransaction

An execution of a key-establishment scheme. It can be either a keyagreement transaction or a key-transport transaction.

Key-transport

A (pair-wise) key-establishment procedure whereby one party (thesender) selects a value for the secret keying material and then securely
distributes that value to another party (the receiver). Contrast with key
agreement.

Key-transporttransaction

An execution of a key-transport scheme.

Key-wrapping

In this Recommendation, key-wrapping is a method of protecting keying
material using a symmetric-key-based authenticated encryption method,such as a block cipher key-wrapping mode specified in [SP 800-38F] thatprovides both confidentiality and integrity protection.

Key-wrapping key

In this Recommendation, a key-wrapping key is a symmetric key
established through a key-agreement transaction and used with a keywrapping algorithm to protect the keying material to be transported.

Keying material

Data that is represented as a binary string such that any non-overlapping
segments of the string with the required lengths can be used as symmetric
cryptographic keys. In this Recommendation, keying material is derived
from a shared secret established during an execution of a keyestablishment scheme or generated by the sender in a key-transport
scheme. As used in this Recommendation, secret keying material may
include keys, secret initialization vectors, and other secret parameters.

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MAC tag

Data obtained from the output of a MAC algorithm that can be used by anentity to verify the integrity and the origination of the information used as
input.

Message
A MAC algorithm is a family of cryptographic functions – parameterized
Authentication Code by a symmetric key – that can be used to provide data origin(MAC) algorithm
authentication, as well as data integrity, by producing a MAC tag on
arbitrary data (the message). In this Recommendation, an approved MAC
algorithm is used for key-confirmation and may also be employed incertain key derivation methods.
Nonce

A time-varying value that has at most an acceptably small chance ofrepeating. For example, the nonce may be a random value that is
generated anew for each use, a timestamp, a sequence number, or somecombination of these.

Owner

For a static public key, static private key and/ or the static key paircontaining those components, the owner is the entity that is authorized touse the static private key corresponding to the static public key, whether
that entity generated the static key pair itself or a trusted party generated
the key pair for the entity. For an ephemeral key pair, ephemeral private
key or ephemeral public key, the owner is the entity that generated theephemeral key pair and uses the ephemeral private key associated withthe public key of that key pair.

Party

See entity.

Public-key
certificate

A data structure that contains an entity’s identifier(s), the entity's publickey (including an indication of the associated set of domain parameters)
and possibly other information, along with a signature on that data set thatis generated by a trusted party, i.e. a certificate authority, thereby binding
the public key to the included identifier(s).

ReceiverThe party that receives secret keying material via a key-transporttransaction. Contrast with sender.

Recipient

A party that (1) receives a public key; or (2) obtains assurance from anassurance provider (e.g., assurance of the validity of a candidate public
key or assurance of possession of the private key corresponding to a
public key); or (3) receives key-confirmation from a key-confirmation
provider.

SchemeA (cryptographic) scheme consists of a set of unambiguously specified
transformations that are capable of providing a (cryptographic) service
when properly implemented and maintained. A scheme is a higher-level
construct than a primitive and a lower-level construct than a protocol.

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3.2

Security strength(Also “Bits of
security”)

A number associated with the amount of work (that is, the number ofoperations) that is required to break a cryptographic algorithm or system.

Sender

The party that sends secret keying material to the receiver in a keytransport transaction. Contrast with receiver.

Shall

This term is used to indicate a requirement of a Federal Information
Processing Standard (FIPS) or a requirement that needs to be fulfilled toclaim conformance to this Recommendation. Note that shall may becoupled with not to become shall not.

Shared secretA secret value that has been computed using a key-agreement scheme andis used as input to a key derivation method.

ShouldThis term is used to indicate an important recommendation. Ignoring therecommendation could result in undesirable results. Note that should may
be coupled with not to become should not.

Static key pair

A key pair, consisting of a private key (i.e., a static private key) and apublic key (i.e., a static public key) that is intended for use for a relatively
long period of time and is typically intended for use in multiple key
establishment transactions. Contrast with an ephemeral key pair.

Symmetric-key
algorithm

A cryptographic algorithm that uses a single secret key for different
operations, such as for encryption and decryption.

Trusted association

Assurance of the integrity of an asserted relationship between items of
information that may be provided by cryptographic or non-cryptographic(e.g., physical) means. Also see Binding.

Trusted party

A trusted party is a party that is trusted by an entity to faithfully perform
certain services for that entity. An entity could be a trusted party for itself.

Trusted third party

A third party, such as a CA, that is trusted by its clients to perform certain
services. (By contrast, in a key establishment transaction, the participants,
parties U and V, are considered to be the first and second parties.)

Symbols and Abbreviations
General:AES

Advanced Encryption Standard (as specified in [FIPS 197]).

ASC

The American National Standards Institute (ANSI) Accredited Standards
Committee.

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ANS

American National Standard.

ASN.1

Abstract Syntax Notation One.

C(ie)Notation for a category of key-establishment schemes, in which iephemeral key pairs are used, where i ∈ {0, 1, 2}.

C(ie,js)

Notation for a subcategory of key-establishment schemes, in which iephemeral key pairs and j static key pairs are used. In thisRecommendation, schemes in the subcategories C(0e, 2s), C(1e, 2s),C(1e, 1s), C(2e, 0s), and C(2e, 2s) are defined.

CA

Certification Authority.

CDH

The cofactor ECC Diffie-Hellman key-agreement primitive.

DH

The (non-cofactor) FFC Diffie-Hellman key-agreement primitive.

DLC

Discrete Logarithm Cryptography, which is comprised of both Finite
Field Cryptography (FFC) and Elliptic Curve Cryptography (ECC).

EC

Elliptic Curve.

ECC

Elliptic Curve Cryptography, the public-key cryptographic methods using
operations in an elliptic curve group.

FF

Finite Field.

FFC

Finite Field Cryptography, the public-key cryptographic methods using
operations in a multiplicative group of a finite field.

H

An auxiliary function used in certain key derivation methods. H is either
an approved hash function, hash, or an HMAC-hash based on an
approved hash function, hash, with a salt value used as the HMAC key.

HMAC-hash

Keyed-hash Message Authentication Code (as specified in [FIPS 198])
with an approved hash function hash.

ID

The bit string denoting the identifier associated with an entity.

KC

Key-Confirmation.

KDF

Key Derivation Function.

MAC

Message Authentication Code.

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MAC(MacKey,
MacData)

A MAC algorithm with MacKey as the key, and MacData as the data.

MacTag

A MAC tag.

MQV

The Menezes-Qu-Vanstone key-agreement primitive.

Null

The empty bit string

SHA

Secure Hash Algorithm (as specified in [FIPS 180]).

TbitLen(X)A truncation function that outputs the most significant (i.e., leftmost)bitLen bits of the input bit string, X, when the bit length of X is greaterthan bitLen; otherwise, the function outputs X. For example, T2(1011) =10, T3(1011) = 101, T4(1011) = 1011, and T5(1011) = 1011.

TTP

Trusted Third Party.

U, V

Represent the two parties in a (pair-wise) key establishment scheme.

{}

In this Recommendation, the curly braces { } are used in the following
three situations: (1) {x} is used to indicate that the inclusion of x is
optional; for example, the notation “Input: w, {x}, y, and z” implies thatthe inclusion of x as an input is optional. (2) If both X and Y are binary
strings, the notation of binary string “Y{||X}” implies that the
concatenation of string X is optional. (3) {x1, x2, …, xk} indicates a set
with elements x1, x2, …, xk.

X || Y

Concatenation of two bit strings X and Y.

[a, b]The set of integers x, such that a ≤ x ≤ b.

x

The ceiling of x; the smallest integer ≥ x. For example, 5 = 5, 5.3 = 6.

ZA shared secret (represented as a byte string) that is used to derive secretkeying material using a key derivation method.

Ze

A component of the shared secret (represented as a byte string) that iscomputed using ephemeral keys in a Diffie-Hellman primitive.

Zs

A component of the shared secret (represented as a byte string) that iscomputed using static keys in a Diffie-Hellman primitive.

The following notations are used for FFC and ECC in this Recommendation. Note that thenotation sometimes differs between the two scheme types, due to the differing notations used in
the two standards on which this Recommendation is based (i.e., ANS X9.42 and ANS X9.63).
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FFC:
GF(p)

The finite field with p elements, where p is an (odd) prime number. The
elements of GF(p) can be represented by the set of integers {0, 1, …, p-1}.
The addition and multiplication operations for GF(p) can be realized by
performing the corresponding integer operations and reducing the results
modulo p.

GF(p)*The multiplicative group of non-zero field elements in GF(p).

g

An FFC domain parameter; the selected generator of the multiplicative
subgroup of prime order q in GF(p)*.

k mod p

The modular reduction of the (arbitrary) integer k by the (positive) integer p(the modulus). For the purposes of this Recommendation, j = k mod p is theunique integer satisfying the following two conditions: 0 ≤ j < p and k − j isa multiple of p.

p

An FFC domain parameter; an odd prime number that determines the sizeof the finite field GF(p).

pgenCounter

An FFC domain parameter; a value that may be output during domain
parameter generation to provide assurance at a later time that the resulting
domain parameters were generated using a canonical process.

q

When used as an FFC domain parameter, q is the (odd) prime number equal
to the order of the multiplicative subgroup of GF(p)* generated by g. Notethat q is a divisor of p – 1.

r U, r V

The ephemeral private keys of party U and party V, respectively. These areintegers in the range [1, q-1].

t U, t V

The ephemeral public keys of party U and party V, respectively. These areintegers in the range [2, p-2], representing elements in the finite field GF(p).

SEEDAn FFC domain parameter; an initialization value that is used during
domain parameter generation that can also be used to provide assurance at alater time that the resulting domain parameters were generated using acanonical process.

xU, xVThe static private keys of party U and party V, respectively. These areintegers in the range [1, q-1].

yU, yVThe static public keys of party U and party V, respectively. These areintegers in the range [2, p-2], representing elements in the finite field GF(p).

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ECC:
a, b

ECC domain parameters; two elements in the finite field GF(q) that define theequation of an elliptic curve, y2 = x3 + ax + b when q is a prime or y2 + xy = x3 +ax2 + b, when q = 2m for an integer m.

avf(Q)

The associate value of the elliptic curve point Q.

de,U, de,V

The ephemeral private keys of party U and party V, respectively. These areintegers in the range [1, n-1].

ds,U, ds,V

The static private keys of party U and party V, respectively. These are integers in
the range [1, n-1].

FR

Field Representation indicator (an ECC domain parameter); an indication of thebasis used for representing field elements. FR is NULL if the field has odd prime
order or if a Gaussian normal basis is used. If a polynomial basis representation
is used for a field of order 2m, then FR indicates the reduction polynomial (atrinomial or a pentanomial).

G

An ECC domain parameter, which is a distinguished (affine) point in an elliptic
curve group that generates a subgroup of prime order n.

GF(q)The finite field with q elements, where either q is an odd prime p or q is equal to2m, for some prime integer m. The elements of GF(q) are represented by the setof integers {0, 1, …, p-1} in the case that q is an odd prime p, or as bit strings oflength m bits in the case that q = 2m.

h

An ECC domain parameter; the cofactor, a positive integer that is equal to theorder of the elliptic curve group, divided by the order of the cyclic subgroup
generated by the distinguished point G. That is, nh is the order of the elliptic
curve, where n is the order of the cyclic subgroup generated by the distinguishedpoint G.

n

An ECC domain parameter; a prime that is the order of the cyclic subgroup
generated by the distinguished point G.

ØThe “point at infinity”; a special element of an elliptic curve group that serves as
the (additive) identity.

q

When used as an ECC domain parameter, q is the field size. It is either an odd
prime p or equal to 2m, for some prime integer m.

Qe,U, Qe,VThe ephemeral public keys of party U and party V, respectively. These are points
on the elliptic curve defined by the domain parameters.

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Qs,U, Qs,V

The static public keys of party U and party V, respectively. These are points on
the elliptic curve defined by the domain parameters.

SEED

An optional ECC domain parameter; an initialization value that is used during
domain parameter generation that can also be used to provide assurance at a latertime that the resulting domain parameters were generated using a canonicalprocess.

xP, yP

Elements of the finite field GF(q), representing the x and y coordinates,
respectively, of a point P.

4.

Overview of Key-Establishment Schemes

Secret cryptographic keying material may be electronically established between parties by using
a key-establishment scheme, that is, by using either a key-agreement scheme or a key-transport
scheme.
During a pair-wise key-agreement scheme, the secret keying material to be established is not sent
directly from one entity to another. Instead, the two parties exchange information from which
they each compute a shared secret that is used (along with other exchanged/known data) to
derive the secret keying material. The method used to combine the information made available toboth parties provides assurance that neither party can control the output of the key-agreement
process.
The key-agreement schemes described in this Recommendation employ public-key techniques
utilizing Discrete Logarithm Cryptography (DLC). The security of these DLC-based keyagreement schemes depends upon the intractability of the discrete logarithm problem in certain
settings.
In this Recommendation, the approved key-agreement schemes are described in terms of theroles played by parties “U” and “V.” These are specific labels that are used to distinguishbetween the two participants engaged in key agreement – irrespective of the actual labels that
may be used by a protocol employing a particular approved key-agreement scheme.
To be in conformance with this Recommendation, a protocol employing any of the approved
pair-wise key-agreement schemes shall unambiguously assign the roles of party U and party V tothe participants by clearly defining which participant performs the actions ascribed by thisRecommendation to party U, and which performs the actions ascribed herein to party V.
During key-transport, one party selects the secret keying material to be transported. The secretkeying material is then wrapped using a shared key-wrapping key and an approved key-wrapping
algorithm (in particular, it is encrypted with integrity protection) and sent to the other party. The
party that selects, wraps, and sends the secret keying material is called the “sender,” and theother party is called the “receiver.” The key-transport techniques described in thisRecommendation combine a DLC key-agreement scheme with a key-wrapping technique. First,
an approved key-agreement scheme is used to establish a key-wrapping key that is shared
between party U and party V. Then, party U (now acting as the key-transport sender) wraps the
keying material that will be transported, using an approved key-wrapping algorithm; party V
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(acting as the key-transport receiver) later uses the same key-wrapping key to unwrap thetransported keying material. (See Section 7 for details, including restrictions on the keyagreement schemes that are approved for such key-transport applications.)
This Recommendation specifies a number of processes that are associated with key
establishment (including processes for generating domain parameters and for deriving secretkeying material from a shared secret). Some of these processes are used to provide assurance (forexample, assurance of the arithmetic validity of a public key or assurance of possession of a
private key associated with a public key). The party that provides the assurance is called the“provider” (of the assurance), and the party that obtains the assurance is called the “recipient” (ofthe assurance). For any of the specified processes, equivalent processes may be used. Twoprocesses are equivalent if, when the same values are input to each process (either as input
parameters or as values made available during the process), the same output is produced.
Sections 4.1, 4.2, and 4.3 describe the various steps that may be performed to establish secretkeying material.
4.1

Key Establishment Preparations

The owner of a private/public key pair is the entity that is authorized to use the private key ofthat key pair. The precise steps required may depend upon the key establishment scheme and thetype of key pair (static or ephemeral).
The first step is to obtain appropriate domain parameters that are generated as specified in
Section 5.5.1; either the owner itself generates the domain parameters, or the owner obtains
domain parameters that another entity (e.g., a trusted third party) has generated. Having obtained
the domain parameters, the owner obtains assurance of the validity of those domain parameters;
approved methods for obtaining this assurance are provided in Section 5.5.2.
If the owner will be using a key establishment scheme that requires that the owner have a static
key pair, the owner obtains this key pair. Either the owner or a trusted third party generates the
key pair as specified in Section 5.6.1. If the key pair is generated by a trusted third party, then the
key pair shall be transported to the owner in a protected manner. If the key establishment scheme
requires an ephemeral key pair, the owner generates it (as close to the time of use as possible) as
specified in Section 5.6.1. Before using a static or ephemeral key pair in a key-establishmenttransaction, its owner is required to confirm its validity by obtaining the assurances specified in
Section 5.6.2.1.
An identifier is used to label the entity that owns a static key pair used in a key establishmenttransaction; an identifier may also be used to label the owner of an ephemeral key pair. This label
may uniquely distinguish the owner from all other entities, in which case it could rightfully beconsidered an identity. However, the label may be something less specific – an organization,
nickname, etc. – hence, the term identifier is used in this Recommendation, rather than the term
identity. For example, an identifier could be “Vegetable.gardener123”, rather than an identifier thatnames a particular person. A key pair’s owner (or an agent trusted to act on the owner’s behalf)
is responsible for ensuring that the identifier associated with its static public key is appropriate
for the applications in which it will be used.
For each static key pair, this Recommendation assumes that there is a trusted association
between the intended owner’s identifier(s) and the intended owner’s static public key. The
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association may be provided using cryptographic mechanisms or by physical means. The use of
cryptographic mechanisms may require the use of a binding authority (i.e., a trusted authority)that binds the information in a manner that can be verified by others; an example of such atrusted authority is a registration authority working with a CA who creates a certificate
containing both the static public key and the identifier. The binding authority shall verify theowner’s intent to associate a specific identifier chosen for the owner and the public key; themeans for accomplishing this is beyond the scope of this Recommendation. The binding
authority shall also obtain assurance of the validity of the domain parameters associated with theowner’s key pair, the arithmetic validity of the owner’s static public key, and the owner’s
possession of the static private key corresponding to that static public key (see Section 5.5.2,
Section 5.6.2.2.1 [method 1], and Section 5.6.2.2.3, respectively.)
As an alternative to reliance upon a binding authority, trusted associations between identifiers
and static public keys may be established by the direct exchange of this information between
entities, using a mutually trusted method (e.g., a trusted courier or a face-to-face exchange). In
this case, each entity receiving an identifier and the associated static public key shall be
responsible for obtaining the same assurances that would have been obtained on their behalf by abinding authority (see the previous paragraph). Entities shall also be responsible for maintaining(by cryptographic or other means) the trusted associations between any identifiers and static
public keys received through such exchanges.
If an entity engaged in a key-establishment transaction owns a static key pair that is employedduring the transaction, then the identifier used to label that party shall be one that has a trusted
association with the static public key of that key pair. If an entity engaged in a key-establishmenttransaction contributes only an ephemeral public key during the transaction, but an identifier isstill desired/required for that party, then a non-null identifier shall be selected/assigned inaccordance with the requirements of the protocol relying upon the transaction.
Figure 1 depicts the steps that may be required of an owner in order to obtain its key pair(s) in
preparation for key establishment.

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Entity itself generates

Obtain domain
parameters (5.5.1)

Another entity generates

Obtain assurance ofvalidity of domain
parameters (5.5.2)

Owner generates
TTP generates

Obtain static
key pair (5.6.1)

Generate
ephemeral key
pair (5.6.1)

Establish trusted
association withidentifier(s)(4.1)

Schemedependent

Obtain required
assurances (5.6.2)

Ready for key establishment

Figure 1: Owner key establishment preparations

4.2

Key Agreement Process

A key agreement process specified in this Recommendation consists of a sequence of ordered
steps. Figure 2 depicts the steps that may be required of an entity when establishing secret keying
material with another entity using one of the key agreement schemes described in thisRecommendation. Some discrepancies in the order of the steps may occur, however, depending
upon the communication protocol in which the key-agreement process is performed. Dependingon the key-agreement scheme and the available keys, the party whose actions are described could
be either of the two participants in the key-agreement scheme (i.e., either party U or party V).
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Note that some of the actions shown in Figure 2 may be absent from certain schemes. For
example, key-confirmation as defined in Section 5.9 is optional, which is indicated with dashedframes and dashed arrows. The specifications of this Recommendation indicate when a particularaction is required.
If required by the key agreement scheme, a party that requires the other entity’s static public key
acquires that key (as well as the associated identifier) and obtains assurance of its validity.
Approved methods for obtaining assurance of the validity of the other entity’s static public key
are provided in Section 5.6.2.2.1. Assurance that the other entity is in possession of thecorresponding static private key must also be obtained prior to using the derived keying material
for purposes beyond those of the key agreement transaction itself. See Section 5.6.2.2.3 forapproved methods for obtaining this assurance. (Note: the restriction above does not, forexample, prohibit the use of derived keying material for key-confirmation performed during the
key agreement transaction.).
If a party receives an ephemeral public key from the other entity for use in the key agreementtransaction, that party must obtain assurance of its validity. Approved methods for obtaining
assurance of the validity of the other entity’s ephemeral public key are provided in Section
5.6.2.2.2.
If required by the key agreement scheme, a party generates an ephemeral key pair (in accordancewith Section 5.6.1) and provides the ephemeral public key of that key pair to the other entity; theephemeral private key is not provided to the other entity.
If required or desired for use in the key agreement transaction, a party generates a nonce (asspecified in Section 5.4) and provides it to the other entity.
Depending upon the circumstances, additional public information (e.g., a party’s static public
key, an identifier, etc.) may be provided to or obtained from the other entity.
If either of the participants in the key agreement transaction requires evidence that the otherparticipant has computed the same shared secret and/or derived the same secret keying material,(unilateral or bilateral) key-confirmation may be performed as specified in Section 5.9.
Figure 2 depicts the steps that may be required of an entity when establishing secret keying
material with another entity by using one of the key-agreement schemes described in thisRecommendation.

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Receive other party’s
static public key

Receive other party’s
ephemeral public key
and/or other public
data

Generate a nonce oran ephemeral key pair
(5.4&5.6.1)

Receive required
assurance for otherparty’s static publickey (5.6.2.2)

Receive required
assurance for otherparty’s ephemeral
public key (5.6.2.2)

Provide ephemeral
public key and/or
other public data toother party

Complete keyconfirmation (5.9 & 6)

Generate sharedsecret and derive
keying material(6&5.8)

Receive assurance ofpossession of private
keys (5.6.2.2.3)
Destroy shared secret,ephemeral private
key, MacKey, etc

Key-agreement completedFigure 2: Key-agreement process
4.3

DLC-based Key-Transport Process

The process begins by establishing a key-wrapping key using an appropriate key-agreement
scheme (see Sections 6 and 7), with the intended key-transport sender acting as party U, and theintended key-transport receiver acting as party V. Key-confirmation may optionally be
performed at the end of the key-agreement process to provide assurance that both parties possess
the same key-wrapping key. Party U then selects secret keying material to be transported, wraps
the keying material using the key-wrapping key and sends the wrapped keying material to party
V. After receiving and unwrapping the transported keying material, party V may optionallyperform key-confirmation to provide assurance to party U that the transported keying material

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has been received and correctly unwrapped. Figure 3 depicts the steps that are performed whentransporting secret keying material from one entity to another using a key-transport scheme.

Key-transport sender
Establish keywrapping key (6&7)

Select keying material
to be transported

Wrap keying
material to betransported (7)Transport wrapped
keying material

Key-transport receiver
Establish keywrapping key (6&7)

Obtain wrapped
keying material

Unwraptransported keying
material (7)Provide keyconfirmation

Key-transport completedFigure 3: Key-transport process

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Cryptographic Elements

This section describes the basic computations that are performed and the assurances that need tobe obtained when performing DLC-based key establishment. The schemes described in Section 6
are based upon the correct implementation of these computations and assurances.
5.1

Cryptographic Hash Functions

In this Recommendation, cryptographic hash functions may be used in key derivation and in
MAC tag computation during key-confirmation. An approved hash function shall be used whena hash function is required. [FIPS 180] specifies approved hash functions.
5.2

Message Authentication Code (MAC) Algorithm

A Message Authentication Code (MAC) algorithm defines a family of MAC functions that isparameterized by a symmetric key. The input to a MAC function includes a symmetric key,called MacKey and a binary data string called MacData. That is, a MAC function is represented
as MAC(MacKey, MacData). In this Recommendation, a MAC function is used in keyconfirmation and may be used for key derivation.
Approved MAC algorithms are specified in [FIPS 198] (i.e., HMAC) and [SP 800-38B] (i.e.,
CMAC). HMAC requires the use of an approved hash function; CMAC requires the use of an
approved block cipher algorithm.
When used for key derivation, HMAC shall be selected as the function H in accordance with
Table 6 and Table 7 of Section 5.8. The selection of CMAC is defined in [800-56C].
When used for key-confirmation, an entity is required to compute a MAC tag on received orderived data using a MAC function determined by a MacKey that is derived from a shared secret.
The MAC tag is sent to the other entity participating in the key-establishment scheme in order toprovide assurance that the shared secret or derived keying material was correctly computed.
MAC tag computation and verification are defined in Sections 5.2.1 and 5.2.2.
5.2.1 MAC Tag Computation for Key-Confirmation
The computation of the MacTag is represented as follows:
MacTag = TMacLen[MAC(MacKey, MacData)].
To compute a MacTag:
1. An approved, agreed-upon MAC algorithm (see [FIPS 198] or [SP 800-38B]) is usedwith a MacKey to compute a MAC on the MacData, where MacKey is a symmetric key,
and MacData represents the data on which the MAC tag is computed. The minimum
length of MacKey is specified in Table 8 and Table 9 of Section 5.9.3.
MacKey is obtained from the DerivedKeyingMaterial (when a key-agreement scheme
employs key-confirmation), as specified in Section 5.9.1.1, or obtained from theTransportedKeyingMaterial (when a key-transport scheme employs key-confirmation),as specified in Section 7.2.

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The resulting MAC consists of MacOutputLen bits, which is the full output length of theselected MAC algorithm.
2. The MacOutputLen bits are input to a truncation function TMacLen to obtain the mostsignificant (i.e., leftmost) MacLen bits, where MacLen represents the intended length of
the MacTag, which is required to be less than or equal to MacOutputLen. (When MacLen
equals MacOutputLen, TMacLen acts as the identity function.) The minimum for MacLen isspecified in Table 8 and Table 9 of Section 5.9.3.
5.2.2 MAC Tag Verification for Key-Confirmation
To check a received MAC tag, MacTag (e.g., received during key-confirmation), a new MAC
tag, MacTag′ is computed using the values of MacKey, MacLen, and MacData possessed by therecipient (as specified in Section 5.2.1). MacTag′ is compared with the received MacTag. If theirvalues are equal, then it may be inferred that the same MacKey, MacLen, and MacData values
were used in the two MAC tag computations.
5.3

Random Number Generation

Whenever this Recommendation requires the use of a randomly generated value (for example,
for obtaining keys or nonces), the values shall be generated using an approved random bit
generator (RBG), such as those specified in [SP800-90A] providing an appropriate security
strength.
5.4

Nonces

A nonce is a time-varying value that has (at most) an acceptably small chance of repeating(where the meaning of “acceptably small” may be application specific). In certain schemes
specified in this Recommendation, a party may be required to provide a (public) nonce that isused for key-agreement and/or key-confirmation purposes. This circumstance arises when ascheme does not require that party to provide an ephemeral public key to the other party as partof the key-establishment process.
This Recommendation requires the use of a nonce (supplied by Party U) in the C(0e, 2s) keyagreement schemes specified in Section 6.3. A nonce (supplied by party V) is also required by
the C(1e, 2s) and C(0e, 2s) schemes when party V obtains key-confirmation from party U in
conformance with this Recommendation (see Section 6.2.1.5 and Section 6.3.3, respectively).
A nonce may be composed of one (or more) of the following components (other components
may also be appropriate):
1. A random value that is generated anew for each nonce, using an approved random bit
generator. A nonce containing a component of this type is called a random nonce.
2. A timestamp of sufficient resolution (detail) so that it is different each time it is used.
3. A monotonically increasing sequence number, or
4. A combination of a timestamp and a monotonically increasing sequence number, suchthat the sequence number is reset when and only when the timestamp changes. (For19

NIST SP 56A: Revision 2, May 2013
example, a timestamp may show the date but not the time of day, so a sequence number
is appended that will not repeat during a particular day.)The specified use of a nonce in key-derivation and/or key-confirmation computations does notprovide the same benefits as the use of an ephemeral key pair in a key-agreement scheme. (Forexample, party U’s contribution of a public nonce during the execution of a C(0e, 2s) scheme
does not protect the secrecy of derived keying material against a future compromise of party U’s
static private key, but the use of an ephemeral key pair by party U during the execution of aC(1e, 2s) scheme can provide such protection.) Still, the contribution of an appropriately formed
nonce can support some of the security goals (e.g., assurance of the freshness of derived keying
material) that might otherwise be supported by the contribution of an ephemeral public key
generated (and used) in conformance with this Recommendation.
Whenever it is required for key-agreement and/or key-confirmation purposes as specified in thisRecommendation, a nonce should be a random nonce. The security strength supported by the
instantiation of this random bit generator and the bit length of the random bit string shall be atleast one-half of the minimum bit length required for the subgroup order, as determined by theapplicable parameter-size set. However, the bit length of the random bit string should be (atleast) equal to the bit length of the subgroup order included among the domain parameters thatare actually employed during key-agreement (i.e., log 2 q in the FFC case, or log 2 n in the
ECC case). The minimum bit lengths required for the subgroup order by various parameter-size
sets are specified in Table 1 of Section 5.5.1.1 (for the FFC case) and in Table 2 of Section
5.5.1.2 (for the ECC case). For details concerning the security strength supported by an
instantiation of a random bit generator, see [SP 800-90A].
As part of the proper implementation of this Recommendation, system users and/or agentstrusted to act on their behalf should determine that the components selected for inclusion in any
required nonces meet their security requirements. The application tasked with performing key
establishment on behalf of a party should determine whether or not to proceed with a keyestablishment transaction based upon the perceived adequacy of the method(s) used to form therequired nonces. Such knowledge may be explicitly provided to the application in some manner,or may be implicitly provided by the operation of the application itself.
5.5

Domain Parameters

Discrete Logarithm Cryptography (DLC), which includes Finite Field Cryptography (FFC) andElliptic Curve Cryptography (ECC), requires that the public and private key pairs be generated
with respect to a particular set of domain parameters. A candidate set of domain parameters is
said to be valid when it conforms to all the requirements specified in this Recommendation. Bothparties executing a key-establishment scheme shall have assurance of domain parameter validity
prior to using them (e.g., to generate key pairs). Although domain parameters are public
information, they shall be managed so that the correct correspondence between a given key pair
and its set of domain parameters is maintained for all parties that use the key pair. Domain
parameters may remain fixed for an extended time period, and one set of domain parameters may
be used with multiple key pairs and with multiple key-establishment schemes.

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NIST SP 56A: Revision 2, May 2013
For this Recommendation, only one set of domain parameters shall be used during any key
establishment transaction. That is, when a key establishment scheme uses both a static key pair
and an ephemeral key pair, they shall be generated using the same set of domain parameters.
5.5.1 Domain Parameter Generation
5.5.1.1 FFC Domain Parameter Generation
If p is a prime number, then GF(p) denotes the finite field with p elements, which can be
represented by the set of integers {0, 1, …, p−1}. The addition and multiplication operations for
GF(p) can be realized by performing the corresponding integer operations and reducing theresults modulo p. The multiplicative group of non-zero field elements is denoted by GF(p)*. In
this Recommendation, an FFC key-establishment scheme requires the use of public keys that are
restricted to a (unique) cyclic subgroup of GF(p)* with prime order q (where q divides p – 1). If
g is a generator of this cyclic subgroup, then its elements can be represented as {1, g mod p,
g2 mod p, …, gq-1 mod p}, and 1 = gq mod p.
Domain parameters for an FFC scheme are of the form (p, q, g{, SEED, pgenCounter}), where pis the (odd) prime field size, q is an (odd) prime divisor of p – 1, g is a generator of the cyclic
subgroup of GF(p)* of order q, and SEED and pgenCounter are optional parameters used in the
canonical process of generating and validating p, q, and possibly g (depending on the method ofgeneration). FFC Domain parameters shall be generated using a method specified in [FIPS 186],
based on a parameter-size set selected from Table 1 (i.e., parameter set FA, FB or FC).
Table 1: FFC parameter-size sets
FFC parameter-size set name

FA

FB

FC

Maximum security strength supported (in bits)

80

112

112

Bit length of field size p (i.e., log 2 p  )

1024

2048

2048 1

Bit length of subgroup order q (i.e., log 2 q  )

160

224

256

As shown in Table 1, there are three parameter-size sets (named FA, FB, and FC). For U.S.
government applications, one or more of the parameter-size sets provided in Table 1 shall be
used for the FFC schemes, based on security strength requirements specified in [SP 800-131A].
See the comparable security table in the Recommendation for Key Management [SP 800-57] toassess the comparable security of any particular parameter-size set. [SP 800-57] provides
guidance on selecting an appropriate security strength and an appropriate FFC parameter set 2. If

1Parameter-size set FC is included with the same bit length for the field size as set FB to allow finite field
applications with a 2048-bit field order to have the option of increasing the private key size to 256 bits without
having to increase the field order (a more substantial change). FC is not intended to provide more security than FB;i.e., parameter sets FB and FC provide keys with the same security strength.
2

In [SP 800-57], l is used to denote the bit length of the field size, and N denotes the bit length of the subgrouporder.

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NIST SP 56A: Revision 2, May 2013
an FFC parameter-size set cannot provide the required security strength, then Elliptic Curve
Cryptography should be used (see Section 5.5.1.2).
For this Recommendation, the size of p (the public key size) is either 1024 or 2048 bits. The sizeof q is a specific bit length, depending on the FFC parameter-size set selected. The leftmost (i.e.,most significant) bit of p and q shall be 1” (e.g., for a 1024-bit prime p, where bit 1 is the mostsignificant bit and bit 1024 is the least significant, bit 1 shall be a 1).
5.5.1.2 ECC Domain Parameter Generation
For ECC, let GF(q) denote the finite field with q elements, where either q is an odd prime p, or q
is equal to 2m, for some prime integer m. An elliptic curve defined over GF(q) is determined byeither an equation of the form y2 = x3+ax+b (when q = p) or by an equation of the form y2+xy =x3+ax2+b (when q = 2m), where a and b are (appropriately chosen) elements of GF(q). In such an
equation, the indicated arithmetic is performed in GF(q). (See Annexes A.2, G.1, and G.2 of[ANS X9.62] for further information concerning arithmetic in finite fields.) For the purposes ofthis Recommendation, an affine point P on the corresponding elliptic curve is one that can be
represented as an ordered pair (xP, yP), whose coordinates are elements of GF(q) that satisfy thegiven equation. Together with an appropriate binary operation “+” (elliptic-curve addition, as
defined by the well-known secant-and-tangent rules) and the introduction of a special point “Ø”(the point at infinity), which serves as the (additive) identity, the set of (affine and infinite)
elliptic curve points forms a group. (See Annexes G.3 and G.4 of [ANS X9.62] for the details of
elliptic-curve group operations.)
As specified in this Recommendation, an ECC key-establishment scheme requires the use of
public keys that are affine elliptic curve points chosen from a specific cyclic subgroup withprime order n. Suppose that the point G is a generator for this cyclic subgroup. If, for eachpositive integer d, dG denotes
G + G +…+ G,dwhere “+” is the elliptic-curve addition operation, then the elements of the cyclic subgroup canbe represented as {Ø, G, 2G,…, (n – 1)G}. Note that nG = Ø. The full elliptic curve group(including Ø and all of the affine points on the curve) has order nh, where the integer h is calleda cofactor of the cyclic subgroup generated by G.
Domain parameters for an ECC scheme have the form (q, FR, a, b{, SEED}, G, n, h). The
parameter q is the field size. As noted above, q may be an odd prime p, or q may be equal to 2m,
for some prime integer m. The field representation parameter FR is used to provide additional
information (as specified in [ANS X9.63]) concerning the method used to represent elements ofthe finite field GF(q). FR is null if q is equal to an odd prime p. In this case, the elements of thefinite field are represented by the integers 0 through p – 1. When q = 2m, the elements of GF(2m)are represented by bit strings of length m, with each bit indicating the coefficient (0 or 1) of a
specific element of a particular basis for GF(2m) viewed as a vector space over GF(2). FR is null
if q = 2m and the representation of field elements corresponds to a Gaussian normal basis forGF(2m) (as specified in Annex A.2.3.3 of [ANS X9.62], and further described in Annexes G.2.4,
G.2.5, and H.1 of that document). If q = 2m and the representation of field elements corresponds
to a polynomial basis (as specified in Annex A.2.3.2 of [ANS X9.62], and further described in
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NIST SP 56A: Revision 2, May 2013
Annexes G.2.2, G.2.3, H.2, and H.3 of that document), then FR specifies the reduction
polynomial – either a trinomial or a pentanomial. The parameters a and b are elements of GF(q)that define the equation of an elliptic curve. G = (xG, yG) is an affine point on the elliptic curve
determined by a and b that is used to generate a cyclic subgroup of prime order n. The parameterh is the cofactor of the cyclic subgroup generated by G. The bit string SEED is an optionalparameter used in the canonical process of generating and validating a, b, and possibly G(depending on the method of generation).
The ECC domain parameters shall either be generated as specified in [ANS X9.62] or selectedfrom the recommended elliptic curve domain parameters specified in [FIPS 186]. (Note: ANS
X9.63 currently refers to the ANS X9.62 for methods of ECC domain parameter generation.)
Table 2: ECC parameter-size sets
ECC parameter-size set name

EA

EB

EC

ED

EE

Maximum security strength supported (in 80
bits)

112

128

192

256

Bit length of ECC subgroup order n(i.e., log 2 n  )Maximum bit length of ECC cofactor h

160 to 224 to 256 to 384 to 512 or
223
255
383
511
longer
10

14

16

24

32

As shown in Table 2, there are five parameter-size sets (named EA, EB, EC, ED and EE) for
ECC. The five different cofactor maximums each ensure that the subgroup of order n is uniqueand that cofactor multiplication is reasonably efficient. For U.S. government applications, one ormore of the parameter-size sets provided in Table 2 shall be used for the ECC schemes, based on
security strength requirements specified in [SP 800-131A]. See the comparable security strengthtable in the Recommendation for Key Management [SP 800-57] to assess the comparable
security of any particular parameter-size set. The Recommendation for Key Management [SP
800-57] provides guidance on selecting the appropriate security strength and an appropriate ECC
key size.
5.5.2 Assurances of Domain Parameter Validity
Secure key establishment depends on the arithmetic validity of the set of domain parameters usedby the parties. Each party shall have assurance of the validity of a candidate set of domain
parameters. Each party shall obtain assurance that the candidate set of domain parameters is
valid in at least one of the following three ways:
1. The party itself generates the set of domain parameters as specified in Section 5.5.1.
2. The party performs an explicit domain parameter validation as specified in:a. [FIPS 186] for FFC, based on a parameter-size set selected from Table 1.
b. [ANS X9.62] for ECC, based on a parameter-size set selected from Table 2.

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NIST SP 56A: Revision 2, May 2013
3. The party has received assurance from a trusted third party (for example, a CA or NIST 3)
that the set of domain parameters was valid at the time that they were generated by reasonof either method 1 or 2 above.
As part of the proper implementation of this Recommendation, system users and/or agentstrusted to act on their behalf should determine which of the methods above meet their security
requirements. The application tasked with performing key establishment on behalf of a party
should determine whether or not to proceed with a key establishment transaction, based upon theperceived adequacy of the method(s) used to obtain assurance of domain parameter validity.
Such knowledge may be explicitly provided to the application in some manner, or may be
implicitly provided by the operation of the application itself.
5.5.3 Domain Parameter Management
The set of domain parameters used shall be protected against modification or substitution until
the set is deactivated (if and when it is no longer needed). Each private/public key pair shall becorrectly associated with its specific set of domain parameters.
5.6 Key-Establishment Key Pairs
This section specifies requirements for the generation of key pairs to be used in keyestablishment transactions, provides methods for obtaining assurances that valid key pairs areused during key establishment, and specifies key management requirements for the static andephemeral key pairs used in key establishment.
5.6.1 Key-Pair Generation
These generation methods assume the use of valid domain parameters (see Section 5.5). Prior toperforming key-pair generation with the selected domain parameters, the party generating the
key pair shall obtain assurance of domain parameter validity in accordance with Section 5.5.2.
5.6.1.1 FFC Key-Pair Generation
For the FFC schemes, each static and ephemeral key pair shall be generated using an approved
method (see Appendix B of [FIPS 186]) and the selected valid domain parameters (i.e.,p, q, g{, SEED, pgenCounter}). Given a set of valid domain parameters, each valid private key isa randomly selected integer in the range [1, q−1]. Each valid static public key y is related to thecorresponding(valid)staticprivate
key
x
by
the
following
formula:y = gx mod p. Similarly, each valid ephemeral public key t is related to the corresponding (valid)ephemeral private key r by the following formula: t = gr mod p.
5.6.1.2 ECC Key-Pair Generation
For the ECC schemes, each static and ephemeral private key d and public key Q shall be
generated using an approved method (see Appendix B of [FIPS 186]) and the selected domain
parameters (i.e., q, FR, a, b{, SEED}, G, n, h). Given valid domain parameters, each valid
private key d is an integer that is randomly selected in the range [1, n−1]. Whether static or

3

NIST-recommended curves can be found in [FIPS 186].

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NIST SP 56A: Revision 2, May 2013
ephemeral, each valid public key Q is related to the corresponding (valid) private key d by thefollowing formula: Q = (xQ, yQ) = dG.
5.6.2 Required Assurances
To explain the assurance requirements associated with key-establishment key pairs, someterminology needs to be introduced. The owner of a static key pair is defined as the entity that is
authorized to use the private key that corresponds to the public key; this is independent of
whether or not the owner generated the key pair. The recipient of a static public key is defined as
the entity that is participating in a key establishment transaction with the owner and obtains the
key before or during the current transaction. The owner of an ephemeral public key is the entitythat generated the key as part of a key-establishment transaction. The recipient of an ephemeral
public key is the entity that receives that public key during a key-establishment transaction with
its owner.
Secure key establishment depends upon the use of valid key-establishment keys. Prior to
obtaining the assurances described in this section, the owner of a key pair and the recipient of the
public key of that key pair shall obtain assurance of the validity of the associated domain
parameters (see Section 5.5.2).
The security of key-agreement schemes also depends on limiting knowledge of the private keys
to those who have been authorized to use them (i.e., their respective owners) and to the trusted
third party that may have generated them. In addition to preventing unauthorized entities fromgaining access to private keys, it is also important that owners have access to their private keys.
Note that as time passes, an owner may lose possession of the correct value of the private key
component of their key pair, either by choice or due to an error; for this reason, current assuranceof possession of a static private key can be of value for some applications, and renewing
assurance of possession may be necessary. See Section 5.6.2.2.3.2 for techniques that therecipient of a static public key can use to directly obtain more current assurance of the owner’s
possession of the corresponding private key.
Prior to or during a key-establishment transaction, the participants in the transaction (i.e., parties
U and V) shall obtain the appropriate assurances about the key pairs used during that transaction.
The types of assurance that may be sought by one or both of the parties (U and/or V) concerning
the components of a key pair (i.e., the private key and public key) are discussed in Sections
5.6.2.1 and 5.6.2.2. The methods that will be specified to provide/obtain these assurances
presuppose the validity of the domain parameters associated with the key pair (see Section 5.5).
The following sections include tables that summarize the types of assurance that are required by
the parties to a key-establishment transaction. Table 4 in Section 5.6.2.1 summarizes assurancesthat a key-pair owner may want to renew periodically. The shaded table entries indicate a type of
key pair (static or ephemeral) and a type of assurance that might be sought for such a key pair.
The unshaded table entries indicate who can perform the actions necessary to obtain theassurance.
5.6.2.1 Assurances Required by the Key Pair Owner
Prior to the use of a static or ephemeral key pair in a key-establishment transaction, the key-pairowner shall confirm the validity of the key pair by obtaining the following assurances:

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NIST SP 56A: Revision 2, May 2013•

Assurance of correct generation – assurance that the key pair was generated as specified
in Section 5.6.1 (see Section 5.6.2.1.1 for the methods for obtaining this assurance).

•

Assurance of private-key validity – assurance that the private key is an integer in thecorrect range, as determined by the domain parameters (see Section 5.6.2.1.2 for the
methods for obtaining this assurance).

•

Assurance of public-key validity – assurance that the public key has the correct
representation for a non-identity element of the correct cryptographic subgroup, as
uniquely determined by the domain parameters (see Section 5.6.2.1.3 for the methods for
obtaining this assurance).

•

Assurance of pair-wise consistency – assurance that the private key and public key havethe correct mathematical relationship to each other (see Section 5.6.2.1.4 for the methods
for obtaining this assurance).

Table 3 indicates the assurances to be obtained by the owner of a key pair for both static andephemeral keys, identifies who can perform the actions necessary for the owner to obtain eachassurance, and indicates the sections of this document where further information is provided.
Table 3: Initial assurances required by the key-pair ownerTypes of assurance
Key-pairtype

StaticCorrect
generation

Private-key
validation

Public-key
validation

Pair-wise
consistency

Ownera or
TTPb

OwnercOwnerd or
TTPe

OwnerfOwneraOwnercOwnerdOwnerfEphemeral
a
b
c
d
e
f

See Section 5.6.2.1.1, method a.
See Section 5.6.2.1.1, method b
See Section 5.6.2.1.2
See Section 5.6.2.1.3, methods a and b.
See Section 5.6.2.1.3, method c.
See Section 5.6.2.1.4.

A static key pair owner may optionally renew certain assurances regarding its key pair at any
time. Table 4 indicates which of the assurances obtained by the owner of a static key pair can be
renewed and indicates the sections of this document where further information is provided. Notethat for ephemeral key pairs, only initial assurances are required; renewed assurance for
ephemeral key pairs is not applicable, since ephemeral key pairs are short-lived. Also, note thatthe assurance of the correct generation of a static key pair is not renewable since, after the fact, it
is not feasible to verify that its private component was randomly selected.

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NIST SP 56A: Revision 2, May 2013
Table 4: Optional renewal of assurances by the key-pair ownerTypes of assurance
Key-pairtype
Statica.
b.
c.

Correct
generation

Private-key
validation

Public-key
validation

Pair-wise
consistency

Infeasible

OwneraOwnerbOwnercSee Section 5.6.2.1.2.
See Section 5.6.2.1.3.
See Section 5.6.2.1.4.

Note that the methods used to obtain the required assurances are not necessarily independent. For
example, the key-pair owner may employ a key-generation routine that is consistent with thecriteria of Section 5.6.1 and also incorporates the actions required to provide (initial) assuranceof the validity and consistency of the private and public components of the resulting key pair.
As part of the proper implementation of this Recommendation, system users and/or agentstrusted to act on their behalf should determine which of the methods above meet their security
requirements. The application tasked with performing key establishment on behalf of a party
should determine whether or not to proceed with a key-establishment transaction, based upon theperceived adequacy of the method(s) used to obtain the above assurances.
5.6.2.1.1 Owner Assurance of Correct Generation
Prior to the use of a key pair in a key-establishment transaction, the owner of a static or
ephemeral key-establishment key pair shall obtain an initial assurance that the key pair has beencorrectly formed (in a manner that is consistent with the criteria of Section 5.6.1) using one of
the following methods:a. The owner generates the key pair as specified in Section 5.6.1 (this applies to both staticand ephemeral key pairs), orb. A trusted third party (trusted by the owner and any recipient of the public key) generates
a static key pair as specified in Section 5.6.1 and provides it to the owner. Note that, in
this case, the TTP needs to be trusted by both the owner and any public-key recipient togenerate the key pair as specified in Section 5.6.1 and not to use the owner’s private key
to masquerade as the owner. This method is not appropriate for ephemeral key pairs,
since the owner generates ephemeral keys.
5.6.2.1.2 Owner Assurance of Private-Key Validity
Prior to the use of a key pair in a key-establishment transaction, the owner of a static or
ephemeral key-establishment key pair shall obtain an initial assurance that the private key is an
integer in the correct range ([1, q−1] for FFC or [1, n−1] for ECC), as determined by the domain
parameters (see Sections 5.5.1.1 and 5.5.1.2) using one of the following methods:a. The owner generates the key pair as specified in Section 5.6.1 (this applies to both staticand ephemeral key pairs), or

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NIST SP 56A: Revision 2, May 2013b. After receiving the static key pair from a trusted third party (trusted by the owner), theowner performs a separate check to determine that the private key is in the correct range;note that only static key pairs may be generated by a TTP.
To renew this assurance for a static key pair, the owner shall perform a separate check todetermine that the private key is in the correct range as determined by the domain parameters.
5.6.2.1.3 Owner Assurance of Public-Key Validity
Prior to a key-establishment transaction, the owner of a key pair shall obtain an initial assurance
that the public key has the (unique) correct representation for a non-identity element of thecorrect cryptographic subgroup, as determined by the domain parameters, using one of thefollowing methods:a. The owner generates the key pair as specified in Section 5.6.1 and performs a full publickey validation or an equivalent procedure as part of its generation process (see Sections
5.6.2.3.1 and 5.6.2.3.2); or
b. The owner performs a full public-key validation as a separate process from the key-pair
generation process (see Sections 5.6.2.3.1 and 5.6.2.3.2); either the owner or a TTP could
have generated the key pair; orc. In the case of static keys, a trusted third party (trusted by the owner) performs a full
public-key validation (see Sections 5.6.2.3.1 and 5.6.2.3.2) and provides the validation
result to the owner. This TTP could, for example, be a binding authority (see Section 4.1)
and/or a TTP that generated the key pair (see method b in Section 5.6.2.1.1). In the caseof TTP generation, the TTP may employ a key-generation routine that performs a full
public-key validation or an equivalent procedure as part of its key-pair generation
process, or the full public-key validation may be performed as a separate process fromthe key-pair generation process following the generation of the key pair.
To renew this assurance for a static public key (if desired), the owner shall perform a successful
full public-key validation (see Sections 5.6.2.3.1 and 5.6.2.3.2). Note that renewed assurance ofvalidity for an ephemeral public key is not applicable, since ephemeral key pairs are short-lived.
5.6.2.1.4 Owner Assurance of Pair-wise Consistency
Prior to a key-establishment transaction, the owner of a key pair shall obtain an initial assurance
that the private key and public key have the correct mathematical relationship to each other byusing one of the following methods:a. The owner generates the key pair as specified in Section 5.6.1 (this applies to both staticand ephemeral key pairs), orb. Subsequent to the generation of a static key pair by the owner or a trusted third party as
specified in Section 5.6.1, the owner performs one of the following alternative
consistency tests. If a consistency test fails, the tested key pair shall not be used.
Alternative 1:
The public key shall be recomputed from the private key and the domain parameters in orderto obtain assurance that the private and public keys are consistent.
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FFC schemes: Check that gx mod p = y, where x is the private key, y is the associatedpublic key of the key pair, and both the generator g and prime modulus p are domain
parameters associated with the key pair.
ECC schemes: Check that dG = Q, where d is the private key Q is the associated public
key of the key pair, and the generator G is a domain parameter associated with the key
pair.
The test fails if the check specified above fails.
Alternative 2:
Using the static key pair to be tested for pair-wise consistency and the domain parameters
associated with that key pair, separately perform the computations that would be required ofboth party U and party V in an approved DLC key-agreement scheme to generate derived
keying material. The static key pair to be tested shall be used as party V’s key pair. Anyother key pairs required to generate the shared secret shall bei.pre-determined (generated as specified in Section 5.6.1 using the same domain
parameters 4 as those used for the static key pair to be tested for consistency);
ii.pair-wise consistent; andiii.used only for pair-wise consistency testing.
The following are the main procedures for the pair-wise consistency test:
1. Select a DLC key-agreement scheme from Section 6 with which the static key pair
will be used to perform the pair-wise consistency test.
2. Use the static key pair to be tested and any pre-determined key pairs required for thescheme with the appropriate DLC primitive for that scheme to generate the sharedsecrets ZU and ZV for party U and party V, respectively.
FCC schemes: Use the DLC key-agreement primitive in either Section 5.7.1.1 or
5.7.2.1, as appropriate for the scheme to be used for the pair-wise-consistency test.
ECC Schemes: Use the DLC primitive in either Section 5.7.1.2 or 5.7.2.3, asappropriate for the scheme to be used for the pair-wise-consistency test.
Depending on the scheme selected, the DH schemes specified in Section 6 may
require one or two pre-determined key pairs for party U. If the key-agreement scheme
requires party V to have only one key pair, then the static key pair to be tested forpair-wise consistency shall be used as that key pair. If the scheme requires party V tohave two key pairs (i.e., an ephemeral key pair and a static key pair), then the static
key pair to be tested shall be used as the static key pair, and the other key pair shallbe a pre-determined key pair distinct from the pre-determined key pair(s) used forparty U.

4

If an implementation allows the use of multiple sets of domain parameters, each set of domain parameters will
require a static key pair for pair-wise consistency testing.

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NIST SP 56A: Revision 2, May 2013
For the schemes using MQV primitives specified in Section 5.7.2, MQV1 and onepass form shall be used for FFC and ECC primitives, respectively. In both cases,
party U requires two pre-determined key pairs. The static key pair to be tested will be
used as both of party V’s contributions to the primitive. Note that the MQV1 and
One-Pass forms of the primitives differ from the MQV2 and Full MQV form of theprimitive only in the number of key pairs required by party V; for the MQV1 and
One-Pass primitives, party V only uses the key pair to be tested, whereas for MQV2and Full MQV form, an additional pre-determined key pair is required for party V.
3. Derive keying material KU using an approved key-derivation function (KDF) as
specified in Section 5.8 for party U using ZU; derive keying material KV using thesame KDF for party V using ZV. The length of KU and KV shall be at least the lengthof the KDF output block.
The test fails if KU ≠ KV.
To renew assurance of pair-wise consistency for a static key pair (if desired), one of the above
alternative consistency tests in method b shall be employed by the owner. Note that renewed
assurance for ephemeral key pairs is not applicable, since ephemeral key pairs are short-lived.
5.6.2.1.5
Owner Assurance of Possession of the Private Key
Prior to a key-establishment transaction, the owner of a key pair shall obtain an initial assuranceof possession of the private key using one of the following methods:a. The owner generates the key pair as specified in Section 5.6.1 (this applies to both static andephemeral key pairs), orb. When a trusted third party (trusted by the owner) generates a static key pair and provides it tothe owner, the owner performs one of the consistency tests in Section 5.6.2.1.4; if a pair-wise
consistency test fails, the tested key pair shall not be used.
To renew this assurance for a static private key (if desired), one of the alternative pair-wise
consistency tests in method b of Section 5.6.2.1.4 shall be employed by the owner. Note that
renewed assurance of the possession of an ephemeral private key is not applicable, sinceephemeral key pairs are short-lived.
5.6.2.2 Assurances Required by a Public Key Recipient
In order to successfully employ any of the schemes specified in this Recommendation, eachparticipant in a key-establishment transaction must receive at least one public key owned by theother participant. The public key(s) may be received during the transaction (which is usually thecase for an ephemeral public key) or prior to the transaction (as is sometimes the case for a static
public key). Regardless of the timing, a transaction participant is said to be acting as a “publickey recipient” when it receives the other participant's public key(s). Note that besides theparticipants (i.e., party U and party V), a binding authority (e.g., a CA) may be a public key
recipient (e.g., when obtaining assurance of possession).
Prior to or during a key-establishment transaction, the recipient of a public key shall obtain thefollowing indicated assurances:

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NIST SP 56A: Revision 2, May 2013•

Assurance of public-key validity – assurance that the public key of the other party (i.e.,the claimed owner of the public key) has the (unique) correct representation for a nonidentity element of the correct cryptographic subgroup, as determined by the domain
parameters (see Sections 5.6.2.2.1 and 5.6.2.2.2). This assurance is required for both
static and ephemeral public keys.

•

Assurance of private-key possession – assurance that the claimed owner of a public keyestablishment key (i.e., the other party) actually has the (correct) private key associatedwith that public key. This assurance is required for static private keys (see Section
5.6.2.2.3). Assurance of private-key possession for ephemeral keys is optional; however,methods for obtaining this assurance are discussed in Section 5.6.2.2.4.

Table 5 summarizes the assurances required by a public-key recipient for both the static andephemeral public keys of the other party, identifying the party that may perform the actions
necessary for the recipient to obtain the assurance and indicating the sections in this document
where further information is provided.
Table 5: Assurances required by a public-key recipientType of assurance
Key-pairtype

Public-key
validation

Private-key
possession

StaticRecipienta or TTPb

Recipientd or TTPe

Ephemeral

Recipientc

Not Requiredf

a
b
c
d.
e.
f

See Section 5.6.2.2.1, method 1.
See Section 5.6.2.2.1, method 2.
See Section 5.6.2.2.2
See Section 5.6.2.2.3.2
See Section 5.6.2.2.3.1
However, see Section 5.6.2.2.4

As part of the proper implementation of this Recommendation, system users and/or agentstrusted to act on their behalf should determine which of the indicated methods for obtaining therequired (and/or desired) assurances meet their security requirements. The application tasked
with performing key establishment on behalf of the recipient should determine whether or not toproceed with a key-establishment transaction, based upon the perceived adequacy of themethod(s) used to obtain the assurances described above.
Once the necessary steps have been taken to provide the recipient of a static public key withassurance of its validity, the assurance obtained by the recipient may endure for a protracted
period of time without the need to reconfirm the validity of that public key. The same may betrue of assurance provided to the recipient that the owner of the static public key possesses thecorresponding static private key. This could be the case, for example, when the source of theassurance is a trusted CA whose (valid) signature on a certificate containing the static public key
indicates to the recipient that the arithmetic validity of the static public key has been confirmed

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NIST SP 56A: Revision 2, May 2013by the CA and that the owner’s possession of the corresponding static private key has been
established to the CA’s satisfaction. Alternatively, a party could maintain an (integrity-protected)
record of previously received static public keys whose validity was confirmed and/or whose
owners have provided assurance of private-key possession.
On the other hand, the recipient of a static public key may choose to obtain renewed assurance ofits validity and/or choose to obtain renewed assurance that the owner of the static public key(i.e., the other party) possesses the corresponding static private key. Deciding how often (if atall) to seek renewed assurance is a determination that should be made by the recipient (or an
agent trusted to act on the recipient’s behalf), based on the recipient’s security needs.
Renewed assurance of the validity of a received ephemeral public key and renewed assurance
that the other party is in possession of the corresponding ephemeral private key are not addressedin this Recommendation, since ephemeral key pairs are short-lived.
5.6.2.2.1
Recipient Assurance of Static Public-Key Validity
The recipient of another party’s static public key shall obtain assurance of the validity of thatpublic key in one or more of the following ways:
1. The recipient performs a successful full public-key validation of the received public key (seeSections 5.6.2.3.1 and 5.6.2.3.2).
2. The recipient receives assurance that a trusted third party (trusted by the recipient) has
performed a successful full public-key validation of the received public key (see Sections
5.6.2.3.1 and 5.6.2.3.2). This TTP could, for example, be a binding authority, such as a CA
(see Section 4.1), and/or the TTP that generated the key pair (in accordance with method b of
Section 5.6.2.1.1) and provided it to the owner (i.e., the other party). In the case of TTP
generation, the TTP may employ a key-generation routine that performs a full public-key
validation or an equivalent procedure as part of its key-pair generation process, or the full
public-key validation may be performed as a separate process from the generation of the key
pair for the other party.
5.6.2.2.2
Recipient Assurance of Ephemeral Public-Key Validity
The recipient of another party’s ephemeral public key is required to obtain assurance of its
validity, unless both of the following conditions are met.
1. The ephemeral public key is provided to the recipient for use in a key-agreement transactionthat employs a scheme from the C(2e, 0s) subcategory; and2. The source (i.e. the putative owner) of the received ephemeral public key is authenticated bymeans of an approved security technique (e.g., a digital signature or a MAC using a preshared key) that binds that ephemeral public key to the transaction and/or provides assurance
to the recipient of active participation in the transaction by the (putative) owner of thereceived ephemeral public key.
If the conditions above are not met, or the recipient desires/requires the assurance for otherreasons, then the recipient of another party’s ephemeral public key shall obtain assurance of thevalidity of that public key by using one of the following methods:

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1. The recipient performs a successful full public-key validation on the received public key (seeSections 5.6.2.3.1 and 5.6.2.3.2).
2. If using an ECC scheme (only), the recipient performs a successful partial public-key
validation on the received public key (see Section 5.6.2.3.3).
5.6.2.2.3 Recipient Assurance of the Owner’s Possession of a Static Private Key
The recipient of another party’s static public key shall obtain an initial assurance that the otherparty (i.e., the claimed owner of the public key) actually possesses the associated private key,either prior to or concurrently with performing a key-agreement transaction with that other party.
Assurance of the validity of the corresponding public key shall be obtained prior to obtaining
this assurance.
As part of the proper implementation of this Recommendation, system users and/or agentstrusted to act on their behalf should determine which of the methods for obtaining assurance ofpossession meet their security requirements. The application tasked with performing key
establishment on behalf of a party should determine whether or not to proceed with a keyestablishment transaction, based upon the perceived adequacy of the method(s) used. Suchknowledge may be explicitly provided to the application in some manner, or may be implicitly
provided by the operation of the application itself.
A binding authority can be used to bind the key pair owner’s identifier to his static public key. In
this case, at the time of binding an owner’s identifier to his static public key, the binding
authority (i.e., a trusted third party, such as a CA) shall obtain assurance that the owner is inpossession of the correct static private key. This assurance shall either be obtained using one of
the methods specified in Section 5.6.2.2.3.2 (e.g., with the binding authority acting as the publickey recipient) or by using an approved alternative (see SP 800-57, Part 1, Sections 5.2 and8.1.5.1.1.2).
Recipients not acting in the role of a binding authority shall obtain this assurance – either
through a trusted third party (see Section 5.6.2.2.3.1) or directly from the owner (i.e., the otherparty) (see Section 5.6.2.2.3.2) – before using the derived keying material for purposes beyond
those required during the key-agreement transaction itself. If the recipient chooses to obtain thisassurance directly from the other party (i.e., the claimed owner of that public key), then tocomply with this Recommendation, the recipient shall use one of the methods specified in
Section 5.6.2.2.3.2.
5.6.2.2.3.1 Recipient Obtains Assurance from a Trusted Third Party

The recipient of a static public key may receive assurance that its owner (i.e., the other party inthe key-agreement transaction) is in possession of the correct static private key from a trusted
third party (trusted by the recipient), either before or during a key-agreement transaction thatmakes use of that static public key. The methods used by a third party trusted by the recipient toobtain that assurance are beyond the scope of this Recommendation (see however, the discussion
in Section 5.6.2.2.3 above and Section 8.1.5.1.1.2 of [SP 800-57]).
5.6.2.2.3.2 Recipient Obtains Assurance Directly from the Claimed Owner (i.e., the Other Party)

When two parties engage in a key-agreement transaction, there is (at least) an implicit claim ofownership made whenever a static public key is provided on behalf of a particular party. That

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party is considered to be a claimed owner of the corresponding static key pair – as opposed tobeing a true owner – until adequate assurance can be provided that the party is actually the one
authorized to use the static private key. The claimed owner can provide such assurance bydemonstrating its knowledge of that private key.
If all of the following conditions are met during a key-agreement transaction that incorporates
key-confirmation as specified in this Recommendation, then in the course of establishing keying
material, the recipient of a static public key may be able to directly obtain (initial or renewed)
assurance of the claimed owner’s (i.e., the other party’s) current possession of the corresponding
static private key:1) The recipient of the static public key contributes an ephemeral public key to the keyagreement process, one that is intended to be arithmetically combined with the claimed
owner’s (i.e., the other party’s) static private key in computations performed by theclaimed owner. (If an appropriate key-agreement scheme is employed, the claimed ownerwill be challenged to demonstrate current knowledge of his static private key by
successfully performing those computations during the transaction.)
2) The recipient of the static public key is also a key-confirmation recipient, with theclaimed owner (i.e., other party) serving as the key-confirmation provider. (Bysuccessfully providing key-confirmation, the claimed owner can demonstrate that he is
the true owner of the received static public key and that he currently knows thecorresponding static private key.)
There are a number of key-agreement schemes specified in this Recommendation that can beused while satisfying both of the conditions above. In order to claim conformance with thisRecommendation, the key-agreement transaction during which the recipient of a static public
key seeks to obtain assurance of its owner’s current possession of the corresponding staticprivate key shall employ one of the following approved key-agreement schemes,
incorporating key-confirmation as specified in the indicated sections, with the recipient ofthat static public key acting as party U and serving as a key-confirmation recipient:•

dhHybridOneFlow (see Section 6.2.1.1, and either Section 6.2.1.5.2 or Section
6.2.1.5.3),•(Cofactor) One-Pass Unified Model (see Section 6.2.1.2, and either Section 6.2.1.5.2 or
Section 6.2.1.5.3),•

MQV1 (see Sections 6.2.1.3, and either Section 6.2.1.5.2 or Section 6.2.1.5.3),•

One-Pass MQV (see Sections 6.2.1.4, and either Section 6.2.1.5.2 or Section 6.2.1.5.3),•

dhOneFlow (see Sections 6.2.2.1 and 6.2.2.3.1), or•(Cofactor) One-Pass Diffie-Hellman (see Sections 6.2.2.2 and 6.2.2.3.1).

5.6.2.2.4

Recipient Assurance of the Owner’s Possession of an Ephemeral Private KeyThis Recommendation does not require the recipient of an ephemeral public key to obtainassurance of the possession of the corresponding ephemeral private key by its claimed owner(i.e., the other participant in a key-establishment transaction). However, such assurance may be
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desired by the recipient, insisted upon by the recipient’s organization, and/or required by aparticular application.
Ephemeral key pairs are generated by their owner when needed (typically for a single use), andtheir private components are destroyed shortly thereafter (See Section 5.6.3.3 for details). Thus,
the opportunity for the recipient of an ephemeral public key to obtain assurance that its claimed
owner is in possession of the corresponding ephemeral private key is limited to the (single) keyestablishment transaction during which it was received.
If all of the following conditions are met during a key-agreement transaction that incorporates
key-confirmation as specified in this Recommendation, then in the course of establishing keying
material, the recipient of an ephemeral public key may be able to obtain assurance that the otherparticipant (i.e., the claimed owner) is in possession of the corresponding ephemeral private key:1) The recipient of the ephemeral public key also receives a static public key that ispresumed to be owned by the other party and is used in the key-agreement transaction.
(Therefore, the other party is the claimed owner of both the received static public key andthe received ephemeral public key.)2) The recipient of the ephemeral public key contributes its own (distinct) ephemeral public
key to the key-agreement process, one that is intended to be arithmetically combined withthe claimed owner’s ephemeral private key in computations performed by the claimed
owner. (If an appropriate key-agreement scheme is employed, the claimed owner will bechallenged to demonstrate current knowledge of his ephemeral private key by
successfully performing those computations during the transaction.)
3) The recipient of the ephemeral public key is also a key-confirmation recipient, with theclaimed owner serving as the key-confirmation provider. (By successfully providing keyconfirmation, the claimed owner can demonstrate that he actually is the owner of thereceived static public key and that he knows the ephemeral private key corresponding to
the received ephemeral public key.)
There are a limited number of key-agreement schemes specified in this Recommendation thatcan be used while satisfying all three of the conditions above. In order to claim conformancewith this Recommendation, the key-agreement transaction during which the recipient of anephemeral public key seeks to obtain assurance of the claimed owner’s possession of thecorresponding ephemeral private key shall employ one of the following approved keyagreement schemes, incorporating key-confirmation as specified in the indicated sections,with the recipient of the ephemeral public key serving as a key-confirmation recipient:• dhHybrid1 (see Section 6.1.1.1, and Section 6.1.1.5) or• (Cofactor) Full Unified Model (see Sections 6.1.1.2 and 6.1.1.5).
Note: If key-confirmation is provided in both directions in a key-agreement transaction
employing one of the schemes above, then each party can obtain assurance of the other party’s
possession of their ephemeral private key.

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5.6.2.3 Public Key Validation Routines
5.6.2.3.1 FFC Full Public-Key Validation Routine
FFC full public-key validation refers to the process of checking all the arithmetic properties of acandidate FFC public-key to ensure that it has the (unique) correct representation in the correct
subgroup (and, therefore, is also in the correct multiplicative group) of the finite field specified
by the associated FFC domain parameters. FFC full public-key validation does not require
knowledge of the associated private key and so may be done at any time by anyone. This routine
shall be used with static and/or ephemeral FFC public keys when assurance of the validity ofsuch keys is to be obtained as specified in Sections 5.6.2.1.3, 5.6.2.2.1, or 5.6.2.2.2.
Input:
1. (p, q, g{, SEED, pgenCounter}): A valid set of FFC domain parameters, and
2. y: A candidate FFC public key.
Process:
1. Verify that 2 ≤ y ≤ p-2.
(This step is to ensure that the public key has the unique correct representation and range
for the field.)
2. Verify that 1= yq mod p.
(This step is to ensure that the public key has the correct order and thus is in the correct
subgroup.)
Output: If any of the above verifications fail, then output an error indicator. Otherwise, output
an indication of successful validation.
5.6.2.3.2 ECC Full Public-Key Validation Routine
ECC full public-key validation refers to the process of checking all the arithmetic properties of acandidate ECC public key to ensure that it has the (unique correct) representation in the correct(additive) subgroup (and, therefore, is also in the correct EC group) specified by the associated
ECC domain parameters. ECC full public-key validation does not require knowledge of theassociated private key and so may be done at any time by anyone. This routine shall be usedwith static and/or ephemeral ECC public keys when assurance of the validity of such keys is tobe obtained in accordance with the requirements of Section 5.6.2.1.3 or Section 5.6.2.2.1, or as
specified in method 1 of Section 5.6.2.2.2.
Input:
1. (q, FR, a, b{, SEED}, G, n, h): A valid set of ECC domain parameters, and
2. Q = (xQ, yQ ): A candidate ECC public key.
Process:
1. Verify that Q is not the point at infinity Ø. This can be done by inspection if the point isentered in the standard affine representation.
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NIST SP 56A: Revision 2, May 2013
(Partial check of the public key for an invalid range in the EC group.)
2. Verify that xQ and yQ are integers in the interval [0, p−1] in the case that q is an odd primep, or that xQ and yQ are bit strings of length m bits in the case that q = 2m.
(Ensures that each coordinate of the public key has the unique correct representation ofan element in the underlying field.)
3. If q is an odd prime p, verify that (yQ)2 = (xQ)3 + axQ + b in GF(p) , where the arithmetic is
performed modulo p.
If q = 2m, verify that (yQ)2 + xQ yQ = (xQ)3 + a(xQ)2 + b in GF(2m), where the arithmetic is
performed as dictated by the field representation parameter FR.
(This step is to ensure that the public key is on the correct elliptic curve.)
4. Verify that nQ = Ø.
(This step is to ensure that the public key has the correct order. Along with theverification in step 1, ensures that the public key is in the correct range in the correct EC
subgroup; that is, it is in the correct EC subgroup and is not the identity element Ø.)
Output: If any of the above verifications fail, then output an error indicator. Otherwise, output
an indication of successful validation.
5.6.2.3.3 ECC Partial Public-Key Validation Routine
ECC partial public-key validation refers to the process of checking some (but not all) of thearithmetic properties of a candidate ECC public key to ensure that it is in the correct group (butnot necessarily the correct subgroup) specified by the associated ECC domain parameters. ECC
partial public-key validation omits the validation of subgroup membership 5, and therefore isusually faster than ECC full public key validation. ECC partial public-key validation does not
require knowledge of the associated private key and so may be done at any time by anyone. Thisroutine shall be used with ephemeral ECC public keys when assurance of the validity of suchkeys is to be obtained in accordance with method 2 of Section 5.6.2.2.2.
Input:
1. (q, FR, a, b{, SEED}, G, n, h): A valid set of ECC domain parameters, and
2. Q = (xQ, yQ): A candidate ECC public key.
Process:
1. Verify that Q is not the point at infinity Ø. This can be done by inspection if the point isentered in the standard affine representation.
(Partial check of the public key for an invalid range in the EC group.)

5

In this Recommendation, co-factor multiplication is included in the ECC primitives for Diffie-Hellman and MQV,
which forces the computed group element into the appropriate subgroup.

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2. Verify that xQ and yQ are integers in the interval [0, p-1] in the case that q is an odd primep, or that xQ and yQ are bit strings of length m bits in the case that q = 2m.
(Ensures that each coordinate of the public key has the unique correct representation ofan element in the underlying field.)
3. If q is an odd prime p, verify that (yQ) 2 = (xQ)3 + axQ + b in GF(p), where the arithmetic is
performed modulo p.
If q = 2m, verify that (yQ)2 + xQ yQ = (xQ)3 + a(xQ)2 + b in GF(2m), where the arithmetic is
performed as dictated by the field representation parameter FR.
(This step is to ensure that the public key is on the correct elliptic curve.)(Note: Since its order is not verified, there is no check that the public key is in the correct
EC subgroup.)
Output: If any of the above verifications fail, then output an error indicator. Otherwise, output
an indication of validation success.
5.6.3 Key Pair Management
5.6.3.1 Common Requirements on Static and Ephemeral Key Pairs
The following are common requirements on static and ephemeral key pairs (see the
Recommendation for Key Management [SP 800-57]):
1. Each private/public key pair shall be correctly associated with its corresponding specific
set of domain parameters. A key pair shall not be used with more than one set of domain
parameters.
2. Each key pair shall be generated as specified in Section 5.6.1.
3. Private keys shall be protected from unauthorized access, disclosure, modification andsubstitution.
4. Public keys shall be protected from unauthorized modification and substitution. This isoften accomplished by using public-key certificates that have been signed by a
Certification Authority (CA).
5.6.3.2 Specific Requirements on Static Key Pairs
The additional specific requirements for static key pairs are as follows:
1. The owner of a static key pair shall confirm the validity of the key pair by obtaining
assurance of the correct generation of the key pair, private and public-key validity, andpair-wise consistency. The owner shall know the methods used to provide/obtain theseassurances. (See Section 5.6.2.1 for further details.)
2. A recipient of a static public key shall be assured of the integrity and correct associationof (a) the public key, (b) the set of domain parameters for that key, and (c) an identifier
for the entity that owns the key pair (that is, the party with whom the recipient intends toestablish a key). This assurance is often provided by verifying a public-key certificate

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NIST SP 56A: Revision 2, May 2013that was signed by a trusted third party (for example, a CA), but may be provided bydirect distribution of the keying material from the owner, provided that the recipient
trusts the owner to do this. See Section 4.1.
3. A recipient of a static public key shall obtain assurance of the validity of the public key.
This assurance may be provided, for example, through the use of a public-key certificate
if the CA obtains sufficient assurance of public-key validity as part of its certification
process. See Section 5.6.2.2.1.
4. A recipient of a static public key shall have assurance of the owner’s possession of thecorresponding private key (see Section 5.6.2.2.3). The recipient shall know the method
used to provide assurance to the recipient of the owner’s possession of the private key.
This assurance may be provided, for example, through the use of a public-key certificate
if the CA obtains sufficient assurance of possession as part of its certification process.
5. A static key pair may be used in more than one key-establishment scheme. However, one
static public/private key pair shall not be used for different purposes (for example, adigital signature key pair is not to be used for key establishment or vice versa) with thefollowing possible exception: when requesting the (initial) certificate for a public static
key-establishment key, the key-establishment private key associated with the public key
may be used to sign the certificate request. See SP 800-57, Part 1 on Key Usage forfurther information.
5.6.3.3 Specific Requirements on Ephemeral Key Pairs
The additional specific requirements on ephemeral key pairs are as follows:
1. An ephemeral private key shall be used in exactly one key-establishment transaction,with one exception: an ephemeral private key may be used in multiple DLC key-transporttransactions that are transporting identical secret keying material simultaneously (orwithin a short period of time; see the broadcast scenario in Section 7). After its use, an
ephemeral private key shall be destroyed.
2. An ephemeral key pair should be generated as close to its time of use as possible. Ideally,an ephemeral key pair is generated just before the ephemeral public key is transmitted.
3. The owner of an ephemeral key pair shall confirm the validity of the key pair byobtaining assurance of correct generation, private- and public-key validity, and pair-wise
consistency. The owner shall know the methods used to provide/obtain these assurances.
These assurances can be obtained by virtue of the technique used by the owner togenerate the key pair. (See Section 5.6.2.1 for further details.)
4. A recipient of an ephemeral public key shall have assurance of the validity of the public
key unless the conditions specified in Section 5.6.2.2.2 are satisfied.
5. If a recipient of an ephemeral public key requires assurance that the claimed owner ofthat public key has possession of the corresponding private key, then, in order to obtainthat assurance in compliance with this Recommendation, such assurance shall beobtained as specified in Section 5.6.2.2.4. Although other methods are sometimes used toprovide such assurance, this Recommendation makes no statement as to their adequacy.

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5.7

DLC Primitives

A primitive is a relatively simple operation that is defined to facilitate implementation in
hardware or in a software subroutine. Each key-establishment scheme shall use exactly one DLC
primitive. Each scheme in Section 6 shall use an appropriate primitive from the following list:
1. The FFC DH primitive (Section 5.7.1.1 of this Recommendation): This primitive shall be
used by the dhHybrid1, dhEphem, dhHybridOneFlow, dhOneFlow and dhStatic schemes,
which are based on finite field cryptography and the Diffie-Hellman algorithm.
2. The ECC CDH primitive (Section 5.7.1.2 of this Recommendation and called theModified Diffie-Hellman primitive in ANS X9.63): This primitive shall be used by the
Full Unified Model, Ephemeral Unified Model, One-Pass Unified Model, One-Pass
Diffie-Hellman and Static Unified Model schemes, which are based on elliptic curve
cryptography and the Diffie-Hellman algorithm.
3. The FFC MQV primitive (Section 5.7.2.1 of this Recommendation): This primitive shallbe used by the MQV2 and MQV1 schemes, which are based on finite field cryptography
and the MQV algorithm.
4. The ECC MQV primitive (Section 5.7.2.3 of this Recommendation): This primitive shallbe used by the Full MQV and One-Pass MQV schemes, which are based on elliptic curve
cryptography and the MQV algorithm.
The shared secret output from these primitives shall be used as input to a key derivation method(see Section 5.8).
5.7.1 Diffie-Hellman Primitives
5.7.1.1 Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive
A shared secret Z is computed using the domain parameters (p, q, g{, SEED, pgenCounter}), theother party’s public key and one’s own private key. This primitive is used in Section 6 by thedhHybrid1, dhEphem, dhHybridOneFlow, dhOneFlow and dhStatic schemes. Assume that the
party performing the computation is party A, and the other party is party B. Note that party A
could be either party U or party V.
Input:
1. (p, q, g{, SEED, pgenCounter}): Domain parameters,
2. xA : One’s own private key, and3. yB : The other party’s public key.
Process:
1. z = y B x A mod p
2. If z = 1, destroy all intermediate values used in the attempted computation of Z (includingz) and output an error indicator.

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3. Else, convert z to Z using the integer-to-byte-string conversion routine defined inAppendix C.1.
4. Destroy the results of all intermediate calculations used in the computation of Z
(including z).
5. Output Z.
Output: The shared secret Z or an error indicator.
5.7.1.2 Elliptic Curve Cryptography Cofactor Diffie-Hellman (ECC CDH) Primitive
A shared secret Z is computed using the domain parameters (q, FR, a, b{, SEED}, G, n, h), theother party’s public key, and one’s own private key. This primitive is used in Section 6 by the
Full Unified Model, Ephemeral Unified Model, One-Pass Unified Model, One-Pass DiffieHellman and Static Unified Model schemes. Assume that the party performing the computation
is party A, and the other party is party B. Note that party A could be either party U or party V.
Input:
1. (q, FR, a, b{, SEED}, G, n, h): Domain parameters,
2. dA : One’s own private key, and
3. QB : The other party’s public key.
Process:
1. Compute the point P = hdAQB.
2. If P = Ø, destroy all intermediate values used in the attempted computation of P and
output an error indicator.
3. Else, convert z = xP to Z, using the field-element-to-byte string conversion routine
defined in Appendix C.2, where xP is the x-coordinate of P.
4. Destroy the results of all intermediate calculations used in the computation of Z
(including P and z).
5. Output Z.
Output: The shared secret Z or an error indicator.
5.7.2 MQV Primitives
5.7.2.1 Finite Field Cryptography MQV (FFC MQV) Primitive
A shared secret Z is computed using the domain parameters (p, q, g{, SEED, pgenCounter}), theother party’s public keys and one’s own public and private keys. Assume that the party
performing the computation is party A, and the other party is party B. Note that party A could beeither party U or party V.
Input:
1. (p, q, g{, SEED, pgenCounter}): Domain parameters,

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2. xA : One’s own static private key,
3. yB : The other party’s static public key,
4. rA : One’s own second private key, 6
5. tA : One’s own second public key, and6. tB : The other party’s second public key.
Process:
1
1. w =  log 2 q  .
2

2. T A = (t A mod 2 w ) + 2 w .
3. S A = (rA + T A x A ) mod q .
4. TB = (t B mod 2 w ) + 2 w .
5. z = ((t B ( y BTB )) S A ) mod p .
6. If z = 1, destroy all intermediate values (including TA, SA, and TB) used in the attempted
computation of z and output an error indicator.
7. Else, convert z to Z using the integer-to-byte-string conversion routine defined inAppendix C.1.
8. Destroy the results of all intermediate calculations used in the computation of Z
(including TA, SA, TB, and z).
9. Output Z.
Output: The shared secret Z or an error indicator.
5.7.2.1.1 MQV2 Form of the FFC MQV Primitive
This form of invoking the FFC MQV primitive is used in Section 6.1.1.3 by the MQV2 scheme.
In this form, each party uses both a static key pair and an ephemeral key pair. Assume that the
party performing the computation is party A, and the other party is party B. Note that party A
could be either party U or party V.
In this form, one’s own second private and public keys (items 4 and 5 of the input list in Section
5.7.2.1) are one’s own ephemeral private and public keys (rA and tA), and the other party’s second
public key (item 6 in Section 5.7.2.1) is the other party’s ephemeral public key (tB).
5.7.2.1.2

MQV1 Form of the FFC MQV Primitive

6

In the FFC MQV primitive, a second key may be either ephemeral or static, depending on which form is being
used, see Sections 5.7.2.1.1 and 5.7.2.1.2.

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NIST SP 56A: Revision 2, May 2013This form of invoking the FFC MQV primitive is used in Section 6.2.1.3 by the MQV1 scheme.
In this form, party U uses a static key pair and an ephemeral key pair, but party V uses only astatic key pair. One-Pass MQV is done using the MQV primitive by using party V’s static key
pair as the second key pair (as party V has no ephemeral key pair).
Party U uses party V’s static public key for the other party’s second public key; that is, whenparty U uses the algorithm in Section 5.7.2.1, item 6 of the input list becomes party V’s static
public key (yB).
Party V uses his/her static private key for his second private key; that is, when party V uses thealgorithm in Section 5.7.2.1, item 4 of the input list becomes party V’s static private key xA, anditem 5 becomes his static public key (yA).
5.7.2.2 ECC MQV Associate Value Function
The associate value function is used by the ECC MQV family of key-agreement schemes tocompute an integer that is associated with an elliptic curve point. This Recommendation defines
avf(Q) to be the associate value function of a public key Q using the domain parameters (q, FR,a, b{, SEED}, G, n, h).
Input:
1. (q, FR, a, b{, SEED}, G, n, h): Domain parameters, and
2. Q: A public key (that is, Q is a point in the subgroup of order n and not equal to the point
at infinity Ø).
Process:
1. Convert xQ to an integer xqi using the convention specified in Appendix C.3.
2. Calculate
xqm = xqi mod 2  f / 2  (where f = log 2 n  ).
3. Calculate the associate value function
avf(Q) = xqm + 2  f / 2  . (See footnote 7).
Output: avf(Q), the associate value of Q.
5.7.2.3 Elliptic Curve Cryptography MQV (ECC MQV) Primitive
The ECC MQV primitive is computed using the domain parameters (q, FR, a, b{, SEED}, G, n,h), the other party’s public keys, and one’s own public and private keys. Assume that the party
performing the computation is party A, and the other party is party B. Note that party A could beeither party U or party V.
Input:
1. (q, FR, a, b{, SEED}, G, n, h): Domain parameters,

7

Note that avf(Q) can be computed using only bit operations.

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NIST SP 56A: Revision 2, May 2013
2. ds,A : One’s own static private key,
3. Qs,B : The other party’s static public key,
4. de,A : One’s own second private key, 8
5. Qe,A : One’s own second public key, and
6. Qe,B : The other party’s second public key.
Process:
1. implicitsigA = (de,A + avf(Qe,A)ds,A ) mod n.
2. P = h(implicitsigA)(Qe,B + avf(Qe,B)Qs,B).
3. If P = Ø, destroy all intermediate values used in the attempted computation of P and
output an error indicator.
4. Else, convert z=xP to Z, using the field-element-to-byte-string conversion routine definedin Appendix C.2, where xP is the x-coordinate of P.
5. Destroy the results of all intermediate calculations used in the computation of Z
(including P and z).
6. Output Z.
Output: The shared secret Z or an error indicator.
5.7.2.3.1 Full MQV Form of the ECC MQV Primitive
This form of invoking the ECC MQV primitive is used in Section 6.1.1.4 by the Full MQV
scheme. In this form, each party has both a static key pair and an ephemeral key pair. Assume
that the party performing the computation is party A, and the other party is party B. Note thatparty A could be either party U or party V.
In this form, one’s own second private and public keys (item 4 and 5 of the input list in Section
5.7.2.3) are one’s own ephemeral private and public keys (de,A and Qe,A), and the other party’s
second public key (item 6 of the input list in Section 5.7.2.3) is the other party’s ephemeral
public key (Qe,B).
5.7.2.3.2 One-Pass Form of the ECC MQV Primitive
This form of invoking the ECC MQV primitive is used in Section 6.2.1.4 by the One-Pass MQV
scheme. In this form, party U has a static key pair and an ephemeral key pair, but party V has
only a static key pair. One-Pass MQV uses the MQV primitive with party V’s static key pair as
the second key pair (as party V has no ephemeral keys).

8

In the ECC MQV primitive, a second key may be either ephemeral or static, depending on which form is being
used, see Sections 5.7.2.3.1 and 5.7.2.3.2.

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Party U uses party V’s static public key as the other party’s second public key. When party U
uses the algorithm in Section 5.7.2.3, item 6 of the input list becomes party V’s static public key(Qs,B).
Party V uses his static private key as his second private key. When party V uses the algorithm in
Section 5.7.2.3, item 4 of the input list becomes party V’s static private key ds,A, and item 5
becomes his static public key (Qs,A).
5.8

Key-Derivation Methods for Key-Agreement Schemes

This section introduces approved key-derivation methods for use in key establishment as
specified in this Recommendation. An approved key-derivation method shall be used to derive
keying material from the shared secret, Z, that is computed during the execution of a keyagreement scheme specified in this Recommendation.
Key-derivation methods that conform to this Recommendation include the use of an approved
single-step key-derivation function (KDF), as well as the use of an approved two-step(extraction-then-expansion) key-derivation procedure – see Sections 5.8.1 and 5.8.2,
respectively. Additional approved application-specific key-derivation methods are discussed in
Section 5.8.3. Other key-derivation methods may be temporarily allowed for backward
compatibility; these other allowable methods – and any restrictions on their use – will be
specified in [FIPS 140 IG].
When employed during the execution of a key-agreement scheme as specified in thisRecommendation, the agreed-upon key-derivation method uses input that includes a newly
computed shared secret, Z. The derived keying material shall be computed in its entirety before
outputting any portion of it, and (all copies of) Z shall be destroyed immediately following its
use.
The output produced by a key-derivation method using input that includes the shared secretcomputed during the execution of any key-agreement scheme specified in this Recommendation
shall only be used as secret keying material – such as a symmetric key used for data encryption
or message integrity, a secret initialization vector, or, perhaps, a master (key derivation) key thatwill be used to generate additional keying material (possibly using a different process – see [SP800-108]). Non-secret keying material (such as a non-secret initialization vector) shall not be
generated from input that includes the shared secret.
5.8.1 The Single-step Key-Derivation Function
This section specifies an approved key-derivation function (KDF) that is executed in a single
step, rather than the two-step procedure discussed in Section 5.8.2. The input to the KDF
includes the shared secret Z (represented as a byte string).
This single-step KDF uses an auxiliary function H, which can be either 1) an approved hashfunction, denoted as hash, as defined in [FIPS 180] or 2) an HMAC with an approved hashfunction, hash, denoted as HMAC-hash, as defined in [FIPS 198]. Table 6 and Table 7 identify
the minimum output block length for the hash functions and HMACs required for each FFC and
ECC parameter-size set.

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Table 6: Minimum output length of H for FFC domain parameter-size sets
FFC parameter set name

FA

FB

FC

Maximum security strength supported (in bits)

80

112

112

Bit length of field size p (i.e., log 2 p  )

1024

2048

2048

Bit length of subgroup order q (i.e., log 2 q  )

160

224

256

Minimum output length of H (in bits)

80

112

112

When an FFC parameter set is selected in a key-establishment scheme, any approved hashfunction, hash, can be used to define the auxiliary function, H, whether H = hash or H = HMAChash (see Section 5.8.1.1).
Table 7: Minimum output length of H for ECC domain parameter sets
ECC parameter set name

EA

EB

EC

ED

EE

Maximum security strength supported (in bits)

80

112

128

192

256

Bit length of ECC subgroup order n

160 to223

224 to255

256 to383

384 to511

512 orlonger

80

112

128

192

256

(i.e., log 2 n  )Minimum output length of H (in bits)

Regardless of the form chosen for the auxiliary function H (i.e., H = hash or H = HMAC hash),the approved hash function, hash, used to define H shall be selected as follows:•••

For ECC parameter sets EA, EB and EC, any approved hash function can be used ashash.
For parameter sets ED and EE, any approved hash function other than SHA-1 can beused as hash; SHA-1 shall not be used, since the output block length of SHA-1 is only
160 bits.
For parameter set EE, any approved hash function with an output block length greaterthan or equal to 256 bits can be used as hash; hash functions with output block lengths
less than 256 bits (e.g., SHA-224 and SHA-512/224) shall not be used.

5.8.1.1 The Single-Step KDF Specification
This section specifies an approved single-step key-derivation function (KDF), whose input
includes the shared secret Z (represented as a byte string) and other information.
The KDF is specified as follows:
Function call: kdf (Z, OtherInput),where OtherInput consists of keydatalen and OtherInfo.
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Auxiliary Function H (two options):
Option 1: H(x) = hash(x), where hash is an approved hash function meeting the selection
requirements specified in this Recommendation (see Sections 5.1 and 5.8.1), andthe input, x, is a bit string.
Option 2: H(x) = HMAC-hash(salt, x), where HMAC-hash is an instantiation of the HMAC
function (as defined in [FIPS 198]) employing an approved hash function, hash(see Section 5.1), and hash meets the selection requirements specified in thisRecommendation (see Sections 5.1 and 5.8.1). An implementation-dependent byte
string, salt, serves as the HMAC key, and x (the input to H) is a bit string thatserves as the HMAC “message” – as specified in [FIPS 198].
Implementation-Dependent Parameters:
1. hashlen: an integer that indicates the length (in bits) of the output block of the hash function,hash, employed by the auxiliary function, H, that is used to derive blocks of secret keying
material.
2. max_H_inputlen: an integer that indicates the maximum permitted length (in bits) of the bit
string, x, that is used as input to the auxiliary function, H.
3. salt: a (public or private) byte string that is only required when an HMAC-based auxiliary
function is implemented (see Option 2 above). The salt could be, for example, a value
computed from nonces exchanged as part of a key-establishment protocol that employs one
or more of the key-agreement schemes specified in this Recommendation, a value alreadyshared by the protocol participants, or a value that is pre-determined by the protocol. In thiscase, the length of the salt can be any agreed-upon length. However, if there is no means of
selecting the salt, then it shall be an all-zero byte string whose bit length equals that specified
as the length of the input block for the hash function, hash.
Input:
1. Z: a byte string that represents the shared secret z.
2. keydatalen: An integer that indicates the length (in bits) of the secret keying material to be
derived; keydatalen shall be less than or equal to hashlen × (232 –1).
3. OtherInfo: A bit string of context-specific data (see Section 5.8.1.2 for details).
Process:
1. reps =  keydatalen / hashlen.
2. If reps > (232 −1), then return an error indicator without performing the remaining actions.
3. Initialize a 32-bit, big-endian bit string counter as 0000000116 (i.e. 0x00000001).
4. If counter || Z || OtherInfo is more than max_H_inputlen bits long,then return an error indicator without performing the remaining actions.
5. For i = 1 to reps by 1, do the following:
5.1 Compute K(i) = H(counter || Z || OtherInfo).
5.2 Increment counter (modulo 232), treating it as an unsigned 32-bit integer.
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6. Let K_Last be set to K(reps) if (keydatalen / hashlen) is an integer; otherwise, let K_Lastbe set to the (keydatalen mod hashlen) leftmost bits of K(reps).
7. Set DerivedKeyingMaterial = K(1) || K(2) || … || K(reps-1) || K_Last.
Output:The bit string DerivedKeyingMaterial of length keydatalen bits (or an error indicator).
Notes:
When an approved key-agreement scheme is used to determine a shared secret Z, theparticipants should know which entity is acting as “party U” and which entity is acting as“party V” to ensure (among other things) that they will derive the same keying material. (See
Section 6 for descriptions of the specific actions required of parties U and V during theexecution of each of the approved key-agreement schemes.) The roles of party U and party
V shall be assigned to the key-establishment participants by the protocol employing the keyagreement scheme.
In step 5.1 above, the entire output of the hash function hash shall be used whether H(x) =hash(x) or H(x) = HMAC-hash(salt, x). Therefore, the bit length of each output block of H ishashlen bits. Some of the hash functions specified in [FIPS 180] are defined with an internal
truncation operation (e.g., SHA-384). In these cases, the “entire output” of hash is the output
value as defined in [FIPS 180] (e.g., for SHA-384, the entire output is defined to be the 384
bits resulting from the internal truncation, so hashlen = 384, in this case). Any truncation
performed by the KDF (external to hash) is done in step 6.
5.8.1.2 OtherInfo
The bit string OtherInfo should be used to ensure that the derived keying material is adequately“bound” to the context of the key-agreement transaction. Although other methods may be used tobind keying material to the transaction context, this Recommendation makes no statement as tothe adequacy of these other methods. Failure to adequately bind the derived keying material to
the transaction context could adversely affect the types of assurance that can be provided bycertain key-agreement schemes.
Context-specific information that may be appropriate for inclusion in OtherInfo:
•
•
•
•
•
•
•
•
•

Public information about parties U and V, such as their identifiers.
The public keys contributed by each party to the key-agreement transaction. (In the caseof a static public key, one could include a certificate that contains the public key.)
Other public and/or private information shared between parties U and V before or during
the transaction, such as nonces or pre-shared secrets.
An indication of the protocol or application employing the key derivation method.
Protocol-related information, such as a label or session identifier.
The desired length of the derived keying material.
An indication of the key-agreement scheme and/or key-derivation method used.
An indication of the domain parameters associated with the asymmetric key pairs
employed for key establishment.
An indication of other parameter or primitive choices (e.g., hash functions, MacTag
lengths, etc.).
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NIST SP 56A: Revision 2, May 2013•

An indication of how the derived keying material should be parsed, including an
indication of which algorithm(s) will use the (parsed) keying material.

For rationale in support of including entity identifiers, scheme identifiers, and/or otherinformation in OtherInfo, see Appendix B.
The meaning of each information item and each item’s position within the bit string OtherInfo
shall be specified. In addition, each item of information included in OtherInfo shall beunambiguously represented, for example, as a fixed-length bit string or in the form Datalen ||Data, where Data is a variable-length string of zero or more (eight-bit) bytes, and Datalen is a
fixed-length, big-endian counter that indicates the length (in bytes) of Data. These requirements
can be satisfied, for example, by using ASN.1 DER encoding as specified in 5.8.1.2.2 for
OtherInfo.
Recommended formats for OtherInfo are specified in Sections 5.8.1.2.1 and 5.8.1.2.2. One ofthese two formats should be used by the single-step KDF specified in Section 5.8.1.1 when theauxiliary function employed is H = hash. When the recommended formats are used, the included
items of information shall be divided into (three, four, or five) subfields as defined below.
AlgorithmID: A required non-null subfield that indicates how the derived keying material
will be parsed and for which algorithm(s) the derived secret keying material will be used. For
example, AlgorithmID might indicate that bits 1-112 are to be used as a 112-bit HMAC key
and that bits 113-240 are to be used as a 128-bit AES key.
PartyUInfo: A required non-null subfield containing public information about party U. At aminimum, PartyUInfo shall include IDU, an identifier for party U, as a distinct item of
information. This subfield could also include information about the public key(s) contributed
to the key-agreement transaction by party U. The nonce provided by party U as required in aC(0e, 2s) scheme (see Section 6.3) shall be included in this subfield.
PartyVInfo: A required non-null subfield containing public information about party V. At aminimum, PartyVInfo shall include IDV, an identifier for party V, as a distinct item of
information. This subfield could also include information about the public key(s) contributed
to the key-agreement transaction by party V. The nonce provided by party V when acting asa key-confirmation recipient in a C(1e, 2s) scheme or a C(0e, 2s) scheme shall be included inthis field (see Sections 6.2.1.5 and 6.3.3).
SuppPubInfo: An optional subfield, which could be null that contains additional, mutuallyknown public information (e.g., keydatalen, the domain parameters associated with the keys
used to derive the shared secret, an identifier for the particular key-agreement scheme thatwas used to form Z, an indication of the protocol or application employing that scheme, asession identifier, etc.; this is particularly useful if these aspects of the key-agreementtransaction can vary – see Appendix B for further discussion).
SuppPrivInfo: An optional subfield, which could be null, that contains additional, mutuallyknown private information (e.g., a shared secret symmetric key that has been communicatedthrough a separate channel).
5.8.1.2.1

The Concatenation Format for OtherInfo

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NIST SP 56A: Revision 2, May 2013This section specifies the concatenation format for OtherInfo. This format has been designed toprovide a simple means of binding the derived keying material to the context of the keyagreement transaction, independent of other actions taken by the relying application. Note: Whenthe single-step KDF specified in Section 5.8.1.1 is used with H = hash as the auxiliary function
and this concatenation format for OtherInfo, the resulting key-derivation method is theConcatenation Key Derivation Function specified in the original version of SP 800-56A.
For this format, OtherInfo is a bit string equal to the following concatenation:
AlgorithmID || PartyUInfo || PartyVInfo {|| SuppPubInfo }{|| SuppPrivInfo },where the five subfields are bit strings comprised of items of information as described in Section
5.8.1.2.
Each of the three required subfields AlgorithmID, PartyUInfo, and PartyVInfo shall be theconcatenation of a pre-determined sequence of substrings in which each substring represents a
distinct item of information. Each such substring shall have one of these two formats: either it isa fixed-length bit string, or it has the form Datalen || Data – where Data is a variable-length
string of zero or more (eight-bit) bytes, and Datalen is a fixed-length, big-endian counter thatindicates the length (in bytes) of Data. (In this variable-length format, a null string of data shallbe represented by a zero value for Datalen, indicating the absence of following data.) A protocol
using this format for OtherInfo shall specify the number, ordering and meaning of theinformation-bearing substrings that are included in each of the subfields AlgorithmID,
PartyUInfo, and PartyVInfo, and shall also specify which of the two formats (fixed-length orvariable-length) is used by each such substring to represent its distinct item of information. The
protocol shall specify the lengths for all fixed-length quantities, including the Datalen counters.
Each of the optional subfields SuppPrivInfo and SuppPubInfo (when allowed by the protocol
employing the one-step KDF) shall be the concatenation of a pre-determined sequence of
substrings representing additional items of information that may be used during key derivation
upon mutual agreement of parties U and V. Each substring representing an item of information
shall be of the form Datalen || Data, where Data is a variable-length string of zero or more(eight-bit) bytes and Datalen is a fixed-length, big-endian counter that indicates the length (inbytes) of Data; the use of this form for the information allows parties U and V to omit a
particular information item without confusion about the meaning of the other information that isprovided in the SuppPrivInfo or SuppPubInfo subfield. The substrings representing items of
information that parties U and V choose not to contribute are set equal to Null, and are
represented in this variable-length format by setting Datalen equal to zero. If a protocol allows
the use of the OtherInfo subfield SuppPrivInfo and/or the subfield SuppPubInfo, then the
protocol shall specify the number, ordering and meaning of additional items of information that
may be used in the allowed subfield(s) and shall specify the fixed-length of the Datalen
counters.
5.8.1.2.2 The ASN.1 Format for OtherInfo
The ASN.1 format for OtherInfo provides an alternative means of binding the derived keying
material to the context of the key-agreement transaction, independent of other actions taken by
the relying application. Note: When the single-step KDF specified in Section 5.8.1.1 is used with
H = hash as the auxiliary function and this ASN.1 format for OtherInfo, the resulting key-

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NIST SP 56A: Revision 2, May 2013derivation method is the ASN.1 Key Derivation Function specified in the original version of SP
800-56A.
For the ASN.1 format, OtherInfo is a bit string resulting from the ASN.1 DER encoding (see[ISO/IEC 8825-1]) of a data structure comprised of a sequence of three required subfields
AlgorithmID, PartyUInfo, and PartyVInfo, and, optionally, a subfield SuppPubInfo and/or asubfield SuppPrivInfo – as described in Section 5.8.1.2. A protocol using this format for
OtherInfo shall specify the type, ordering and number of distinct items of information includedin each of the (three, four, or five) subfields employed.
5.8.1.2.3 Other Formats for OtherInfo
Formats other than those provided in Sections 5.8.1.2.1 and 5.8.1.2.2 (e.g., those providing theitems of information in a different arrangement) may be used for OtherInfo, but the contextspecific information described in the preceding sections should be included (see the discussion
in Section 5.8.1.2). This Recommendation makes no statement as to the adequacy of otherformats.
5.8.2 The Extraction-then-Expansion Key-Derivation Procedure
This Recommendation permits the use of an approved extraction-then-expansion key-derivation
procedure as an alternative to the single-step key-derivation function specified in Section 5.8.1.
When the extraction-then-expansion key-derivation procedure is employed in an approved keyagreement scheme, the secret keying material is derived in two steps. The first step is called(randomness) extraction, and the second step is called (key) expansion. In the extraction step, theinput is the shared secret Z (represented as a byte string) along with a salt. The output of theextraction step is a key-derivation key. In the expansion step, the key-derivation key is used withadditional data, such as that included in the OtherInfo used by the KDFs specified above; also
see Appendix B for guidance) to derive the keying material with required length. The details of
the approved extraction-then-expansion key-derivation procedure are specified in [SP 800-56C].
5.8.3 Application-Specific Key-Derivation Methods
Additional approved application-specific key-derivation methods are enumerated in [SP 800135]. Unless an explicit exception is made in that document, any hash function employed by the
key-derivation methods enumerated in [SP 800-135] shall be approved and shall also meet the
selection requirements specified in this Recommendation for the auxiliary function H (seeSections 5.1 and 5.8.1).
5.9 Key-Confirmation
The term key-confirmation (KC) refers to actions taken to provide assurance to one party (thekey-confirmation recipient) that another party (the key-confirmation provider) is in possession ofa (supposedly) shared secret and/or confirm that the other party has the correct version of keying
material that was derived or transported during a key-establishment transaction. (Correct, that is,from the perspective of the key-confirmation recipient.) Such actions are said to provide
unilateral key-confirmation when they provide this assurance to only one of the participants inthe key-establishment transaction; the actions are said to provide bilateral key-confirmation

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when this assurance is provided to both participants (i.e., when unilateral key-confirmation isprovided in both directions).
Oftentimes, key-confirmation is obtained (at least implicitly) by some means external to the keyestablishment scheme employed during a transaction (e.g., by using a symmetric key that wasestablished during the transaction to decrypt an encrypted message sent later by the keyconfirmation provider), but this is not always the case. In some circumstances, it may beappropriate to incorporate the exchange of explicit key-confirmation information as an integral
part of the key-establishment scheme itself. The inclusion of key-confirmation may enhance thesecurity services that can be offered by a key-establishment scheme. For example, when certain
key-agreement schemes incorporate key-confirmation (as described in this Recommendation),they can be used to provide the recipient with assurance that the provider is in possession of theprivate key corresponding to a particular public key, from which the recipient may infer that theprovider is the owner of that key pair (see Sections 5.6.2.2.3 and 5.6.2.2.4).
For key-confirmation to comply with this Recommendation, key-confirmation shall beincorporated into an approved key-establishment scheme as specified in the sections that follow.
5.9.1 Unilateral Key-Confirmation for Key-Agreement Schemes
As specified in this Recommendation, unilateral key-confirmation occurs when one participant in
the execution of a key-agreement scheme (the key-confirmation “provider”) demonstrates to thesatisfaction of the other participant (the key-confirmation “recipient”) that both the provider andthe recipient have possession of the same secret MacKey.
MacKey is a symmetric key derived using the (shared) secret Z that was computed by each party
during that particular execution of the key-agreement scheme (see Section 5.8 for key
derivation). MacKey and certain context-specific MacData (as specified in Sections 5.9.1.1) areused by the provider as input to an approved MAC algorithm to obtain a MacTag that is sent to
the recipient. The recipient performs an independent computation of the MacTag. If the MacTag
value computed by the key-confirmation recipient matches the MacTag value received from the
key-confirmation provider, then key-confirmation is successful. (See Section 5.2 for MacTag
generation and verification, and Section 5.9.3 for MAC security discussion.)
Successful key-confirmation provides assurance to the recipient that the same Z value has been
computed by both parties and that the two parties have used Z in the same way to derive sharedkeying material.
Unilateral key-confirmation is an optional feature that can be incorporated into any keyagreement scheme in which the key-confirmation provider is required to own a static keyestablishment key pair that is used in the key-establishment process. If the intended keyconfirmation recipient is not required to contribute an ephemeral public key to the keyestablishment process, then the recipient shall instead contribute a nonce that is used as part ofthe input to the key-derivation method employed by the scheme. Each party is required to havean identifier, chosen in accordance with the assumptions stated for the key-agreement scheme.
5.9.1.1 Adding Unilateral Key-Confirmation to a Key-Agreement Scheme
To include unilateral key-confirmation from a provider (who has a static key pair) to a recipient,the following steps shall be incorporated into the scheme. Additional details will be provided for
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each scheme in the appropriate subsections of Section 6. In the discussion that follows, the keyconfirmation provider, P, may be either party U or party V, as long as P has a static key pair. The
key-confirmation recipient, R, is the other party.
1. If the recipient, R, is not required to generate an ephemeral key pair as part of the keyagreement scheme, then R shall contribute a random nonce to be used (in addition to theshared secret Z) as input to the key-derivation method employed by the scheme; see Section
5.4 for a discussion of the length and security strength required for the nonce.
2. The provider, P, computes
MacDataP = message_stringP || IDP || IDR || EphemDataP || EphemDataR {|| TextP}
where- message_stringP is a six byte string with a value of “KC_1_U” when party U isproviding the MacTag, or “KC_1_V” when party V is providing the MacTag. (Notethat these values will be changed for bilateral key-confirmation, as specified in
Section 5.9.2.)- IDP is the identifier used to label the key-confirmation provider.
- IDR is the identifier used to label the key-confirmation recipient.
- EphemDataP and EphemDataR are ephemeral values (corresponding to ephemeral
public keys or nonces) contributed by the provider and recipient, respectively. The
particular type of ephemeral data is specified in the subsections of Section 6 that
describe how key-confirmation can be incorporated into the particular schemes
included in this Recommendation. EphemDataP is null only in the case that theprovider has contributed neither an ephemeral public key nor a nonce during thescheme. For example, in a C(1e, 2s) scheme with unilateral key-confirmation fromparty V to party U as introduced in Section 6.2.1.5.2, party V only contributes a static
key pair; in this case, EphemDataV can be null.
- TextP is an optional bit string that may be used during key-confirmation and that is
known by both parties.
The content of each of the components that are concatenated to form MacDataP shall be
precisely defined and unambiguously represented. A particular component’s content may
be represented, for example, as a fixed-length bit string or in the form Datalen || Data,
where Data is a variable-length string of zero or more (eight-bit) bytes, and Datalen is a
fixed-length, big-endian counter that indicates the length (in bytes) of Data. These
requirements could also be satisfied by using a specific ASN.1 DER encoding of eachcomponent. It is imperative that the provider and recipient have agreed upon on thecontent and format that will be used for each component of MacDataP.
3. After computing the shared secret Z and applying the key-derivation method to obtain
DerivedKeyingMaterial (see Section 5.8), the provider uses agreed-upon bit lengths toparse DerivedKeyingMaterial into two parts, MacKey and KeyData, in the pre-agreed
lengths:
MacKey || KeyData = DerivedKeyingMaterial.
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4. Using an agreed-upon bit length MacLen, the provider computes MacTagP (see Sections
5.2.1 and 5.9.3):
MacTagP = TMacLen[MAC (MacKey, MacDataP)],and sends it to the recipient.
5. The recipient forms MacDataP, determines MacKey, computes MacTagP in the samemanner as the provider, and then compares its computed MacTagP to the value received
from the provider. If the received value is equal to the derived value, then the recipient isassured that the provider has derived the same value for MacKey and that the provider
shares the recipient’s value of MacTagP. The assurance of a shared value for MacKey
provides assurance to the recipient that the provider also shares the secret value (Z) fromwhich MacKey and KeyData are derived. Thus, the recipient also has assurance that theprovider could compute KeyData correctly.
Both parties shall destroy the MacKey once it is no longer needed to provide or obtain keyconfirmation.
If, during a particular key-agreement transaction, it happens that MacTagP cannot be verified by
the recipient, then key-confirmation has failed and all of the derived keying material (MacKey
and KeyData) shall be destroyed by each participant. In particular, DerivedKeyingMaterial shallnot be revealed by either participant to any other party (not even to the other participant), and thederived keying material shall not be used for any further purpose. In the case of a keyconfirmation failure, the key-agreement transaction shall be discontinued.
Unilateral key-confirmation may be added in either direction to any of the C(2e, 2s), C(1e, 2s)
and C(0e, 2s) schemes; it may also be added to the C(1e, 1s) schemes, but only when party V(the party contributing the static key pair) is the key-confirmation provider, and party U is the
key-confirmation recipient. (See the relevant subsections of Section 6.)
5.9.2 Bilateral Key-Confirmation for Key-Agreement Schemes
Bilateral key-confirmation is an optional feature that can be incorporated into any key-agreement
scheme in which each party is required to own a static key-establishment key pair that is used in
the key-establishment process. Bilateral key-confirmation is accomplished by performing
unilateral key-confirmation in both directions (with party U providing MacTagU to recipient
party V, and party V providing MacTagV to recipient party U) during the same scheme. If a party
is not also required to contribute an ephemeral public key to the key-establishment process, thenthat party shall instead contribute a random nonce that is used as part of the input to the keyderivation method employed by the scheme; see Section 5.4 for a discussion of the length andsecurity strength required for the nonce. Each party is required to have an identifier, chosen inaccordance with the assumptions stated for the key-agreement scheme.
5.9.2.1 Adding Bilateral Key-Confirmation to a Key-Agreement Scheme
To include bilateral key-confirmation, two instances of unilateral key-confirmation (as specified
in Section 5.9.1.1, subject to the modifications listed below) shall be incorporated into thescheme, once with party U as the key-confirmation provider (i.e., P = U and R = V) and once

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with party V as the provider (i.e., P = V and R = U). Additional details will be provided for eachscheme in the appropriate subsections of Section 6.
In addition to setting P = U and R = V in one instance of the unilateral key-confirmation
procedure described in Section 5.9.1.1 and setting P = V and R = U in a second instance, thefollowing changes/clarifications apply when using the procedure for bilateral key-confirmation:
1. When computing MacTagU, the value of the six-byte message_stringU that forms theinitial segment of MacDataU is “KC_2_U”.
2. When computing MacTagV, the value of the six-byte message_stringV that forms theinitial segment of MacDataV is “KC_2_V”.
3. If used at all, the value of the (optional) byte string TextU used to form the final segmentof MacDataU can be different than the value of the (optional) byte string TextV used toform the final segment of MacDataV, provided that both parties are aware of the value(s)used.
Bilateral key-confirmation may be added to the C(2e, 2s), C(1e, 2s) and C(0e, 2s) schemes, as
specified in the relevant subsections of Section 6.
5.9.3 Security Strength of the Mac Tag
In this Recommendation, a MagTag used for key-confirmation shall be generated using anapproved MAC algorithm, which can be an HMAC [FIPS 198] with an approved hash function
or a CMAC [SP 800-38B] with an approved block cipher (see Section 5.2). The domain
parameter set shall be used to determine the minimum length of the key (MacKey length) used tocompute the MacTag and the minimum length of the MacTag, the (possibly truncated) result
from the MAC computation using the MAC algorithm with MacKey, as shown in Table 8 andTable 9 for the FFC and ECC domain parameter sets, respectively.
Table 8: Minimum MacKey length and MacLen for FFC domain parameter sets
FFC parameter set name

FA

FB

FC

Maximum security strength supported (in bits)

80

112

112

Bit length of the field size p (i.e., log 2 p  )

1024

2048

2048

Bit length of the subgroup order q (i.e., log 2 q  )

160

224

256

Minimum MacKey length (in bits)

80

112

112

Minimum MacLen, i.e., the MacTag length (in bits) 9

64

64

64

9

The minimum MacTag length is specified for key-confirmation in this Recommendation. For other applications,please refer to the relevant specifications for the minimum length required for message authentication code.

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HMAC using any approved hash function or AES CMAC with any approved AES key length(128, 192 or 256 bits) can be used with all the FFC domain parameter sets described in Table 8.
Table 9: Minimum MacKey length and MacLen for ECC domain parameter sets
ECC parameter set name

EA

EB

EC

ED

128

192

EE

Maximum security strength supported (in bits)

80

112

256

Bit length of ECC the subgroup order n (i.e.,
log 2 n  )160 to223

224 to 256 to 384 to 512 or
255
383
511
longer

Minimum MacKey length (in bits)

80

112

128

192

256

Minimum MacLen, i.e., the MacTag length (inbits) 10

64

64

64

64

64

As shown in Table 9, when an HMAC is used for key-confirmation for the ECC schemes, any
approved hash function can be used for the ECC domain parameter sets EA, EB, and EC;
however, HMAC with the SHA-1 hash function shall not be used for domain parameter sets ED
and EE, because the maximum length of the MacTag that an HMAC with SHA-1 can generate is
only 160 bits. For the same reason, an HMAC with SHA-224 and SHA-512/224 shall not be
used for key-confirmation with ECC parameter set EE. AES CMAC can be used for keyconfirmation with any AES key length (128, 192, 256 bits) for the ECC domain parameter sets
EA, EB, and EC; however, AES CMAC shall not be used for domain parameter sets ED and EE,
because the maximum length of the MacTag that an AES CMAC can generate is only 128 bits,
the AES output block length.

6.

Key-Agreement

This Recommendation provides three categories of key-agreement schemes (see Table 10). The
classification of the categories is based on the number of ephemeral keys used by the two parties
to the key-agreement process, parties U and V. In category C(ie), parties U and V have a total ofi ephemeral key pairs. The first category, C(2e), consists of schemes requiring the generation ofephemeral key pairs by both parties; a C(2e) scheme is suitable for an interactive key
establishment protocol. The second category, C(1e), consists of schemes requiring the generationof an ephemeral key pair by only one party; a C(1e) scheme is suitable for a store and forward
scenario, but may also be used in an interactive key establishment protocol. The third category,C(0e), consists of schemes that do not use ephemeral keys.
Key-confirmation may be added to many of these schemes to provide assurance that theparticipants share the same keying material; see Section 5.9 for details on key-confirmation.
10

The minimum MacTag length is specified for key-confirmation in this Recommendation. For other applications,please refer to the relevant specifications for the minimum length required for message authentication code.

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Each party should have such assurance. Although other methods are often used to provide thisassurance, this Recommendation makes no statement as to the adequacy of these other methods.
Table 10: Key-agreement scheme categories
Category

Comment

C(2e): Two ephemeral key pairs

Each party generates an ephemeral key pair.

C(1e): One ephemeral key pair

Only party U generates an ephemeral key pair.

C(0e): Zero ephemeral key pairs

No ephemeral keys are used.

Each category is comprised of one or more subcategories that are classified by the use of static
keys by the parties (see Table 11). In subcategory C(ie, js), parties U and V have a total of iephemeral key pairs and j static key pairs. The suitability for interactive or store-and-forward
protocols of each subcategory is discussed in Section 8.
Table 11: Key-agreement scheme subcategories
Category

SubcategoryC(2e): Two ephemeral key
pairs

C(2e, 2s): Each party generates an ephemeral key pair and uses astatic key pair.
C(2e, 0s): Each party generates an ephemeral key pair; no static
key pairs are used.

C(1e): One ephemeral key
pair

C(1e, 2s): Party U generates an ephemeral key pair and uses astatic key pair; party V uses only a static key pair.
C(1e, 1s): Party U generates an ephemeral key pair, but uses nostatic key pair; party V uses only a static key pair.

C(0e): Zero ephemeral key
pairs

C(0e, 2s): Each party uses only a static key pair.

The schemes may be further classified by whether they use finite field cryptography (FFC) or
elliptic curve cryptography (ECC). A scheme may use either Diffie-Hellman or MQV primitives(see Section 5.7). Thus, for example, notation C(2e, 2s, FFC DH) completely classifies thedhHybrid1 scheme of Section 6.1.1.1 as a scheme with two ephemeral keys and two static keys
that uses finite field cryptography and a Diffie-Hellman primitive (see Table 12). The names ofthese schemes are taken from [ANS X9.42] and [ANS X9.63].
Table 12: Key-agreement schemes
Category

Subcategory

Primitive

C(2e)C(2e, 2s)

FFC DH

SchemedhHybrid1

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NotationC(2e, 2s, FFC DH)NIST SP 56A: Revision 2, May 2013
Category

Subcategory

Primitive

SchemeNotationC(2e)C(2e, 2s)

ECC CDH(Cofactor) Full Unified ModelC(2e, 2s, ECC
CDH)

C(2e)

C(2e, 2s)

FFC MQV

MQV2C(2e, 2s, FFC
MQV)

C(2e)

C(2e, 2s)

ECC MQV Full MQV

C(2e, 2s, ECC
MQV)

C(2e)

C(2e, 0s)

FFC DHdhEphemC(2e, 0s, FFC DH)

C(2e)C(2e, 0s)

ECC CDH(Cofactor) Ephemeral Unified C(2e, 0s, ECC
Model
CDH)

C(1e)

C(1e, 2s)

FFC DHdhHybridOneFlow

C(1e)

C(1e, 2s)

ECC CDH(Cofactor) One-Pass Unified C(1e, 2s, ECC
Model
CDH)

C(1e)

C(1e, 2s)

FFC MQV

MQV1

C(1e)

C(1e, 2s)

ECC MQV One-Pass MQV

C(1e, 2s, ECC
MQV)

C(1e)

C(1e, 1s)

FFC DH

dhOneFlowC(1e, 1s, FFC DH)

C(1e)C(1e, 1s)

ECC CDH(Cofactor) One-Pass Diffie- C(1e, 1s, ECC
Hellman
CDH)

C(0e)

C(0e, 2s)

FFC DHdhStatic

C(0e)C(0e, 2s)

ECC CDH(Cofactor)
Model

C(1e, 2s, FFC DH)

C(1e, 2s, FFC
MQV)

C(0e, 2s, FFC DH)
StaticUnified C(0e, 2s, ECC
CDH)

Each party in a key-agreement process shall use the same set of valid domain parameters. These
parameters shall be established, and assurance of their validity shall be obtained prior to thegeneration of key pairs and the initiation of the key-agreement process. See Section 5.5 for a
discussion of domain parameters.
If party U uses a static key pair in a given key-agreement transaction, then party U shall have an
identifier, IDU, that has an association with the static key pair that is known (or discoverable) andtrusted by party V (i.e., there shall be a trusted association between IDU and party U’s static
public key). If party U does not contribute a static public key as part of a key-agreementtransaction, then IDU (if required for that transaction) is a non-null identifier selected inaccordance with the relying application/protocol. Similar rules apply to Party V’s identifier, IDV.
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A general flow diagram is provided for each subcategory of schemes. The dotted-line arrows
represent the distribution of static public keys that may be distributed by the parties themselves
or by a third party, such as a Certification Authority (CA). The solid-line arrows represent the
distribution of ephemeral public keys or nonces that occur during the key-agreement or keyconfirmation process. Note that the flow diagrams in this Recommendation omit explicit mentionof various validation checks that are required. The flow diagrams and descriptions in thisRecommendation assume a successful completion of the key-establishment process. The error
conditions are handled in the process text.
For each scheme, there are conditions that must be satisfied to enable proper use of that scheme.
These conditions are listed as the assumptions. Failure to meet all such conditions could yield
undesirable results, such as the inability to communicate or the loss of security. As part of theproper implementation of this Recommendation, system users and/or agents trusted to act on
their behalf (including application developers, system installers, and system administrators) areresponsible for ensuring that all assumptions are satisfied at the time a key-establishmenttransaction takes place.
6.1

Schemes Using Two Ephemeral Key Pairs, C(2e)In this category, each party generates an ephemeral key pair and sends the ephemeral public key
to the other party. This category consists of two subcategories that are determined by the use ofstatic keys by the parties. In the first subcategory, each party contributes both static andephemeral keys (see Section 6.1.1), while in the second subcategory, each party contributes only
ephemeral keys (see Section 6.1.2).
6.1.1 C(2e, 2s) Schemes
Figure 4 depicts a typical flow for a C(2e, 2s). For these schemes, each party (U and V)
contributes a static key pair and generates an ephemeral key pair during the key-agreement
process. All key pairs shall be generated using the same domain parameters. Party U and party V
obtain each other’s static public keys, which have been generated prior to the key-establishment
process. Both parties generate ephemeral private/public key pairs and exchange the ephemeral
public keys. Using the static and ephemeral keys, both parties generate a shared secret. The
secret keying material is derived from the shared secret.

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U’s Static Public Key
V’s Static Public Key

Obtain static publickeys

U’s Ephemeral Public Key
V’s Ephemeral Public Key

Exchangeephemeralpublic keys

Figure 4: C(2e, 2s) schemes: each party contributes a static and an ephemeral key pair

Assumptions: In order to execute a C(2e, 2s) key-establishment scheme in compliance with thisRecommendation, the following assumptions shall be true.
1. Each party has an authentic copy of the same set of domain parameters, D, that have been
generated as specified in Section 5.5.1. For FFC schemes, D = (p, q, g{, SEED,
pgenCounter}); for ECC schemes, D = (q, FR, a, b{, SEED}, G, n, h). Furthermore, eachparty has obtained assurance of the validity of these domain parameters as specified in
Section 5.5.2.
2. Each party has been designated as the owner of a static key pair that was generated as
specified in Section 5.6.1 using the set of domain parameters, D. For FFC schemes, thestatic key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Each party has
obtained assurance of the validity of its own static public key as specified in Section
5.6.2.1.3 and has obtained assurance of its possession of the correct value for its own
private key as specified in Section 5.6.2.1.5.
3. The parties have agreed upon an approved key-derivation method, as well as an
approved algorithm to be used with that method (e.g., a hash function) and otherassociated parameters to be used for key derivation (see Section 5.8).
4. If key confirmation is used, the parties have also agreed upon an approved MAC andassociated parameters, including the lengths of MacKey and MacTag, as specified in Step
3 and Step 4 in Section 5.9.1.1 (for the minimum lengths of MacKey and MacTag, see
Table 8 and Table 9 in Section 5.9.3).
5. Prior to or during the key-agreement process, each party receives the other party’s static
public key in a trusted manner (e.g., from a certificate signed by a trusted CA or directly

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from the other party, who is trusted by the recipient). Each party has obtained assuranceof the validity of the other party’s static public key as specified in Section 5.6.2.2.
6. The recipient of a static public key has obtained assurance that its (claimed) owner is (orwas) in possession of the corresponding static private key, as specified in Section
5.6.2.2.3.
7. When an identifier is used to label a party during the key-agreement process, that
identifier has a trusted association to that party’s static public key. (In other words,
whenever both the identifier and static public key of one participant are employed in the
key-agreement process, they are associated in a manner that is trusted by the otherparticipant.) When an identifier is used to label a party during the key-agreement process,both parties are aware of the particular identifier employed for that purpose.
6.1.1.1 dhHybrid1, C(2e, 2s, FFC DH) Scheme
This section describes the dhHybrid1 scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.1.1 are true. In particular, it isassumed that party U has obtained the static public key yV of party V, and party V has obtained
the static public key yU of party U.
With the exception of key derivation, dhHybrid1 is “symmetric” in the actions of parties U and
V. Only the actions performed by party U are specified here; a specification of the actions
performed by party V may be obtained by systematically replacing the letter “U” by “V” (andvice versa) in the description of the key-agreement transformation. Note, however, that parties U
and V must use identical orderings of the bit strings that are input to the key-derivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in
Section 5.6.1.1. Send the public key tU to party V. Receive an ephemeral public key tV(purportedly) from party V. If tV is not received, destroy the ephemeral private key rU,
and return an error indicator without performing the remaining actions.
2. Verify that tV is a valid public key for the parameters D as specified in Section 5.6.2.3. If
assurance of public key validity cannot be obtained, destroy the ephemeral private key rU,
and return an error indicator without performing the remaining actions.
3. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Zs from the set of
domain parameters D, party U’s static private key xU, and party V’s static public key yV.
If the call to the FFC DH primitive outputs an error indicator, destroy the ephemeral
private key rU, destroy the results of all intermediate calculations used in the attempted
computation of Zs, and return an error indicator without performing the remaining
actions.
4. Use the FCC DH primitive to derive a shared secret Ze from the set of domain parametersD, party U’s ephemeral private key rU, and party V’s ephemeral public key tV. If this callto the FFC DH primitive outputs an error indicator, destroy Zs and the ephemeral private
key rU,, destroy the results of all intermediate calculations used in the attempted
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computation of Ze, and return an error indicator without performing the remaining
actions.
5. Compute the shared secret Z = Ze || Zs. Destroy Ze and Zs.
6. Use the agreed-upon key derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key rU, and return an error indicator without performing the remaining
actions.
7. If the ephemeral private key rU will not be used in a broadcast scenario (see Section 7) for
subsequent key-establishment transactions using this scheme, then destroy rU.
8. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note 1: Key-confirmation can be incorporated into this scheme. See Section 6.1.1.5 for details.
Note 2: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(rU, tU) may be used in other key-establishment transactions occurring during the same broadcast(i.e., step 1 above would not be repeated). After the final broadcast transaction, the ephemeral
private key rU shall be destroyed (see step 7 above).
dhHybrid1 is summarized in Table 13.

Table 13: dhHybrid1 key-agreement scheme summary
Party U

Party V

Domainparameters

D = (p, q, g{, SEED, pgenCounter})

D = (p, q, g{, SEED, pgenCounter})

Static Data

Static private key xU

Static private key xV

Static public key yU

Static public key yV

Ephemeral Data Ephemeral private key rU

Ephemeral private key rV

Ephemeral public key tU

Ephemeral public key tVComputation

1. Compute Zs by calling FFC DH 1. Compute Zs by calling FFC DH
using xU and yV
using xV and yU
2. Compute Ze by calling FFC DH 2. Compute Ze by calling FFC DH
using rU and tVusing rV and tU
3. Compute Z = Ze || Zs

3. Compute Z = Ze || Zs

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Party U

Party V

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
Derive SecretKeying Material
2. Destroy Z
2. Destroy Z
6.1.1.2 (Cofactor) Full Unified Model, C(2e, 2s, ECC CDH) Scheme
This section describes the Full Unified Model scheme. Assurance of secure key establishment
using this scheme can only be obtained when the assumptions in Section 6.1.1 are true. In
particular, it is assumed that party U has obtained the static public key Qs,V of party V, and party
V has obtained the static public key Qs,U of party U.
With the exception of key derivation, the Full Unified Model is “symmetric” in the actions ofparties U and V. Only the actions performed by party U are specified here; a specification of theactions performed by party V may be obtained by systematically replacing the letter “U” by “V”(and vice versa) in the description of the key-agreement transformation. Note, however, thatparties U and V must use identical orderings of the bit strings that are input to the key-derivation
function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (de,U, Qe,U) from the domain parameters D as specified in
Section 5.6.1.2. Send the public key Qe,U to party V. Receive an ephemeral public key
Qe,V (purportedly) from party V. If Qe,V is not received, destroy the ephemeral private key
de,U, and return an error indicator without performing the remaining actions.
2. Verify that Qe,V is a valid public key for the parameters D as specified in Section 5.6.2.3.
If assurance of public key validity cannot be obtained, destroy the ephemeral private key
de,U, and return an error indicator without performing the remaining actions.
3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Zs from the set of
domain parameters D, party U’s static private key ds,U, and party V’s static public key
Qs,V. If the call to the ECC CDH primitive outputs an error indicator, destroy theephemeral private key de,U, destroy the results of all intermediate calculations used in theattempted computation of Zs, and return an error indicator without performing theremaining actions.
4. Use the ECC CDH primitive to derive a shared secret Ze from the set of domain
parameters D, party U’s ephemeral private key de,U, and party V’s ephemeral public key
Qe,V. If this call to the ECC CDH primitive outputs an error indicator, destroy Zs and theephemeral private key de,U, destroy the results of all intermediate calculations used in theattempted computation of Ze, and return an error indicator without performing theremaining actions.
5. Compute the shared secret Z = Ze || Zs. Destroy Ze and Zs.
6. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the63

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key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key de,U, and return an error indicator without performing theremaining actions.
7. If the ephemeral private key de,U will not be used in a broadcast scenario (see Section 7)
for subsequent key-establishment transactions using this scheme, then destroy de,U.
8. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note 1: Key-confirmation can be incorporated into this scheme. See Section 6.1.1.5 for details.
Note 2: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(de,U, Qe,U) may be used in other key-establishment transactions occurring during the samebroadcast (i.e., step 1 above would not be repeated). After the final broadcast transaction, theephemeral private key de,U shall be destroyed (see step 7 above).
The Full Unified Model is summarized in Table 14.
Table 14: Full unified model key-agreement scheme summary
Party U

Party V

Domainparameters

D = (q, FR, a, b{, SEED}, G, n, h)

D = (q, FR, a, b{, SEED}, G, n, h)

Static data

Static private key ds,U

Static private key ds,VStatic public key Qs,U

Static public key Qs,V

Ephemeral private key de,U

Ephemeral private key de,V

Ephemeral public key Qe,U

Ephemeral public key Qe,V

Ephemeral data

Computation

1. Compute Zs by calling ECC 1. Compute Zs by calling ECC
CDH using ds,U and Qs,V
CDH using ds,V and Qs,U
2. Compute Ze by calling ECC 2. Compute Ze by calling ECC
CDH using de,U and Qe,V
CDH using de,V and Qe,U
3. Compute Z = Ze || Zs

Derive secretkeying material

3. Compute Z = Ze || Zs

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.1.1.3 MQV2, C(2e, 2s, FFC MQV) Scheme
This section describes the MQV2 scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.1.1 are true. In particular, it is64

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assumed that party U has obtained the static public key yV of party V, and party V has obtained
the static public key yU of party U.
With the exception of key derivation, MQV2 is “symmetric” in the actions of parties U and V.
Only the actions performed by party U are specified here; a specification of the actions
performed by party V may be obtained by systematically replacing the letter “U” by “V” (andvice versa) in the description of the key-agreement transformation. Note, however, that parties U
and V must use identical orderings of the bit strings that are input to the key-derivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in
Section 5.6.1.1. Send the public key tU to party V. Receive an ephemeral public key tV(purportedly) from party V. If tV is not received, destroy the ephemeral private key rU,
and return an error indicator without performing the remaining actions.
2. Verify that tV is a valid public key for the parameters D as specified in Section 5.6.2.3. If
assurance of public key validity cannot be obtained, destroy the ephemeral private key rU,
and return an error indicator without performing the remaining actions.
3. Use the MQV2 form of the FFC MQV primitive in Section 5.7.2.1 to derive a sharedsecret Z from the set of domain parameters D, party U’s static private key xU, party V’sstatic public key yV, party U’s ephemeral private key rU, party U’s ephemeral public key
tU, and V’s ephemeral public key tV. If the call to the FFC MQV primitive outputs anerror indicator, destroy the ephemeral private key rU, destroy the results of allintermediate calculations used in the attempted computation of Z, and return an error
indicator without performing the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key rU, and return an error indicator without performing the remaining
actions.
5. If the ephemeral private key rU will not be used in a broadcast scenario (see Section 7) for
subsequent key-establishment transactions using this scheme, then destroy rU.
6. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note 1: Key-confirmation can be incorporated into this scheme. See Section 6.1.1.5 for details.
Note 2: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(rU, tU) may be used in other key-establishment transactions occurring during the same broadcast(i.e., step 1 above would not be repeated). After the final broadcast transaction, the ephemeral
private key rU shall be destroyed (see step 5 above).
MQV2 is summarized in Table 15.

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Table 15: MQV2 key-agreement scheme summary
Party U

Party V

Domainparameters

D =(p, q, g{, SEED, pgenCounter})D =(p, q, g{, SEED, pgenCounter})

Static data

Static private key xU

Static private key xV

Static public key yU

Static public key yV

Ephemeral private key rU

Ephemeral private key rV

Ephemeral public key tU

Ephemeral public key tV

Ephemeral data

Computation

Compute Z by calling FFC MQV Compute Z by calling FFC MQVusing xU, yV, rU, tU, and tVusing xV, yU, rV, tV, and tU

Derive secretkeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.1.1.4 Full MQV, C(2e, 2s, ECC MQV) Scheme
This section describes the Full MQV scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.1.1 are true. In particular, it isassumed that party U has obtained the static public key Qs,V of party V, and party V has obtained
the static public key Qs,U of party U.
With the exception of key derivation, the Full MQV scheme is “symmetric” in the actions ofparties U and V. Only the actions performed by party U are specified here; a specification of theactions performed by party V may be obtained by systematically replacing the letter “U” by “V”(and vice versa) in the description of the key-agreement transformation. Note, however, thatparties U and V must use identical orderings of the bit strings that are input to the key derivation
function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (de,U, Qe,U) from the domain parameters D as specified in
Section 5.6.1.2. Send the public key Qe,U to party V. Receive an ephemeral public key
Qe,V (purportedly) from party V. If Qe,V is not received, destroy the ephemeral private key
de,U, and return an error indicator without performing the remaining actions.
2. Verify that Qe,V is a valid public key for the parameters D as specified in Section 5.6.2.3.
If assurance of public key validity cannot be obtained, destroy the ephemeral private key
de,U, and return an error indicator without performing the remaining actions.
3. Use the Full MQV form of the ECC MQV primitive in Section 5.7.2.3.1 to derive ashared secret value Z from the set of domain parameters D, party U’s static private key
ds,U, party V’s static public key Qs,V, party U’s ephemeral private key de,U, party U’s
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ephemeral public key Qe,U, and party V’s ephemeral public key Qe,V. If the call to the
ECC MQV primitive outputs an error indicator, destroy the ephemeral private key de,U,destroy the results of all intermediate calculations used in the attempted computation ofZ, and return an error indicator without performing the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key de,U, and return an error indicator without performing theremaining actions.
5. If the ephemeral private key de,U will not be used in a broadcast scenario (see Section 7)
for subsequent key-establishment transactions using this scheme, then destroy de,U.
6. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note 1: Key-confirmation can be incorporated into this scheme. See Section 6.1.1.5 for details.
Note 2: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(de,U, Qe,U)may be used in other key-establishment transactions occurring during the samebroadcast (i.e., step 1 above would not be repeated). After the final broadcast transaction, theephemeral private key de,U shall be destroyed (see step 5 above).
The Full MQV is summarized in Table 16.
Table 16: Full MQV key-agreement Scheme Summary
Party U

Party V

Domainparameters

D = (q, FR, a, b{, SEED}, G, n, h)

D = (q, FR, a, b{, SEED}, G, n, h)

Static data

1. Static private key ds,U

1. Static private key ds,V

2. Static public key Qs,U

2. Static public key Qs,V

1. Ephemeral private key de,U

1. Ephemeral private key de,V

2. Ephemeral public key Qe,U

2. Ephemeral public key Qe,V

Ephemeral data

Computation

Compute Z by calling ECC MQV Compute Z by calling ECC MQVusing ds,U, Qs,V, de,U, Qe,U, and Qe,V
using ds,V, Qs,U, de,V, Qe,V, and Qe,U

Derive secretkeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

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6.1.1.5 Incorporating Key-Confirmation into a C(2e, 2s) Scheme
The subsections that follow illustrate how to incorporate key-confirmation (as described in
Section 5.9) into the C(2e, 2s) key-agreement schemes described above.
The flow depictions separate the key-establishment flow from the key-confirmation flow. The
depictions and accompanying discussions presume that the assumptions of the scheme have beensatisfied, that the key-agreement transaction has proceeded successfully through key derivation,and that the received MacTags are successfully verified as specified in Section 5.2.2.
6.1.1.5.1

C(2e, 2s) Scheme with Unilateral Key-Confirmation Provided by party U toparty V
Figure 5 depicts a typical flow for a C(2e, 2s) scheme with unilateral key-confirmation fromparty U to party V. In this scenario, party U and party V assume the roles of key-confirmation
provider and recipient, respectively. The successful completion of this process provides party V
with a) assurance that party U has derived the same secret Z value; and b) assurance that party U
has actively participated in the process.

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U’s Static Public Key
V’s Static Public Key

Obtain static
public keys

U’s Ephemeral Public Key
V’s Ephemeral Public Key
MacTagU

Exchangeephemeralpublic keys

U’s keyconfirmation

Figure 5: C(2e, 2s) scheme with unilateral key-confirmation from party U to party V
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), party U (and partyV) setEphemDataU = EphemPubKeyU, and EphemDataV = EphemPubKeyV.
Party U provides MacTagU to party V (as specified in Section 5.9.1.1, with P = U and R = V),
where MacTagU is computed (as specified in Section 5.2.1) using
MacDataU = “KC_1_U” || IDU || IDV || EphemPubKeyU || EphemPubKeyV {|| TextU}.
Party V (the key-confirmation recipient) uses the same format for MacDataU to compute its own
version of MacTagU, and then verifies that the newly computed MacTagU matches the value
provided by party U.
6.1.1.5.2C(2e, 2s) Scheme with Unilateral Key-Confirmation Provided by party V toparty U
Figure 6 depicts a typical flow for a C(2e, 2s) scheme with unilateral key-confirmation fromparty V to party U. In this scenario, party V and party U assume the roles of key-confirmation
provider and recipient, respectively. The successful completion of the key-confirmation processprovides party U with a) assurance that party V has derived the same secret Z value; and b)
assurance that party V has actively participated in the process.

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U’s Static Public Key
V’s Static Public Key

U’s Ephemeral Public Key

V’s Ephemeral Public Key, MacTagVObtain static publickeys

U sends anephemeralpublic key
V sends anephemeral publickey with key
confirmation

Figure 6: C(2e, 2s) scheme with unilateral key-confirmation from party V to party U
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), party V (and partyU) setEphemDataV = EphemPubKeyV, and EphemDataU = EphemPubKeyU.
Party V provides MacTagV to party U (as specified in Section 5.9.1.1, with P = V and R = U),
where MacTagV is computed (as specified in Section 5.2.1) using
MacDataV = “KC_1_V” || IDV || IDU || EphemPubKeyV || EphemPubKeyU {|| TextV}.
Party U (the key-confirmation recipient) uses the same format for MacDataV to compute its own
version of MacTagV and then verifies that the newly computed MacTagV matches the value
provided by party V.
Note that in Figure 6, party V’s ephemeral public key (EphemPubKeyV) and the MacTag(MacTagV) are depicted as being sent in the same message (to reduce the number of passes in the
combined key-agreement/key-confirmation process). They may also be sent separately.
6.1.1.5.3 C(2e, 2s) Scheme with Bilateral Key-Confirmation
Figure 7 depicts a typical flow for a C(2e, 2s) scheme with bilateral key-confirmation. In thismethod, party U and party V assume the roles of both the provider and the recipient in order toobtain bilateral key-confirmation. The successful completion of the key-confirmation processprovides each party with a) assurance that the other party has derived the same secret Z value,and b) assurance that the other party has actively participated in the process.

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U’s Static Public Key
V’s Static Public Key

Obtain static publickeys

U’s Ephemeral Public Key

U sends an ephemeral
public key

V’s Ephemeral Public Key, MacTagV

V sends an ephemeral
public key with key
confirmation

MacTagU

U’s key
confirmation

Figure 7: C(2e, 2s) scheme with bilateral key-confirmation
To provide bilateral key-confirmation (as described in Section 5.9.2.1), party U and party V
exchange and verify MacTags that have been computed (as specified in Section 5.2.1) using
EphemDataU = EphemPubKeyU, and EphemDataV = EphemPubKeyV.
Party V provides MacTagV to party U (as specified in Sections 5.9.1.1 and 5.9.2.1, with P = V
and
R = U); MacTagV is computed by party V (and verified by party U) using
MacDataV = “KC_2_V” || IDV || IDU || EphemPubKeyV || EphemPubKeyU {|| TextV}.
Party U provides MacTagU to party V (as specified in Sections 5.9.1.1 and 5.9.2.1, with P = U
and
R = V); MacTagU is computed by party U (and verified by party V) using
MacDataU = “KC_2_U” || IDU || IDV || EphemPubKeyU || EphemPubKeyV {|| TextU}.
Note that in Figure 7, party V’s ephemeral public key (EphemPubKeyV) and the MacTag(MacTagV) are depicted as being sent in the same message (to reduce the number of passes in the
combined key-agreement/key-confirmation process). They may also be sent separately, and if
sent separately, then the order in which the MacTags are sent could be reversed.
6.1.2 C(2e, 0s) Schemes
For this category, only Diffie-Hellman schemes are specified. Each party generates ephemeral
key pairs with the same domain parameters. The two parties exchange ephemeral public keys

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and then compute the shared secret. The secret keying material is derived using the shared secret(see Figure 8).

UV
U’s Ephemeral Public Key
V’s Ephemeral Public Key

Exchangeephemeralpublic keys

Figure 8: C(2e, 0s) schemes: each party contributes only an ephemeral key pair
Assumptions: In order to execute a C(2e, 0s) key-establishment scheme in compliance with thisRecommendation, the following assumptions shall be true.
1. Each party has an authentic copy of the same set of domain parameters, D. These
parameters have been generated as specified in Section 5.5.1. For FFC schemes, D = (p,q, g{, SEED, pgenCounter}); for ECC schemes, D = (q, FR, a, b{, SEED}, G, n, h).
Furthermore, each party has obtained assurance of the validity of these domain
parameters as specified in Section 5.5.2.
2. The parties have agreed upon an approved key-derivation method, as well as an
approved algorithm to be used with that method (e.g., a hash function) and otherassociated parameters to be used (see Section 5.8).
3. When an identifier is used to label a party during the key-agreement process, it has been
selected/assigned in accordance with the requirements of the protocol relying upon theuse of the key-agreement scheme, and its value is known to both parties.
6.1.2.1 dhEphem, C(2e, 0s, FFC DH) Scheme
This section describes the dhEphem scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.1.2 are true.
With the exception of key derivation, dhEphem is “symmetric” in the actions of parties U and V.
Only the actions performed by party U are specified here; a specification of the actions
performed by party V may be obtained by systematically replacing the letter “U” by “V” (andvice versa) in the description of the key-agreement transformation. Note, however, that parties U
and V must use identical orderings of the bit strings that are input to the key-derivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
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1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in
Section 5.6.1.1. Send the public key tU to party V. Receive an ephemeral public key tV(purportedly) from party V. If tV is not received, destroy the ephemeral private key rU,
and return an error indicator without performing the remaining actions.
2. Verify that tV is a valid public key for the parameters D as specified in Section 5.6.2.3. If
assurance of public key validity cannot be obtained, destroy the ephemeral key rU, andreturn an error indicator without performing the remaining actions.
3. Use the FCC DH primitive in Section 5.7.1.1 to derive a shared secret Z from the set of
domain parameters D, party U’s ephemeral private key rU, and party V’s ephemeral
public key tV. Then destroy the ephemeral private key rU. If the call to the FFC DH
primitive outputs an error indicator, destroy the results of all intermediate calculations
used in the attempted computation of Z, and return an error indicator without performing
the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
dhEphem is summarized in Table 17.
Table 17: dhEphem key-agreement scheme summary
Party U

Party V

Domainparameters(p, q, g{, SEED, pgenCounter})(p, q, g{, SEED, pgenCounter})

Static data

N/A

N/A

Ephemeral data

Ephemeral private key rU

Ephemeral private key rV

Ephemeral public key tU

Ephemeral public key tVComputation

Compute Z by calling FFC DH using Compute Z by calling FFC DH usingrV and tUrU and tV

Derive secretkeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

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6.1.2.2 (Cofactor) Ephemeral Unified Model, C(2e, 0s, ECC CDH)
This section describes the Ephemeral Unified Model scheme. Assurance of secure key
establishment using this scheme can only be obtained when the assumptions in Section 6.1.2 aretrue.
With the exception of key derivation, Ephemeral Unified Model is “symmetric” in the actions ofparties U and V. Only the actions performed by party U are specified here; a specification of theactions performed by party V may be obtained by systematically replacing the letter “U” by “V”(and vice versa) in the description of the key-agreement transformation. Note, however, thatparties U and V must use identical orderings of the bit strings that are input to the key derivation
function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (de,U, Qe,U) from the domain parameters D as specified in
Section 5.6.1.2. Send the public key Qe,U to party V. Receive an ephemeral public key
Qe,V (purportedly) from party V. If Qe,V is not received, destroy the ephemeral private key
de,U, and return an error indicator without performing the remaining actions.
2. Verify that Qe,V is a valid public key for the parameters D as specified in Section 5.6.2.3.
If assurance of public key validity cannot be obtained, destroy the ephemeral private key
de,U, and return an error indicator without performing the remaining actions.
3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z from the set of
domain parameters D, party U’s ephemeral private key de,U, and party V’s ephemeral
public key Qe,V. Then destroy the ephemeral private key de,U. If the call to the ECC CDH
primitive outputs an error indicator, destroy the results of all intermediate calculations
used in the attempted computation of Z, and return an error indicator without performing
the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
The Ephemeral Unified Model is summarized in Table 18.
Table 18: Ephemeral unified model key-agreement scheme
Party U

Party V

Domainparameters(q, FR, a, b{, SEED}, G, n, h)(q, FR, a, b{, SEED}, G, n, h)

Static data

N/A

N/A

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Party U
Ephemeral data

Party V

Ephemeral private key de,U

Ephemeral private key de,V

Ephemeral public key Qe,U

Ephemeral public key Qe,V

Computation

Compute Z by calling ECC CDH Compute Z by calling ECC CDH
using de,U and Qe,V
using de,V and Qe,U

Derive secretkeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.1.2.3 Key-Confirmation for C(2e, 0s)
In a C(2e, 0s) key-agreement scheme, none of the parties contributes a static key pair. Only
ephemeral key pairs are used to derive the secret value Z. Without a trusted association with an
identifier of either party, key-confirmation cannot achieve the expected purposes. Therefore, inthis Recommendation, key-confirmation is not incorporated for the C(2e, 0s) key-agreement
schemes.
6.2

Schemes Using One Ephemeral Key Pair, C(1e)This category consists of two subcategories that are determined by the use (or non-use) of a static
key pair by each of the parties. Only party U generates an ephemeral key pair. In the firstsubcategory, both party U and party V use a static key pair, and party U also generates an
ephemeral key pair (see Section 6.2.1). In the second subcategory, party U generates an
ephemeral key pair, but uses no static key pair; party V uses only a static key pair (see Section
6.2.2).
6.2.1 C(1e, 2s) Schemes
Figure 9 depicts typical flow for a C(1e, 2s) scheme. For these schemes, party U uses both staticand ephemeral private/public key pairs. Party V uses only a static private/public key pair. Party
U and party V obtain each other’s static public keys in a trusted manner. Party U also sends its
ephemeral public key to party V. A shared secret is generated by both parties using the available
static and ephemeral keys. The secret keying material is derived using the shared secret.

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U’s Static Public Key
V’s Static Public Key

U’s Ephemeral Public Key

Obtain static publickeys

U sends an ephemeral
public key

Figure 9: C(1e, 2s) schemes: party U contributes a static and an ephemeral key pair while
party V contributes only a static key pair
Assumptions: In order to execute a C(1e, 2s) key-establishment scheme in compliance with thisRecommendation, the following assumptions shall be true.
1. Each party has an authentic copy of the same set of domain parameters, D. These
parameters have been generated as specified in Section 5.5.1. For FFC schemes, D = (p,q, g{, SEED, pgenCounter}); for ECC schemes, D = (q, FR, a, b{, SEED}, G, n, h).
Furthermore, each party has obtained assurance of the validity of these domain
parameters as specified in Section 5.5.2.
2. Each party has been designated as the owner of a static key pair that was generated as
specified in Section 5.6.1 using the set of domain parameters, D. For FFC schemes, thestatic key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Each party has
obtained assurance of the validity of its own static public key as specified in Section
5.6.2.1.3. Each party has also obtained assurance of its possession of the correct value forits own private key as specified in Section 5.6.2.1.5.
3. The parties have agreed upon an approved key-derivation method, as well as an
approved algorithm to be used with that method (e.g., a hash function) and otherassociated parameters to be used for key derivation (see Section 5.8).
4. If key-confirmation is used, the parties have also agreed upon an approved MAC andassociated parameters, including the lengths of MacKey and MacTag, as specified in Step
3 and Step 4 in Section 5.9.1.1 (for the minimum lengths of MacKey and MacTag, see
Table 8 and Table 9 in Section 5.9.3).
5. Prior to or during the key-agreement process, each party receives the other party’s static
public key in a trusted manner (e.g., from a certificate signed by a trusted CA or directlyfrom the other party, who is trusted by the recipient). Each party has obtained assuranceof the validity of the other party’s static public key as specified in Section 5.6.2.2.1.

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6. The recipient of a static public key has obtained assurance that its (claimed) owner is (orwas) in possession of the corresponding static private key, as specified in Section
5.6.2.2.3.
7. When an identifier is used to label a party during the key-agreement process, that
identifier has a trusted association to that party’s static public key. (In other words,
whenever both the identifier and static public key of one participant are employed in the
key-agreement process, they are associated in a manner that is trusted by the otherparticipant.) When an identifier is used to label a party during the key-agreement process,both parties are aware of the particular identifier employed for that purpose.
6.2.1.1 dhHybridOneFlow, C(1e, 2s, FFC DH) Scheme
This section describes the dhHybridOneFlow scheme. Assurance of secure key establishment
using this scheme can only be obtained when the assumptions in Section 6.2.1 are true. In
particular, it is assumed that party U has obtained the static public key yV of party V, and party V
has obtained the static public key yU of party U.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the keyderivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in
Section 5.6.1.1. Send the public key tU to party V.
2. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Zs from the set of
domain parameters D, party U’s static private key xU, and party V’s static public key yV.
If the call to the FFC DH primitive outputs an error indicator, destroy the ephemeral
private key rU, destroy the results of all intermediate calculations used in the attempted
computation of Zs, and return an error indicator without performing the remaining
actions.
3. Use the FCC DH primitive to derive a shared secret Ze from the set of domain parametersD, party U’s ephemeral private key rU, and party V’s static public key yV. If this call to
the FFC DH primitive outputs an error indicator, destroy Zs and the ephemeral private
key rU, destroy the results of all intermediate calculations used in the attempted
computation of Ze, and return an error indicator without performing the remaining
actions.
4. Compute the shared secret Z = Ze || Zs. Destroy Ze and Zs.
5. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key rU, and return an error indicator without performing the remaining
actions.

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6. If the ephemeral private key rU will not be used in a broadcast scenario (see Section 7) for
subsequent key-establishment transactions using this scheme, then destroy rU.
7. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(rU, tU) may be used in other key-establishment transactions occurring during the same broadcast(i.e., step 1 above would not be repeated). After the final broadcast transaction, the ephemeral
private key rU shall be destroyed (see step 6 above).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Receive an ephemeral public key tU (purportedly) from party U. If tU is not received,
return an error indicator without performing the remaining actions.
2. Verify that tU is a valid public key for the parameters D as specified in Section 5.6.2.3. If
assurance of public key validity cannot be obtained, return an error indicator without
performing the remaining actions.
3. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret value Zs from theset of domain parameters D, party V’s static private key xV, and party U’s static public
key yU. If the call to the FFC DH primitive outputs an error indicator, destroy the resultsof all intermediate calculations used in the attempted computation of Zs, and return anerror indicator without performing the remaining actions.
4. Use the FCC DH primitive to derive a shared secret Ze from the set of domain parametersD, party V’s static private key xV, and party U’s ephemeral public key tU. If this call to
the FFC DH primitive outputs an error indicator, destroy Zs, destroy the results of allintermediate calculations used in the attempted computation of Ze, and return an error
indicator without performing the remaining actions.
5. Compute the shared secret Z = Ze || Zs. Destroy Ze and Zs.
6. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
7. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: Key-confirmation can be incorporated into this scheme. See Section 6.2.1.5 for details.
dhHybridOneFlow is summarized in Table 19.

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Table 19: dhHybridOneFlow key-agreement scheme summary
Party U

Party V

Domainparameters(p, q, g{, SEED, pgenCounter})(p, q, g{, SEED, pgenCounter})

Static data

Static private key xU

Static private key xV

Static public key yU

Static public key yV

Ephemeral private key rU

N/A

Ephemeral data

Ephemeral public key tU
Computation

1. Compute Zs by calling FFC DH 1. Compute Zs by calling FFC DH
using xU and yV
using xV and yU
2. Compute Ze by calling FFC DH 2. Compute Ze by calling FFC DH
using rU and yV
using xV and tU
3. Compute Z = Ze || Zs

Derive secretkeying material

3. Compute Z = Ze || Zs

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.2.1.2 (Cofactor) One-Pass Unified Model, C(1e, 2s, ECC CDH) Scheme
This section describes the One-Pass Unified Model scheme. Assurance of secure key
establishment using this scheme can only be obtained when the assumptions in Section 6.2.1 aretrue. In particular, it is assumed that party U has obtained the static public key Qs,V of party V,
and party V has obtained the static public key Qs,U of party U.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the key
derivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (de,U, Qe,U) from the domain parameters D as specified in
Section 5.6.1.2. Send the public key Qe,U to V.
2. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Zs from the set of
domain parameters D, party U’s static private key ds,U, and party V’s static public key
Qs,V. If the call to the ECC CDH primitive outputs an error indicator, destroy theephemeral private key de,U, destroy the results of all intermediate calculations used in theattempted computation of Zs, and return an error indicator without performing theremaining actions..

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3. Use the ECC CDH primitive to derive a shared secret Ze, from the set of domain
parameters D, party U’s ephemeral private key de,U, and party V’s static public key Qs,V.
If this call to the ECC CDH primitive outputs an error indicator, destroy Zs and theephemeral private key de,U, destroy the results of all intermediate calculations used in theattempted computation of Ze, and return an error indicator without performing theremaining actions.
4. Compute the shared secret Z = Ze || Zs. Destroy Ze and Zs.
5. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key de,U, and return an error indicator without performing theremaining actions.
6. If the ephemeral private key de,U will not be used in a broadcast scenario (see Section 7)
for subsequent key-establishment transactions using this scheme, then destroy de,U.
7. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(de,U, Qe,U) may be used in other key-establishment transactions occurring during the samebroadcast (i.e., step 1 above would not be repeated). After the final broadcast transaction, theephemeral private key de,U shall be destroyed (see step 6 above).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Receive an ephemeral public key Qe,U (purportedly) from party U. If Qe,U is not received,
return an error indicator without performing the remaining actions.
2. Verify that Qe,U is a valid public key for the parameters D as specified in Section 5.6.2.3.
If assurance of public key validity cannot be obtained, return an error indicator without
performing the remaining actions.
3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Zs from the set of
domain parameters D, party V’s static private key ds,V, and party U’s static public key
Qs,U. If the call to the ECC CDH primitive outputs an error indicator, destroy the resultsof all intermediate calculations used in the attempted computation of Zs, and return anerror indicator without performing the remaining actions.
4. Use the ECC CDH primitive to derive a shared secret Ze from the set of domain
parameters D, party V’s static private key ds,V, and party U’s ephemeral public key Qe,U.
If this call to the ECC CDH primitive outputs an error indicator, destroy Zs, destroy the
results of all intermediate calculations used in the attempted computation of Ze, andreturn an error indicator without performing the remaining actions.
5. Compute the shared secret Z = Ze || Zs. Destroy Ze and Zs.
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6. Use the agreed-upon key derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
7. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: Key-confirmation can be incorporated into this scheme. See Section 6.2.1.5 for details.
The One-Pass Unified Model is summarized in Table 20.
Table 20: One-pass unified model key-agreement scheme summary
Party U

Party V

Domainparameters(q, FR, a, b{, SEED}, G, n, h)(q, FR, a, b{, SEED}, G, n, h)

Static data

Static private key ds,U

Static private key ds,VStatic public key Qs,U

Static public key Qs,V

Ephemeral private key de,U

N/A

Ephemeral data

Ephemeral public key Qe,U
Computation

1. Compute Zs by calling ECC 1. Compute Zs by calling ECC
CDH using ds,U and Qs,V
CDH using ds,V and Qs,U
2. Compute Ze by calling ECC 2. Compute Ze by calling ECC
CDH using de,U and Qs,V
CDH using ds,V and Qe,U
3. Compute Z = Ze || Zs

Derive secretkeying material

3. Compute Z = Ze || Zs

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.2.1.3 MQV1, C(1e, 2s, FFC MQV) Scheme
This section describes the MQV1 scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.2.1 are true. In particular, it isassumed that party U has obtained the static public key yV of party V, and party V has obtained
the static public key yU of party U.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the keyderivation function.
Party U shall execute the following key-agreement transformation in order to a) establish ashared secret value Z with party V, and b) derive secret keying material from Z.

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Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in
Section 5.6.1.1. Send the public key tU to V.
2. Use the MQV1 form of the FFC MQV primitive in Section 5.7.2.1.2 to derive a sharedsecret Z from the set of domain parameters D, party U’s static private key xU, party V’sstatic public key yV, party U’s ephemeral private key rU, party U’s ephemeral public key
tU, and (for a second time) party V’s static public key yV. If the call to the FFC MQVprimitive outputs an error indicator, destroy the ephemeral private key rU, destroy the
results of all intermediate calculations used in the attempted computation of Z, and return
an error indicator without performing the remaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key rU, and return an error indicator without performing the remaining
actions.
4. If the ephemeral private key rU will not be used in a broadcast scenario (see Section 7) for
subsequent key-establishment transactions using this scheme, then destroy rU.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(rU, tU) may be used in other key-establishment transactions occurring during the same broadcast(i.e., step 1 above would not be repeated). After the final broadcast transaction, the ephemeral
private key rU shall be destroyed (see step 4 above).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Receive an ephemeral public key tU (purportedly) from party U. If tU is not received,
return an error indicator without performing the remaining actions.
2. Verify that tU is a valid public key for the parameters D as specified in Section 5.6.2.3. If
assurance of public key validity cannot be obtained, return an error indicator without
performing the remaining actions.
3. Use the MQV1 form of the FFC MQV primitive in Section 5.7.2.1.2 to derive a sharedsecret Z from the set of domain parameters D, party V’s static private key xV, party U’s
static public key yU, party V’s static private key xV (for a second time), party V’s static
public key yV, and party U’s ephemeral public key tU. If the call to the FFC MQVprimitive outputs an error indicator, destroy the results of all intermediate calculations
used in the attempted computation of Z, and return an error indicator without performing
the remaining actions.

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4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
5. Destroy all copies of the shared secret Z and output DerivedKeyingMaterial.
Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator.
Note: Key-confirmation can be incorporated into this scheme. See Section 6.2.1.5 for details.
MQV1 is summarized in Table 21.
Table 21: MQV1 Key-agreement scheme summary
Party U

Party V

Domainparameters(p, q, g{, SEED, pgenCounter})(p, q, g{, SEED, pgenCounter})

Static data

Static private key xU

Static private key xV

Static public key yU

Static public key yV

Ephemeral private key rU

N/A

Ephemeral data

Ephemeral public key tUCComputation

Derive secretkeying material

Compute Z by calling FFC MQV Compute Z by calling FFC MQVusing xU, yV, rU, tU, and yV (again)using xV, yU, xV (again), yV, and tU
1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.2.1.4 One-Pass MQV, C(1e, 2s, ECC MQV) Scheme
This section describes the One-Pass MQV scheme. Assurance of secure key establishment using
this scheme can only be obtained when the assumptions in Section 6.2.1 are true. In particular, it
is assumed that party U has obtained the static public key Qs,V of party V, and party V has
obtained the static public key Qs,U of party U.
In this scheme, each party has different actions, which are presented separately below. However,note that party U and party V must use identical orderings of the bit strings that are input to the
key-derivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (de,U, Qe,U) from the domain parameters D as specified in
Section 5.6.1.2. Send the public key Qe,U to party V.
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2. Use the One-Pass MQV form of the ECC MQV primitive in Section 5.7.2.3.2 to derive ashared secret value Z from the set of domain parameters D, party U’s static private key
ds,U, party V’s static public key Qs,V, party U’s ephemeral private key de,U, party U’s
ephemeral public key Qe,U, and (for a second time) party V’s static public key Qs,V. If thecall to the ECC MQV primitive outputs an error indicator, destroy the ephemeral private
key de,U, destroy the results of all intermediate calculations used in the attempted
computation of Z, and return an error indicator without performing the remaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key de,U, and return an error indicator without performing theremaining actions.
4. If the ephemeral private key de,U will not be used in a broadcast scenario (see Section 7)
for subsequent key-establishment transactions using this scheme, then destroy de,U.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(de,U, Qe,U) may be used in other key-establishment transactions occurring during the samebroadcast (i.e., step 1 above would not be repeated). After the final broadcast transaction, theephemeral private key de,U shall be destroyed (see step 4 above).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive shared secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Receive an ephemeral public key Qe,U (purportedly) from party U. If Qe,U is not received,
return an error indicator without performing the remaining actions.
2. Verify that Qe,U is a valid public key for the parameters D as specified in Section
5.6.2.3.2 or 5.6.2.3.3. If assurance of public key validity cannot be obtained, return anerror indicator without performing the remaining actions.
3. Use the One-Pass MQV form of the ECC MQV primitive in Section 5.7.2.3.2 to derive ashared secret value Z from the set of domain parameters D, party V’s static private key
ds,V, party U’s static public key Qs,U, party V’s static private key ds,V (for a second time),
party V’s static public key Qs,V, and party U’s ephemeral public key Qe,U. If the call to the
ECC MQV primitive outputs an error indicator, destroy the results of all intermediate
calculations used in the attempted computation of Z, and return an error indicator without
performing the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length bits from the shared secret value Z and other input. (See Section 5.8.) If
the key-derivation method outputs an error indicator, destroy all copies of Z, and return
an error indicator without performing the remaining action.

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5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: Key-confirmation can be incorporated into this scheme. See Section 6.2.1.5 for details.
The One-Pass MQV scheme is summarized in Table 22.
Table 22: One-pass MQV model key-agreement scheme summary
Party U

Party V

Domainparameters(q, FR, a, b{, SEED}, G, n, h)(q, FR, a, b{, SEED}, G, n, h)

Static data

Static private key ds,U

Static private key ds,VStatic public key Qs,U

Static public key Qs,V

Ephemeral private key de,U

N/A

Ephemeral data

Ephemeral public key Qe,U
Computation

Compute Z by calling ECC MQV Compute Z by calling ECC MQVusing ds,U, Qs,V, de,U, Qe,U, and Qs,V using ds,V, Qs,U, ds,V (again), Qs,V, and(again)Qe,UDerive secretKeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.2.1.5 Incorporating Key-Confirmation into a C(1e, 2s) Scheme
The subsections that follow illustrate how to incorporate key-confirmation (as described in
Section 5.9) into the C(1e, 2s) key-agreement schemes described above. Note that party V cannot
act as a key-confirmation recipient unless a nonce (NonceV) is provided by party V to party U
and is used (in addition to the shared secret Z) as input to the key-derivation method employedby the scheme. In terms of the preceding descriptions of C(1e, 2s) schemes, this would beaccomplished by including NonceV (in a protocol-specific fashion) in the input to key derivation.
The flow depictions separate the key-establishment flow from the key-confirmation flow. The
depictions and accompanying discussions presume that the assumptions of the scheme have beensatisfied, that the key-agreement transaction has proceeded successfully through key derivation,and that the received MacTags are successfully verified as specified in Section 5.2.2.
6.2.1.5.1C(1e, 2s) Scheme with Unilateral Key-Confirmation Provided by party U toparty V
Figure 10 depicts a typical flow for a C(1e, 2s) scheme with unilateral key-confirmation fromparty U to party V. In this situation, party U and party V assume the roles of key-confirmation
provider and recipient, respectively. Since party V does not contribute an ephemeral public key
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during the key-agreement process, a nonce (NonceV) shall be provided by party V to party U andused (in addition to the shared secret Z) as input to the key-derivation method employed by thescheme. NonceV is used as EphemDataV during MacTag computations. The successful
completion of the key-confirmation process provides party V with assurance that party U has
derived the same secret Z value. If NonceV is a random nonce, then party V also obtains
assurance that party U has actively participated in the process; see Section 5.4 for a discussion of
the length and security strength required for the nonce.

UV
U’s Static Public Key
V’s Static Public Key

U’s Ephemeral Public Key

Obtain static publickeys

U sends an ephemeral
public key

NonceVV sends a nonce
MacTagU
U’s key
confirmation

Figure 10: C(1e, 2s) scheme with unilateral key-confirmation from party U to party V
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), party U (and partyV) setEphemDataU = EphemPubKeyU, and EphemDataV = NonceV.
Party U provides MacTagU to party V (as specified in Section 5.9.1.1, with P = U and R = V),
where MacTagU is computed (as specified in Section 5.2.1) using
MacDataU = “KC_1_U” || IDU || IDV || EphemPubKeyU || NonceV {|| TextU}.
Party V (the key-confirmation recipient) uses the same format for MacDataU to compute its own
version of MacTagU and then verifies that the newly computed MacTag matches the value
provided by party U.

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6.2.1.5.2

C(1e, 2s) Scheme with Unilateral Key-Confirmation Provided by party V toparty U
Figure 11 depicts a typical flow for a C(1e, 2s) scheme with unilateral key-confirmation from party
V to party U. In this scenario, party V and party U assume the roles of key-confirmation provider
and recipient, respectively. The successful completion of the key-confirmation process provides
party U with a) assurance that party V has derived the same secret Z value; and b) assurance thatparty V has actively participated in the process.

U

V
U’s Static Public Key
V’s Static Public Key

U’s Ephemeral Public Key

Obtain static publickeys

U sends an ephemeral
public key

MacTagVV’s keyconfirmation

Figure 11: C(1e, 2s) scheme with unilateral key-confirmation from party V to party U
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), both parties set
EphemDataV = Null, and EphemDataU = EphemPubKeyU.
Party V provides MacTagV to party U (as specified in Section 5.9.1.1, with P = V and R = U),
where MacTagV is computed (as specified in Section 5.2.1) using
MacDataV = “KC_1_V” || IDV || IDU || Null || EphemPubKeyU {|| TextV}.
Party U (the key-confirmation recipient) uses the same format for MacDataV to compute its own
version of MacTagV, and then verifies that the newly computed MacTag matches the value
provided by party V.
6.2.1.5.3 C(1e, 2s) Scheme with Bilateral Key-Confirmation
Figure 12 depicts a typical flow for a C(1e, 2s) scheme with bilateral key-confirmation. In thismethod, party U and party V assume the roles of both the provider and the recipient in order toobtain bilateral key-confirmation. Since party V does not contribute an ephemeral public key
during the key-agreement process, a nonce (NonceV) shall be provided by party V to party U andused (in addition to the shared secret Z) as input to the key-derivation method employed by thescheme. NonceV is used as the EphemDataV during MacTag computations. The successful
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completion of the key-confirmation process provides each party with assurance that the otherparty has derived the same secret Z value. Party U obtains assurance that party V has actively
participated in the process; if NonceV is a random nonce, then party V also obtains assurance thatparty U has actively participated in the process; see Section 5.4 for a discussion of the length andsecurity strength required for the nonce.

UV
U’s Static Public Key

Obtain static publickeys

V’s Static Public Key

U’s Ephemeral Public Key
NonceV, MacTagV

U sends an ephemeral
public key
V’s key confirmation
with a nonce

MacTagU
U’s key confirmation

Figure 12: C(1e, 2s) scheme with bilateral key-confirmation
To provide bilateral key-confirmation (as described in Section 5.9.2.1), party U and party V
exchange and verify MacTags that have been computed (as specified in Sections 5.2.1) using
EphemDataU = EphemPubKeyU and EphemDataV = NonceV.
Party V provides MacTagV to party U (as specified in Sections 5.9.1.1 and 5.9.2.1, with P = V
and
R = U); MacTagV is computed by party V (and verified by U) using
MacDataV = “KC_2_V” || IDV || IDU || NonceV || EphemPubKeyU {|| TextV}.
Party U provides MacTagU to party V (as specified in Sections 5.9.1.1 and 5.9.2.1, with P = U
and R = V); MacTagU is computed by party U (and verified by party V) using
MacDataU = “KC_2_U” || IDU || IDV || EphemPubKeyU || NonceV {|| TextU}.
Note that in Figure 12 party V’s nonce (NonceV) and the MacTag (MacTagV) are depicted as
being sent in the same message (to reduce the number of passes in the combined keyagreement/key-confirmation process). They may also be sent separately (as long as NonceV issent before the MacTags are exchanged). The MacTagV and MacTagU can be sent in any order,
as long as NonceV is available to generate and verify both MAC tags.

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6.2.2 C(1e, 1s) Schemes
For each of C(1e, 1s) schemes, party U generates an ephemeral key pair, but uses no static key
pair; party V has only a static key pair. Party U obtains party V’s static public key in a trusted
manner (for example, from a certificate signed by a trusted CA or directly from party V, who istrusted) and sends its ephemeral public key to party V. The parties compute a shared secret using
their private keys and the other party’s public key. Each party uses the shared secret to derive
secret keying material (see Figure 13).

UV
V’s Static Public Key

U’s Ephemeral Public Key

U obtains V’s static
public key

U sends an ephemeral
public key

Figure 13: C(1e, 1s) schemes: party U contributes an ephemeral key pair, and party V
contributes a static key pair
Assumptions: In order to execute a C(1e, 1s) key-establishment scheme in compliance with thisRecommendation, the following assumptions shall be true.
1. Each party has an authentic copy of the same set of domain parameters, D. These
parameters have been generated as specified in Section 5.5.1. For FFC schemes, D = (p,q, g{, SEED, pgenCounter}); for ECC schemes, D = (q, FR, a, b{, SEED}, G, n, h).
Furthermore, each party has obtained assurance of the validity of these domain
parameters as specified in Section 5.5.2.
2. Party V has been designated as the owner of a static key pair that was generated as
specified in Section 5.6.1 using the set of domain parameters, D. For FFC schemes, thestatic key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Party V has
obtained assurance of the validity of its own static public key as specified in Section
5.6.2.1. Party V has obtained assurance of its possession of the correct value of its own
private key as specified in Section 5.6.2.1.5.
3. The parties have agreed upon an approved key-derivation method, as well as an
approved algorithm to be used with that method (e.g., a hash function) and otherassociated parameters to be used (see Section 5.8).
4. If key-confirmation is used, the parties have also agreed upon an approved MAC andassociated parameters, including the lengths of MacKey and MacTag, as specified in Step

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3 and Step 4 in Section 5.9.1.1 (for the minimum lengths of MacKey and MacTag, see
Table 8 and Table 9 in Section 5.9.3).
5. Prior to or during the key-agreement process, party U receives party V’s static public key
in a trusted manner (e.g., from a certificate signed by a trusted CA or directly from party
V, who is trusted by the recipient) Party U has obtained assurance of the validity of party
V’s static public key as specified in Section 5.6.2.2.1.
6. When an identifier is used to label either party during the key-agreement process, bothparties are aware of the particular identifier employed for that purpose. In particular,when an identifier is used to label party V during the key-agreement process, that
identifier has a trusted association to that party V’s static public key. (In other words,
whenever both the identifier and static public key of one participant are employed in the
key-agreement process, they are associated in a manner that is trusted by the otherparticipant.) When an identifier is used to label party U during the key-agreement
process, it has been selected/assigned in accordance with the requirements of the protocol
relying upon the use of the key-agreement scheme.
The following is an assumption for using the derived keying material for purposes beyond theC(1e,1s) scheme itself.
Party U has obtained assurance that party V is (or was) in possession of the appropriate
static private key, as specified in Section 5.6.2.2.3.
6.2.2.1 dhOneFlow, C(1e, 1s, FFC DH) Scheme
This section describes the dhOneFlow scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.2.2 are true. In particular, it isassumed that party U has obtained the static public key yV of party V.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the keyderivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in
Section 5.6.1.1. Send the public key tU to party V.
2. Use the FCC DH primitive in Section 5.7.1.1 to derive a shared secret Z from the set of
domain parameters D, party U’s ephemeral private key rU, and party V’s static public key
yV. If the call to the FFC DH primitive outputs an error indicator, destroy the ephemeral
private key rU, destroy the results of all intermediate calculations used in the attempted
computation of Z, and return an error indicator without performing the remaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z and the90

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ephemeral private key rU, and return an error indicator without performing the remaining
actions.
4. If the ephemeral private key rU will not be used in a broadcast scenario (see Section 7) for
subsequent key-establishment transactions using this scheme, then destroy rU.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(rU, tU) may be used in other key-establishment transactions occurring during the same broadcast(i.e., step 1 above would not be repeated). After the final broadcast transaction, the ephemeral
private key rU shall be destroyed (see step 4 above).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Receive an ephemeral public key tU (purportedly) from party U. If tU is not received,
return an error indicator without performing the remaining actions.
2. Verify that tU is a valid public key for the parameters D as specified in Section 5.6.2.3. If
assurance of public key validity cannot be obtained, return an error indicator without
performing the remaining actions.
3. Use the FCC DH primitive in Section 5.7.1.1 to derive a shared secret Z from the set of
domain parameters D, party V’s static private key xV, and party U’s ephemeral public key
tU. If the call to the FFC DH primitive outputs an error indicator, destroy the results of allintermediate calculations used in the attempted computation of Z, and return an error
indicator without performing the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.

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Note: Key-confirmation can be incorporated into this scheme. See Section 6.2.2.3 for details.
dhOneFlow is summarized in Table 23.
Table 23: dhOneFlow key-agreement scheme summary
Party U

Party V

Domainparameters(p, q, g{, SEED, pgenCounter})(p, q, g{, SEED, pgenCounter})

Static data

N/A

Static private key xV
Static public key yV

Ephemeral data

N/A

Ephemeral private key rU
Ephemeral public key tU

Computation

Compute Z by calling FFC DH using Compute Z by calling FFC DH usingrU and yV
xV and tU

Derive secretmaterial

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.2.2.2 (Cofactor) One-Pass Diffie-Hellman, C(1e, 1s, ECC CDH) Scheme
This section describes the One-Pass Diffie-Hellman scheme. Assurance of secure key
establishment using this scheme can only be obtained when the assumptions in Section 6.2.2 aretrue. In particular, it is assumed that party U has obtained the static public key Qs,V of party V.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the key
derivation function.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Generate an ephemeral key pair (de,U, Qe,U) from the domain parameters D as specified in
Section 5.6.1.2. Send the public key Qe,U to party V.
2. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z from the set of
domain parameters D, party U’s ephemeral private key de,U, and party V’s static public
key Qs,V. If this call to the ECC CDH primitive outputs an error indicator, destroy theephemeral private key de,U, destroy the results of all intermediate calculations used in theattempted computation of Z, and return an error indicator without performing theremaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the92

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key-derivation method outputs an error indicator, destroy all copies of Z and theephemeral private key de,U, and return an error indicator without performing theremaining actions.
4. If the ephemeral private key de,U will not be used in a broadcast scenario (see Section 7)
for subsequent key-establishment transactions using this scheme, then destroy de,U.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If the ephemeral key pair is used in a broadcast scenario by party U (see Section 7) for
subsequent key-establishment transactions using this scheme, then the same ephemeral key pair(de,U, Qe,U) may be used in other key-establishment transactions occurring during the samebroadcast (i.e., step 1 above would not be repeated). After the final broadcast transaction, theephemeral private key de,U shall be destroyed (see step 4 above).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Receive an ephemeral public key Qe,U (purportedly) from party U. If Qe,U is not received,
return an error indicator without performing the remaining actions.
2. Verify that Qe,U is a valid public key for the parameters D as specified in Section 5.6.2.3.
If assurance of public key validity cannot be obtained, return an error indicator without
performing the remaining actions.
3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z from the set of
domain parameters D, party V’s static private key ds,V, and party U’s ephemeral public
key Qe,U. If this call to the ECC CDH primitive outputs an error indicator, destroy the
results of all intermediate calculations used in the attempted computation of Z, and return
an error indicator without performing the remaining actions.
4. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z and other input. (See Section 5.8.) If the
key-derivation method outputs an error indicator, destroy all copies of Z, and return anerror indicator without performing the remaining action.
6. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.

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Note: Key-confirmation can be incorporated into this scheme. See Section 6.2.2.3 for details.
The One-Pass Diffie-Hellman is summarized in Table 24.
Table 24: One-pass Diffie-Hellman key-agreement scheme summary
Party U

Party V

Domainparameters(q, FR, a, b{, SEED}, G, n, h)(q, FR, a, b{, SEED}, G, n, h)

Static data

N/A

Static private key ds,VStatic public key Qs,V

Ephemeral data

Ephemeral private key de,U

N/A

Ephemeral public key Qe,U
Computation

Compute Z by calling ECC CDH Compute Z by calling ECC CDH
using de,U and Qs,V
using ds,V and Qe,U

Derive secretkeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterial
2. Destroy Z

2. Destroy Z

6.2.2.3 Incorporating Key-Confirmation into a C(1e, 1s) Scheme
The subsection that follows illustrates how to incorporate key-confirmation (as described in
Section 5.9) into the C(1e, 1s) key-agreement schemes described above. Note that only unilateral
key-confirmation from party V to party U is specified, since only party V has a static key pairthat is used in the key-establishment process.
The flow depiction separates the key-establishment flow from the key-confirmation flow. The
depiction and accompanying discussion presumes that the assumptions of the scheme have beensatisfied, that the key-agreement transaction has proceeded successfully through key derivation,and that the received MacTag is successfully verified as specified in Section 5.2.2.
6.2.2.3.1

C(1e, 1s) Scheme with Unilateral Key-Confirmation Provided by party V toparty U
Figure 14 depicts a typical flow for a C(1e, 1s) scheme with unilateral key-confirmation from party
V to party U. In this scenario, party V and party U assume the roles of the key-confirmation provider
and recipient, respectively. The successful completion of the key-confirmation process provides
party U with a) assurance that party V has derived the same secret Z value; and b) assurance thatparty V has actively participated in the process.

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UV
V’s Static Public Key

U’s Ephemeral Public Key

U obtains V’s static
public key
U sends an ephemeral
public key

MacTagVV’s key-confirmation

Figure 14: C(1e, 1s) scheme with unilateral key-confirmation from party V to party U
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), both parties set
EphemDataV = Null, and EphemDataU = EphemPubKeyU.
Party V provides MacTagV to party U (as specified in Section 5.9.1.1, with P = V and R = U),
where MacTagV is computed (as specified in Section 5.2.1) using
MacDataV = “KC_1_V” || IDV || IDU || Null || EphemPubKeyU {|| TextV}.
Party U (the key-confirmation recipient) uses the same format for MacDataV to compute its own
version of MacTagV and then verifies that the newly computed MacTag matches the value
provided by V.
6.3

C(0e, 2s) Schemes

In this category, the parties use only static key pairs. Each party obtains the other party’s static
public key. A nonce, NonceU, is sent by party U to party V to ensure that the derived keying
material is different for each key-establishment transaction. NonceU should be a random nonce.
The parties calculate the shared secret using their own static private key and the other party’s
static public key. Secret keying material is derived using the key-derivation method, the sharedsecret, and the nonce (see Figure 15).

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U’s Static Public Key
V’s Static Public Key

Obtain static public
keys

NonceUU sends a nonce

Figure 15: C(0e, 2s) schemes: each party contributes only a static key pair
Assumptions: In order to execute a C(0e, 2s) key-establishment scheme in compliance with thisRecommendation, the following assumptions shall be true.
1. Each party has an authentic copy of the same set of domain parameters, D. These
parameters have been generated as specified in Section 5.5.1. For FFC schemes, D = (p,q, g{, SEED, pgenCounter}); for ECC schemes, D = (q, FR, a, b{, SEED}, G, n, h).
Furthermore, each party has assurance of the validity of these domain parameters as
specified in Section 5.5.2.
2. Each party has been designated as the owner of a static key pair that was generated as
specified in Section 5.6.1 using the set of domain parameters, D. For FFC schemes, thestatic key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Each party has
obtained assurance of the validity of its own static public key as specified in Section
5.6.2.1. Each party has obtained assurance of its possession of the correct value for its
own private key as specified in Section 5.6.2.1.5.
3. The parties have agreed upon an approved key-derivation method (see Section 5.8), as
well as an approved algorithm used with the method (e.g., a hash function) and otherassociated parameters to be used (see Section 5.8.1). In addition, the parties have agreed
on the form of the nonce (see Section 5.4).
4. If key-confirmation is used, the parties have also agreed upon an approved MAC andassociated parameters, including the lengths of MacKey and MacTag, as specified in Step
3 and Step 4 in Section 5.9.1.1 (for the minimum lengths of MacKey and MacTag, see
Table 8 and Table 9 in Section 5.9.3).
5. Prior to or during the key-agreement process, each party receives the other party’s static
public key in a trusted manner (e.g., from a certificate signed by a trusted CA or directlyfrom the other party, who is trusted by the recipient). Each party has obtained assuranceof the validity of the other party’s static public key as specified in Section 5.6.2.2.

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6. The recipient of a static public key has obtained assurance that its (claimed) owner is (orwas) in possession of the corresponding static private key, as specified in Section 5.6.3.2.
7. When an identifier is used to label a party during the key-agreement process, that
identifier has a trusted association to that party’s static public key. (In other words,
whenever both the identifier and static public key of one participant are employed in the
key-agreement process, they are associated in a manner that is trusted by the otherparticipant.) When an identifier is used to label a party during the key-agreement process,both parties are aware of the particular identifier employed for that purpose.
6.3.1 dhStatic, C(0e, 2s, FFC DH) Scheme
This section describes the dhStatic scheme. Assurance of secure key establishment using thisscheme can only be obtained when the assumptions in Section 6.3 are true. In particular, it isassumed that party U has obtained the static public key yV of party V, and party V has obtained
the static public key yU of party U.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the keyderivation method.
Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Obtain a nonce, NonceU (see Section 5.4). Send NonceU to party V.
2. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Z from the set of
domain parameters D, party U’s static private key xU, and party V’s static public key yV.
If the call to the FFC DH primitive outputs an error indicator, destroy NonceU, destroy
the results of all intermediate calculations used in the attempted computation of Z, andreturn an error indicator without performing the remaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z, NonceU and other input. (See Section
5.8.) If the key-derivation method outputs an error indicator, destroy all copies of Z, andreturn an error indicator without performing the remaining actions.
4. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material bits or an error indicator.
Note: If NonceU is used in a broadcast scenario by party U (see Section 7) for subsequent keyestablishment transactions using this scheme, then the same NonceU may be used in other keyestablishment transactions occurring during the same broadcast (i.e., step 1 above would not be
repeated).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party U, and b) derive secret keying material from Z.
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1. Obtain party U’s nonce, NonceU, from party U. If NonceU is not available, return an error
indicator without performing the remaining actions.
2. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret from the set of
domain parameters D, party V’s static private key xV, and party U’s static public key yU. If
the call to the FFC DH primitive outputs an error indicator, destroy the results of allintermediate calculations used in the attempted computation of Z, and return an error
indicator without performing the remaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z, NonceU, and other input. (See Section
5.8.) If the key-derivation method outputs an error indicator, destroy all copies of Z, andreturn an error indicator without performing the remaining action.
4. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: Key-confirmation can be incorporated into this scheme. See Section 6.3.3 for details.
dhStatic is summarized in Table 25.
Table 25: dhStatic key-agreement scheme summary
Party U

Party V

Domainparameters(p, q, g{, SEED, pgenCounter})(p, q, g{, SEED, pgenCounter})

Static data

Static private key xU

Static private key xV

Static public key yU

Static public key yV

Ephemeral data

NonceU

Computation

Compute Z by calling FFC DH Compute Z by calling FFC DH using
using xU, and yV
xV, and yU

Derive secretkeying material

1. Compute DerivedKeyingMaterialusing NonceU

1. Compute DerivedKeyingMaterialusing NonceU

2. Destroy Z

2. Destroy Z

6.3.2 (Cofactor) Static Unified Model, C(0e, 2s, ECC CDH) Scheme
This section describes the Static Unified Model scheme. Assurance of secure key establishment
using this scheme can only be obtained when the assumptions in Section 6.3 are true. In
particular, it is assumed that party U has obtained the static public key Qs,V of party V, and party
V has obtained the static public key Qs,U of party U.
In this scheme, each party has different actions, which are presented separately below. However,note that parties U and V must use identical orderings of the bit strings that are input to the keyderivation function.
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Party U shall execute the following key-agreement transformation to a) establish a shared secretvalue Z with party V, and b) derive secret keying material from Z.
Actions: Party U generates a shared secret and derives secret keying material as follows:
1. Obtain a nonce, NonceU (see Section 5.4). Send NonceU to party V.
2. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z from the set of
domain parameters D, party U’s static private key ds,U, and party V’s static public key
Qs,V. If the call to the ECC CDH primitive outputs an error indicator, destroy the resultsof all intermediate calculations used in the attempted computation of Z, and return anerror indicator without performing the remaining actions.
3. Use the agreed-upon key-derivation method to derive secret keying material with thespecified length from the shared secret value Z, NonceU, and other input. (See Section
5.8.) If the key-derivation method outputs an error indicator, destroy all copies of Z, andreturn an error indicator without performing the remaining actions.
4. If NonceU will not be used in a broadcast scenario (see Section 7) for subsequent keyestablishment transactions using this scheme, then destroy NonceU.
5. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: If NonceU is used in a broadcast scenario by party U (see Section 7) for subsequent keyestablishment transactions using this scheme, then the same NonceU may be used in other keyestablishment transactions occurring during the same broadcast (i.e., step 1 above would not be
repeated).
Party V shall execute the following key-agreement transformation to a) establish a shared secretvalue, Z, with party U, and b) derive secret keying material from Z.
Actions: Party V derives secret keying material as follows:
1. Obtain party U’s nonce, NonceU, from party U. If NonceU is not available, return an error
indicator without performing the remaining actions.
2. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z from the set of
domain parameters D, party V’s static private key ds,V, and party U’s static public key
Qs,U. If the call to the ECC CDH primitive outputs an error indicator, destroy the resultsof all intermediate calculations used in the attempted computation of Z, and return anerror indicator without performing the remaining actions.
3. Use the agreed-upon key derivation method to derive secret keying material with thespecified length from the shared secret value Z, NonceU, and other input. (See Section
5.8.) If the key-derivation method outputs an error indicator, destroy all copies of Z, andreturn an error indicator without performing the remaining action.
4. Destroy all copies of the shared secret Z and output the derived keying material.
Output: The derived keying material or an error indicator.
Note: Key-confirmation can be incorporated into this scheme. See Section 6.3.3 for details.

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Static Unified Model is summarized in Table 26.
Table 26: Static unified model key-agreement scheme summary
Party U

Party V

Domainparameters(q, FR, a, b{, SEED}, G, n, h)(q, FR, a, b{, SEED}, G, n, h)

Static data

Static private key ds,U

Static private key ds,VStatic public key Qs,U

Static public key Qs,V

Ephemeral data

NonceU

Computation

Compute Z by calling ECC CDH Compute Z by calling ECC CDH
using ds,U, and Qs,V
using ds,V, and Qs,U

Derive secretkeying material

1. Compute DerivedKeyingMaterial 1. Compute DerivedKeyingMaterialusing NonceU
using NonceU
2. Destroy Z

2. Destroy Z

6.3.3 Incorporating Key-Confirmation into a C(0e, 2s) Scheme
The subsections that follow illustrate how to incorporate key-confirmation (as described in
Section 5.9) into the C(0e, 2s) key-agreement schemes described above. Note that party V cannot
act as a key-confirmation recipient unless a nonce (NonceV) is provided by party V to party U
and is used (in addition to the shared secret Z) as input to the key-derivation method employedby the scheme. In terms of the preceding descriptions of C(0e, 2s) schemes, this would beaccomplished by including NonceV (in a protocol-specific fashion) in the input to the key
derivation.
The flow depictions separate the key-establishment flow from the key-confirmation flow. The
depictions and accompanying discussions presume that the assumptions of the scheme have beensatisfied, that the key-agreement transaction has proceeded successfully through key derivation,and that the received MacTags are successfully verified as specified in Section 5.2.2.
6.3.3.1 C(0e, 2s) Scheme with Unilateral Key-Confirmation Provided by party U toparty V
Figure 16 depicts a typical flow for a C(0e, 2s) scheme with unilateral key-confirmation from party
U to party V. In this scenario, party U and party V assume the roles of key-confirmation provider
and recipient, respectively. A nonce (NonceV) shall be provided by party V to party U and used (inaddition to the shared secret Z and the nonce provided by party U) as input to the key-derivation
method employed by the scheme. NonceV is used as the EphemDataV during MacTag computations.
The successful completion of the key-confirmation process provides party V with assurance thatparty U has derived the same secret Z value. If NonceV is a random nonce, then party V also obtains
assurance that party U has actively participated in the process; see Section 5.4 for a discussion of thelength and security strength required for the nonce.

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U’s Static Public Key
V’s Static Public Key
NonceU

Obtain static
public keys

U sends a nonce

NonceVV sends a nonce
MacTagU
U’s key confirmation

Figure 16: C(0e, 2s) scheme with unilateral key-confirmation from party U to party V
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), party U (and partyV) setEphemDataU = NonceU, and EphemDataV = NonceV.
Party U provides MacTagU to party V (as specified in Section 5.9.1.1, with P = U and R = V),
where MacTagU is computed (as specified in Section 5.2.1) using
MacDataU = “KC_1_U” || IDU || IDV || NonceU || NonceV {|| TextU}.
Party V (the key-confirmation recipient) uses the same format for MacDataU to compute its own
version of MacTagU and then verifies that the newly computed MacTag matches the value provided
by party U.
6.3.3.2 C(0e, 2s) Scheme with Unilateral Key-Confirmation Provided by party V toparty U
Figure 17 depicts a typical flow for a C(0e, 2s) scheme with unilateral key-confirmation fromparty V to party U. In this situation, party V and party U assume the roles of key-confirmation
provider and recipient, respectively. The successful completion of the key-confirmation processprovides party U with assurance that party V has derived the same secret Z value; if NonceU is arandom nonce, then party U also obtains assurance that party V has actively participated in theprocess; see Section 5.4 for a discussion of the length and security strength required for thenonce.

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U’s Static Public Key
V’s Static Public Key

Obtain static public
keys

NonceUU sends a nonce
MacTagVV’s key confirmation

Figure 17: C(0e, 2s) scheme with unilateral key-confirmation from party V to party U
To provide (and receive) key-confirmation (as described in Section 5.9.1.1), both parties set
EphemDataV = Null, and EphemDataU = NonceU.
Party V provides MacTagV to party U (as specified in 5.9.1.1, with P = V and R = U), where
MacTagV is computed (as specified in Section 5.2.1) using
MacDataV = “KC_1_V” || IDV || IDU || Null || NonceU {|| TextV}.
Party U (the key-confirmation recipient) uses the same format for MacDataV to compute its own
version of MacTagV, and then verifies that the newly computed MacTag matches the value
provided by party V.
6.3.3.3 C(0e, 2s) Scheme with Bilateral Key-Confirmation
Figure 18 depicts a typical flow for a C(0e, 2s) scheme with bilateral key-confirmation. In thismethod, party U and party V assume the roles of both the provider and the recipient in order toobtain bilateral key-confirmation. A nonce (NonceV) shall be provided by party V to party U andused (in addition to the shared secret Z and the nonce, NonceU, provided by party U) as input tothe key-derivation method employed by the scheme. NonceV is used as the EphemDataV during
MacTag computations. The successful completion of the key-confirmation process provides eachparty with assurance that the other party has derived the same secret Z value. If NonceU is arandom nonce, then party U obtains assurance that party V has actively participated in theprocess; if NonceV is a random nonce, then party V obtains assurance that party U has actively
participated in the process. See Section 5.4 for a discussion about the length and security strength
required for the nonce.

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U’s Static Public Key
V’s Static Public Key

Obtain static public
keys

NonceUU sends a nonce

NonceV, MacTagV
V sends a nonce with
key-confirmation
MacTagU
U’s key-confirmation

Figure 18: C(0e, 2s) scheme with bilateral key-confirmation
To provide bilateral key-confirmation (as described in Section 5.9.2.1), party U and party V
exchange and verify MacTags that have been computed (as specified in Sections 5.2.1) using
EphemDataU = NonceU, and EphemDataV = NonceV.
Party V provides MacTagV to party U (as specified in Sections 5.9.1.1 and 5.9.2.1, with P = V
and R = U); MacTagV is computed by party V (and verified by party U) using
MacDataV = “KC_2_V” || IDV || IDU || NonceV || NonceU {|| TextV}.
Party U provides MacTagU to party V (as specified in Sections 5.9.1.1 and 5.9.2.1, with P = U
and R = V); MacTagU is computed by party U (and verified by party V) using
MacDataU = “KC_2_U” || IDU || IDV || NonceU || NonceV {|| TextU}.
Note that in Figure 18, party V’s nonce (NonceV ) and the MacTag (MacTagV) are depicted as
being sent in the same message (to reduce the number of passes in the combined keyagreement/key-confirmation process). They can also be sent in other orders and combinations (aslong as NonceU and NonceV are available to generate and verify both MAC tags).

7.

DLC-Based Key-Transport

A DLC-based key-transport scheme uses both a key-agreement scheme and a key-wrapping
algorithm in a single transaction to establish keying material. During this transaction, a keywrapping key shall be established using an approved key-agreement scheme. This key shall be
used by party U to wrap secret keying material using an approved key-wrapping algorithm; the103

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wrapped keying material is then sent to party V (i.e., party U in the key-agreement scheme willbe the key-transport sender, and party V will be the key-transport receiver). [SP 800-38F]
specifies the approved key-wrapping algorithms.
To comply with this Recommendation, the key-transport transaction shall use only approved
key-agreement schemes that employ party V’s static key pair 11 and require an ephemeral
contribution by party U 12. In particular, a C(2e, 2s), C(1e, 2s), C(1e, 1s) or C(0e, 2s) keyagreement scheme shall be used in which party U is the intended key-transport sender; a C(2e,0s) scheme shall not be used to establish the key-wrapping key (regardless of which party is theintended key-transport sender). Although other methods may be used by protocols thatincorporate key-transport, this Recommendation makes no statement as to the adequacy of thosemethods.
Key-confirmation may optionally be provided by party V following the unwrapping of thereceived keying material, either instead of or in addition to any key-confirmation that may be
performed as part of the key-agreement scheme. When key-confirmation is performed following
the unwrapping process in accordance with this Recommendation, parties U and V shall haveagreed upon an approved MAC algorithm and associated parameters (see Table 8 and Figure 9
in Section 5.9.3).
7.1 Key-Transport Scheme
The DLC-based key-transport scheme is as follows:
1. An agreed-upon C(2e, 2s), C(1e, 2s), C(1e, 1s) or C(0e, 2s) key-agreement scheme isused between party U and party V to establish DerivedKeyingMaterial, which includes a
KeyWrappingKey that will subsequently be used by party U for key-transport. Key confirmation (as specified in Section 5.9 and Section 6) may optionally be incorporated
in the key-agreement scheme to provide assurance that the DerivedKeyingMaterial is thesame for both parties.
2. Party U obtains the KeyWrappingKey from the DerivedKeyingMaterial.
3. Party U selects secret keying material, TransportedKeyingMaterial, to transport to party
V, the receiver. If key-confirmation is to be performed following key-transport, thisTransportedKeyingMaterial shall include a fresh (i.e., not previously used) MacKey to be
used for key-confirmation and the KeyData to be used subsequent to key-transport (seeSection 7.2).
4. Party U calculates WrappedKeyingMaterial = KeyWrap(KeyWrappingKey,
TransportedKeyingMaterial) using KeyWrap( ), an approved key-wrapping algorithm.
5. Party U sends WrappedKeyingMaterial to party V.
6. Party V receives WrappedKeyingMaterial from party U.

11

To prevent receiver identifier spoofing; since the receiver has used a static key pair during key-agreement, thesender has assurance of the identifier of the intended receiver.
12

To provide the key-transport sender with assurance of the freshness of the key-wrapping key.

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7. Party V obtains the KeyWrappingKey from the DerivedKeyingMaterial.
8. Party V calculates TransportedKeyingMaterial = KeyUnwrap(KeyWrappingKey,
WrappedKeyingMaterial) using KeyUnwrap( ), the key-unwrapping algorithm that
corresponds to KeyWrap( ).
9. If key-confirmation is to be performed subsequent to key-transport to provide assurance
to party U that the correct TransportedKeyingMaterial has been obtained by party V, thenboth parties U and V shall proceed as specified in Section 7.2.
Note that if the key-agreement scheme used in Step 1 is such that party V does not contribute anephemeral key pair to the calculation of the shared secret (that is, a C(1e, 2s), C(1e, 1s), or C(0e,2s) scheme has been used) and key confirmation is not included in the key-agreement scheme,
then Steps 1 through 5 can be performed by party U without direct involvement of party V. Thiscan be useful in a store-and-forward environment, such as e-mail.
Key-transport schemes can be used in broadcast scenarios. In a broadcast scenario, an exception
is made to the rule in this Recommendation that ephemeral keys shall not be reused (see Section
5.6.3.3). That is, party U may use the same ephemeral key pair in step 1 above in multiple
instances of DLC-based key-agreement (employing the same scheme) if the same secret keying
material is being transported to multiple entities for use following key-transport 13, and if all these
instances of key transport occur “simultaneously” (or within a short period of time). However,the security properties of the key-establishment scheme may be affected by reusing theephemeral key in this manner.
7.2 Key-Confirmation for Transported Keying Material
If key-confirmation is to be provided in compliance with this Recommendation following thetransport of keying material (as specified in Section 7.1), party U shall generate a fresh MacKey
and include it as part of the TransportedKeyingMaterial to be transported (see Section 7.1). The
transported MacKey shall be used for the computation and verification of the MacTag provided
by party V to party U.
For each instance of key-confirmation following key-transport, this MacKey shall be generated
anew using an approved random bit generator that is instantiated at or above the security
strength required for the key establishment transaction. In broadcast scenarios, a different
MacKey shall be included in the TransportedKeyingMaterial for each key-transport receiver thatis expected to provide key-confirmation to party U.
The domain parameter set used by the key-agreement scheme employed to establish the keywrapping key shall be used to determine the minimum MacKey length and the length of the
MacTag, as specified in Tables 8 and 9 in Section 5.9.3.
The transported keying material shall be formatted as follows:

13

Note that when key-confirmation is performed after key-transport, the MacKey is different for each instance of
key-confirmation, but KeyData is the same for each key-transport receiver participating in the broadcast (see Section
7.2).

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TranportedKeyingMaterial = MacKey || KeyData.
The KeyData may be null, or may contain keying material to be used subsequent to keytransport. The MacKey shall be used during key-confirmation and then immediately destroyedby both party U and party V.
The MacKey portion of the transported keying material and an approved MAC algorithm (seeSections 5.2 and 5.9.3) are used by each party to compute a MacTag (of an appropriate, agreedupon length) on the MacData (see Section 5.9.1.1) represented as
MacData = “KC_KT” || IDV || IDU || EphemDataV || EphemDataU || WrappedKeyingMaterial { ||Text},
where IDV is the identifier associated with party V, and IDU is the identifier associated with party
U. These identifiers shall be the same as those used to label parties U and V during the keyagreement portion of the key-transport transaction. EphemDataV is the ephemeral public key ornonce contributed by party V during the establishment of the key-wrapping key used for keytransport; if no ephemeral data was contributed by party V, then Null shall be used. EphemDataU
is the ephemeral public key or nonce that was contributed by party U during the establishment of
the key-wrapping key. WrappedKeyingMaterial is the ciphertext of the keying material that has
been transported, and Text is an optional bit string that may be used during key-confirmation thatis known by both parties.
Party V (the MacTag sender) computes a MacTag (using the MacKey obtained from the
TranportedKeyingMaterial and MacData formed as described above) and provides it to Party U.
Party U (the MacTag receiver) computes a MacTag (using the MacKey that was included in the
TranportedKeyingMaterial and the MacData formed as described above). Party U then verifies
that this newly computed MacTag matches the MacTag value provided by party V.

8.

Rationale for Selecting a Specific Scheme

The subsections that follow present possible justifications for selecting schemes from eachsubcategory, C(ie, js). The proffered rationales are intended to provide the user and/or developer
with some information that may help when making a decision as to which key-agreement scheme
to use. The rationales include brief discussions of basic security properties, but do not constitutean in-depth analysis of all possible security properties of all schemes under all adversary models.
The specific security properties that are cited will depend on such considerations as whether astatic key is used, whether an ephemeral key is used, how the shared secret is calculated, andwhether key-confirmation can be incorporated into a scheme. In general, the security properties
cited for a particular subcategory of schemes are exhibited by each scheme within thatsubcategory; when this is not the case, the exceptions are identified.
A scheme should not be chosen based solely on the number of security properties it may possess.
Rather, a scheme should be selected based on how well the scheme fulfills system requirements.
For instance, if messages are exchanged over a large-scale network where each exchange
consumes a considerable amount of time, a scheme with fewer exchanges during a single keyagreement transaction might be preferable to a scheme with more exchanges, even though thelatter may possess more security benefits. It is important to keep in mind that a key-agreement
scheme may be a component of a larger protocol that offers additional security-related
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assurances beyond those provided by the key-agreement scheme alone. For example, theprotocol may include specific features that limit opportunities for accidental or intentional
misuse of the key-agreement component of the protocol. Protocols, per se, are not specified inthis Recommendation.
Important Note: In order to provide concise descriptions of security properties possessed by thevarious schemes, it is necessary to make some assumptions concerning the format and type ofdata that is used as input during key derivation. These assumptions are made solely for the
purposes of Sections 8.1 through 8.6; they are not intended to preclude the options specified
elsewhere in this Recommendation. When discussing the security properties of a subcategory ofschemes, it is assumed that the OtherInfo input to a key-derivation function employed during aparticular key-agreement transaction uses either the concatenation format or the ASN.1 format(see Section 5.8.1.2). It is also assumed that OtherInfo includes sufficiently specific identifiers
for the participants in the transaction, an identifier for the key-agreement scheme being usedduring the transaction, and additional input (e.g., a nonce, ephemeral public key, and/or session
identifier) that may provide assurance to one or both participants that the derived keying material
will reflect the specific context in which the transaction occurs (see Section 5.8.1.2 andAppendix B for further discussion concerning context-specific information that may beappropriate for inclusion in OtherInfo). In general, OtherInfo may include pre-shared secrets, butthat is not assumed to be the case in the analysis of security properties that follows. In cases
where an approved extraction-then-expansion key-derivation procedure is employed, it isassumed that this OtherInfo is used as the Context input during the key-expansion step, as
specified in [SP 800-56C]. Finally, it is assumed that all required nonces employed during thetransaction are random nonces that contain a component consisting of a random bit string formedin accordance with the recommendations of Section 5.4.
8.1 Rationale for Choosing a C(2e, 2s) Scheme
These schemes require each participant to own a static key pair that is used in their keyagreement transaction. Static key pairs can provide the participants with some level of assurance
that they have correctly identified the party with whom they will be establishing keying material
if the transaction is successfully completed.
In the case of a key-agreement transaction based on the Full Unified model or dhHybrid1, eachparticipant has assurance that no unintended entity (i.e., no entity other than the owners of thestatic key pairs involved in the transaction) could employ a Diffie-Hellman primitive (seeSection 5.7.1) to compute Zs, the static component of the shared secret, Z, without knowledge of
one of the static private keys employed during the transaction. Absent the compromise of Zs or
one of those static private keys, each participant can be confident of correctly identifying theother participant in the key-establishment transaction. The level of confidence is commensuratewith the specificity of the identifiers that are associated with the static public keys (and are used
as input during the key derivation process), the degree of trust in the association between those
identifiers and static public keys, the assurance of validity of the domain parameters and static
public keys, and the availability of evidence that the keying material has been correctly derived.
Similarly, in the case of a key-agreement transaction based on Full MQV or MQV2, eachparticipant has assurance that no unintended entity could use a DLC primitive to compute theshared secret, Z, without knowledge of either a static private key or a private-key-dependent
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implicit signature employed during the transaction. (The term “implicit signature” refers to thosequantities denoted SA and implicitsigA in the descriptions of the MQV primitives in Section
5.7.2.1 and Section 5.7.2.3, respectively.) Absent the compromise of Z, a static private key, or an
implicit signature, each participant can be confident of correctly identifying the other participant
in the key-establishment transaction. As above, the level of confidence is commensurate with thespecificity of the identifiers that are associated with the static public keys (and are used as input
during the key-derivation process), the degree of trust in the association between those identifiers
and static public keys, the assurance of validity of the domain parameters and static public keys,
and the availability of evidence that the keying material has been correctly derived.
These schemes also require each participant to generate an ephemeral key pair that is used intheir transaction, providing each participant with assurance that the resulting shared secret (andthe keying material derived from it) will vary from one of their C(2e, 2s) transactions to the next.
Each participant in a C(2e, 2s) transaction has assurance that the value of the resulting sharedsecret, Z, will not be completely revealed to an adversary who is able to compromise (only) theirstatic private keys at some time after the transaction is completed. (The adversary would,
however, be able to compute Zs, the static component of the shared secret, if the key-agreementtransaction was based on the Full Unified model or dhHybrid1.) This assurance is commensuratewith the confidence that a participant has that neither of the ephemeral private keys employed in
the transaction will be compromised. By generating their ephemeral key pairs as close to the timeof use as possible and destroying the ephemeral private keys after their use, the participants
reduce the risk of such a compromise.
If a particular entity’s static private key is acquired by an adversary, then the adversary could
masquerade as that entity while engaging in any C(2e, 2s) key-agreement transaction that permits
the use of the compromised key pair. If an MQV scheme (MQV2 or Full MQV) will be
employed during a transaction with an adversary who is in possession of a compromised staticprivate key (or a compromised implicit signature corresponding to that static private key), theadversary is limited to masquerading as the owner of the compromised key pair (or as the ownerof the static key pair corresponding to the compromised implicit signature). The use of the Full
Unified model or dhHybrid1 scheme, however, offers the adversary additional opportunities formasquerading: If an adversary compromises a particular entity’s static private key, then theadversary may be able to impersonate any other entity during a Full Unified model- or
dhHybrid1-based key-agreement transaction with that entity. Also, the compromise of Zs, thestatic component of a shared secret that was (or would be) formed by two parties using the Full
Unified Model or dhHybrid1 scheme will permit an adversary to masquerade as either party tothe other party in key-agreement transactions that rely on the same scheme and the same twostatic key pairs.
Key-confirmation can be provided in either or both directions as part of a C(2e, 2s) scheme byusing the methods specified in Section 6.1.1.5. This allows the key-confirmation recipient toobtain assurance that the key-confirmation provider has possession of the MacKey derived fromthe shared secret Z and has used it with the appropriate MacData to compute the received
MacTag. In the absence of some compromise of secret information (e.g., a static private key or astatic component of Z), a key-confirmation recipient can obtain assurance that the appropriate
identifier has been used to label the key-confirmation provider and that the provider is the ownerof the static public key associated with that identifier. A key-confirmation recipient can also108

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receive assurance of active (and successful) participation by the key-confirmation provider in the
key-agreement transaction.
8.2 Rationale for Choosing a C(2e, 0s) Scheme
These schemes require each participant to generate an ephemeral key pair that is used in their
key-agreement transaction. No static key pairs are employed. Because the ephemeral private
keys used in the computation of their shared secret are destroyed immediately after use, theseschemes offer assurance to each party that the shared secret, Z, computed during a legitimateC(2e, 0s) transaction (i.e., one that involves two honest parties and is not influenced by anadversary) is protected against any compromise of shared secrets and/or private keys associatedwith other (prior or future) transactions.
Unlike a static public key, which is assumed to have a trusted association with an identifier for
its owner, there is no assumption of a trusted association between an ephemeral public key andan identifier. Thus, these schemes, by themselves, offer no assurance to either party of theaccuracy of any identifier that may be used to label the entity with whom they have established ashared secret. The use of C(2e, 0s) schemes may be appropriate in applications where any trusted
association desired/required between an identifier and an ephemeral public key is enforced bymethods external to the scheme (e.g., in the protocol incorporating the key-agreement scheme).
This Recommendation does not specify incorporation of key-confirmation in a C(2e, 0s) scheme.
8.3 Rationale for Choosing a C(1e, 2s) Scheme
These schemes require each participant to own a static key pair that is used in their keyagreement transaction; in addition, the participant acting as party U is required to generate anduse an ephemeral key pair. Different assurances are provided to the participants by the utilizationof a C(1e, 2s) scheme, depending upon which one acts as party U and which one acts as party V.
The use of static key pairs in the key-agreement transaction can provide the participants withsome level of assurance that they have correctly identified the party with whom they will beestablishing keying material if the transaction is successfully completed.
In the case of a transaction based on the One-Pass Unified model or dhHybridOneflow, eachparticipant has assurance that no unintended entity (i.e., no entity other than the owners of thestatic key pairs involved in the key-establishment transaction) could employ a Diffie-Hellman
primitive (see Section 5.7.1) to compute Zs, the static component of the shared secret, Z, without
knowledge of one of the static private keys employed during the transaction. Absent thecompromise of Zs or one of those static private keys, each participant can be confident ofcorrectly identifying the other participant in the key-establishment transaction. The level ofconfidence is commensurate with the specificity of the identifiers that are associated with thestatic public keys (and are used as input during the key-derivation process), the degree of trust in
the association between those identifiers and static public keys, the assurance of validity of thedomain parameters and static public keys, and the availability of evidence that the keying
material has been correctly derived.
Similarly, in the case of a key-agreement transaction based on One-Pass MQV or MQV1, eachparticipant has assurance that no unintended entity could use a DLC primitive to compute theshared secret, Z, without knowledge of either the static private key of one of the participants in
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the transaction or the private-key dependent implicit signature employed by party U during thetransaction. (The term “implicit signature” refers to those quantities denoted SA and implicitsigA
in the descriptions of the MQV primitives in Section 5.7.2.1 and Section 5.7.2.3, respectively.)
Absent the compromise of Z, a static private key, or party U’s implicit signature, each participant
can be confident of correctly identifying the other participant in the key-establishmenttransaction. As above, the level of confidence is commensurate with the specificity of theidentifiers that are associated with the static public keys (and are used as input during the keyderivation process), the degree of trust in the association between those identifiers and static
public keys, the assurance of validity of the domain parameters and static public keys, and theavailability of evidence that the keying material has been correctly derived.
The party U, whose ephemeral key pair is used in the computations, has assurance that theresulting shared secret will vary from one transaction of C(1e, 2s) with the same party V to thenext. The participant acting as party V cannot obtain such assurance, in general, since party V’s
contribution to the computation of Z is static. Party V can, however, obtain assurance that thederived keying material will vary if, for example, party V contributes a nonce that is used as
input to the key-derivation method employed during these transactions (as is required when party
V is a recipient in a key-confirmation process performed as specified in this Recommendation).
The assurance of freshness of the derived keying material that can be obtained in this way by theparticipant acting as party V is commensurate with the participant’s assurance that a different
nonce will be contributed during each such transaction.
The compromise of the static private key used by party U does not, by itself, compromise theshared secret computed during any legitimate C(1e, 2s) transaction (i.e., one involving two
honest parties). Likewise, the compromise of only the ephemeral private key used by party U
would not compromise the shared secret Z for that transaction. However, the compromise of anentity’s static private key may lead to the compromise of the shared secrets computed during
past, current, and future C(1e, 2s) transactions in which that entity acts as party V (regardless ofthe static or ephemeral keys used by the entity acting as party U); to compromise those sharedsecrets, the adversary must also acquire the public keys contributed by whomever acts as party U
in those transactions.
If an adversary learns a particular entity’s static private key, then, in addition to masquerading asthat particular entity, the adversary may be able to impersonate any other entity while acting as
party U in a C(1e, 2s) transaction in which the owner of the compromised static private key acts
as party V. Similarly, the compromise of the static component, Zs, of a shared secret formed by
two entities using the One-Pass Unified Model or dhHybrid1OneFlow scheme will permit anadversary to masquerade as either entity (while acting as party U) to the other entity (acting as
party V) in future key-agreement transactions that rely on the same scheme and the same twostatic key pairs. If MQV1 or One-Pass MQV will be employed during a key-agreementtransaction with an adversary who is in possession of a compromised implicit signaturecorresponding to a particular static private key, the adversary may be able to masquerade as theowner of that static key pair while acting as party U (provided that the static key pair iscompatible with the domain parameters employed during the transaction).
Key-confirmation can be provided in either or both directions as part of a C(1e, 2s) scheme byusing the methods specified in Section 6.2.1.5. This allows the key-confirmation recipient toobtain assurance that the key-confirmation provider has possession of the MacKey derived from110

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the shared secret Z and has used it with the appropriate MacData to compute the received
MacTag. In the absence of a compromise of secret information (e.g., a static private key or astatic component of Z), a key-confirmation recipient can obtain assurance that the appropriate
identifier has been used to label the key-confirmation provider and that the provider is the ownerof the static public key associated with that identifier. A key-confirmation recipient can also
receive assurance of active (and successful) participation by the key-confirmation provider in the
key-agreement transaction.
8.4 Rationale for Choosing a C(1e, 1s) Scheme
In these schemes, the participant acting as party U is required to generate and use an ephemeral
key pair, while the participant acting as party V is required to own a static key pair that is used in
the key-agreement transaction. Different assurances are provided to the participants by theutilization of a C(1e, 1s) scheme, depending upon which one acts as party U and which one acts
as party V.
The use of a static public key attributed to party V can provide the participant acting as party U
with some level of assurance that he has correctly identified the party with whom he will beestablishing keying material if the transaction is successfully completed.
Whether the transaction is based on the One-Pass Diffie-Hellman or dhOneflow, the participant
acting as party U has assurance that no unintended entity (i.e., no entity other than himself andthe owner of the static public key attributed to party V) could employ a Diffie-Hellman primitive(see Section 5.7.1) to compute the shared secret, Z, without knowledge of one of the private keys
employed during the transaction. Absent the compromise of Z or one of those private keys, theparticipant acting as party U can be confident of correctly identifying the other participant in the
key-establishment transaction as the owner of the static public key attributed to party V. The
level of confidence is commensurate with the specificity of the identifier that is associated withthe static public key attributed to party V (and is used as input during the key derivation process),the degree of trust in the association between that identifier and the static public key, theassurance of validity of the domain parameters and static public key, and the availability ofevidence that the keying material has been correctly derived.
The participant acting as party V has no such assurance, in general, since he has no assurance
concerning the accuracy of any identifier that may be used to label party U (unless the protocol
using this scheme includes additional elements that establish a trusted association between an
identifier for party U and the ephemeral public key that party U contributes to the transaction).
The participant acting as party U, whose ephemeral key pair is used in the computations, hasassurance that the resulting shared secret will vary from one C(1e, 1s) transaction to the next.
The participant acting as party V has no such assurance, since party V’s contribution to thecomputation of Z is static.
There is no assurance provided to either participant that the security of the shared secret isprotected against the compromise of a private key. A compromise of the ephemeral private key
used in a C(1e, 1s) transaction only compromises the shared secret resulting from that particulartransaction (and by generating the ephemeral key pair as close to the time of use as possible anddestroying the ephemeral private key after its use, the participant acting as party U reduces therisk of such a compromise). However, the compromise of an entity’s static private key may lead111

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to the compromise of shared secrets resulting from past, current, and future C(1e, 1s)transactions in which that entity acts as party V (no matter what party plays the role of party U);
to compromise those shared secrets, the adversary must also acquire the ephemeral public keys
contributed by whomever acts as party U in those transactions. In addition, if an adversary learns
a particular entity’s static private key, the adversary may be able to impersonate that particular
entity while acting as party V in a C(1e, 1s) transaction that employs compatible domain
parameters.
The participant acting as party V may provide key-confirmation to party U as specified in
Section 6.2.2.3. This allows the participant acting as party U (who is the key-confirmation
recipient) to obtain assurance that party V has possession of the MacKey derived from the sharedsecret Z and has used it with the appropriate MacData to compute the received MacTag. In theabsence of a compromise of secret information (e.g., a private key), the participant acting as
party U can obtain assurance that the appropriate identifier has been used to label party V, andthat the participant acting as party V is indeed the owner of the static public key associated withthat identifier. Under such circumstances, the participant acting as party U can also receive
assurance of the active (and successful) participation in the key-agreement transaction by theowner of the static public key attributed to party V.
This Recommendation does not specify the incorporation of key-confirmation from party U toparty V in a C(1e, 1s) scheme.
8.5 Rationale for Choosing a C(0e, 2s) Scheme
These schemes require each participant to own a static key pair that is used in their keyagreement transaction; in addition, the participant acting as party U is required to generate anonce, which is sent to party V and used (by both participants) as input to their chosen keyderivation method.
The use of static key pairs in the key-agreement transaction can provide the participants withsome level of assurance that they have correctly identified the party with whom they will beestablishing keying material if the transaction is successfully completed.
Whether the transaction is based on the Static Unified Model or dhStatic scheme, eachparticipant has assurance that no unintended entity (i.e., no entity other than the owners of thestatic key pairs employed in the transaction) could employ a Diffie-Hellman primitive (seeSection 5.7.1) to compute the static shared secret, Z, without knowledge of one of the staticprivate keys employed during the transaction. Absent the compromise of Z or one of those staticprivate keys, each participant can be confident of correctly identifying the other party in the keyestablishment transaction. The level of confidence is commensurate with the specificity of theidentifiers that are associated with the static public keys (and are used as input during the keyderivation process), the degree of trust in the association between those identifiers and static
public keys, the assurance of validity of the domain parameters and static public keys, and theavailability of evidence that the keying material has been correctly derived.
Although the value of Z is the same in all C(0e, 2s) key-establishment transactions between thesame two parties (as long as the two participants employ the same static key pairs), theparticipant acting as party U, whose (required) nonce is used in the key-derivation computations,has assurance that the derived keying material will vary from one of their C(0e, 2s) transactions
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to the next. In general, the participant acting as party V has no such assurance – unless, forexample, party V also contributes a nonce that is used as input to the key-derivation method
employed during the transaction (as is required when party V is a recipient of key-confirmation
performed as specified in this Recommendation). The assurance of freshness of the derived
keying material that can be obtained by a participant in a C(0e, 2s) transaction is commensuratewith the participant’s assurance that a different nonce will be contributed during each suchtransaction.
If the static Z value formed by the two participants is ever compromised, then all of the keying
material derived in past, current, and future C(0e, 2s) key-agreement transactions between thesesame two entities that employ these same static key pairs may be compromised as well, since thesame Z value is used to derive keying material in each instance. However, to compromise thekeying material from a particular transaction, the adversary must also acquire (at least) the nonce
contributed by the participant that acted as party U in that transaction. The compromise of thestatic Z value may also permit an adversary to masquerade as either entity to the other entity infuture C(0e, 2s) key-agreement transactions.
If a particular entity’s static private key is compromised, then shared secrets resulting fromcurrent, prior and future C(0e, 2s) transactions involving that entity’s static key pair may becompromised, irrespective of the role (whether party U or party V) played by the compromised
entity. Regardless of what entity acts in the other role when interacting with the compromised
entity, the adversary may be able to compute the shared secret Z and proceed to compromise thederived keying material, as described above. To complete the attack against a particulartransaction, the adversary must acquire (at least) the static public key contributed by the otherentity participating in that transaction with the compromised entity, as well as the nonce
contributed by whichever entity acted as party U during the transaction.
Of course, if a static private key has been compromised by an adversary, then (as long as thecompromised key pair is of the type permitted by the scheme and domain parameters) theadversary may masquerade as the owner of the compromised static key pair in key-agreementtransactions with any other party. In addition, the adversary may masquerade as any entity(whether acting as party U or party V) while engaging in a C(0e, 2s) key-agreement transaction
with the owner of the compromised key pair.
Keyconfirmation can be provided in either or both directions as part of a C(0e, 2s) scheme byusing the methods specified in Section 6.3.3.1. This allows the key-confirmation recipient toobtain assurance that the key-confirmation provider has possession of the MacKey derived fromthe shared secret Z and has used it with the appropriate MacData to compute the received
MacTag. In the absence of a compromise of private information (e.g., a static private key or thestatic shared secret, Z), a key-confirmation recipient can obtain assurance that the appropriate
identifier has been used to label the key-confirmation provider, and that the provider is the ownerof the static public key associated with that identifier. A key-confirmation recipient can also
receive assurance of active (and successful) participation by the key-confirmation provider in the
key-agreement transaction.
8.6 Choosing a Key-Agreement Scheme for use in Key-Transport
The key-agreement scheme employed while performing DLC-based key-transport as specified inthis Recommendation is required to be a C(2e, 2s), C(1e, 2s), C(1e, 1s) or C(0e, 2s) scheme in
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which the intended key-transport sender acts as party U, and the intended key-transport receiveracts as party V. The basic security properties of these schemes have been described in theprevious sections. The following discussion emphasizes the effects that the properties of the keyagreement scheme used to establish a key-wrapping key may have on assurances that can beprovided to the sender and/or receiver of the wrapped keying material.
Note: Unless it is explicitly stated otherwise, the analysis that follows is restricted to keytransport transactions that involve only two parties – the sender (acting as party U) and one
receiver (acting as party V). The broadcast scenario (involving multiple receivers) will be
addressed briefly at the conclusion of the section.)
Each of the schemes that can be used during the key-agreement phase of the transaction requires
the use of a static public key owned by the participant acting as party V. Unless there is acompromise of some secret information (e.g., a static component of Z or a private key), the keytransport sender (who acts as party U) has assurance that no unintended entity (i.e., no parties
other than himself and the owner of the static public key attributed to party V) could employ a
DLC primitive to compute the shared secret Z that is used to derive the key-wrapping key usedduring the key-transport process. Absent such a compromise, the key-transport sender can be
confident that he has correctly identified the key-transport receiver (assumed to have been acting
as party V). The level of confidence is commensurate with the specificity of the identifier that is
associated with the static public key attributed to party V, the degree of trust in the association
between that identifier and that static public key, the assurance of validity of the domain
parameters and public keys employed during the key-agreement phase of the transaction, and theavailability of evidence that the key-wrapping key has been correctly derived by the keytransport receiver.
When a C(2e, 2s), C(1e, 2s), or C(1e, 1s) scheme is employed during the key-agreement portionof the transaction, the key-transport sender (i.e., party U) generates an ephemeral key pair that isused in the computation of Z. This provides assurance to the party U (the key-transport sender)that both the shared secret and the derived key-wrapping key will vary from one key-transporttransaction to the next. Assurance of the freshness of the derived key-wrapping key may also be
obtained by party U when a C(0e, 2s) scheme is employed. In that case, party U is required tocontribute a nonce (see Section 5.4) that is used in the derivation of the key-wrapping key; theassurance of freshness that party U (the key-transport sender) can obtain is commensurate withthe probability that the contributed nonce has not been previously employed in the keyderivation process of the key-agreement portion of some other transaction. Assurance that a fresh
key-wrapping key is used during each instance of key-transport provides commensurate
assurance to party U (the key-transport sender) that the confidentiality of the wrapped keying
material transported during a particular transaction with party V will not be threatened by thepossibility that the key-wrapping key has been (or will be) compromised as a result of its use insome other transaction and/or application.
Assuming that no key pairs and/or static Z values are compromised, the required use of a static
public key attributed to party V (the intended key-transport receiver) during the key-agreement
portion of the transaction, together with each scheme’s required ephemeral contribution fromparty U, provides assurance to party U (the key-transport sender) that the owner of the staticprivate key attributed to party V is the only other party who will be able to acquire the (fresh)
key-wrapping key and use it to unwrap the transported keying material.

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If a C(2e, 2s), C(1e, 2s), or C(0e, 2s) scheme is employed during the key-agreement portion of
the transaction, the use of a static public key attributed to party U (the key-transport sender)
provides the participant acting as party V (the key-transport receiver) with a means of identifying
the entity with whom he will be establishing keying material if the transaction is successfullycompleted. The trusted association of an identifier with a static public key attributed to party U
provides party V with a method for accurately labeling the (purported) key-transport sender (i.e.,
party U). Absent the compromise of some secret information (e.g., a static component of Z or a
private key), party V can be confident that no unintended entity (i.e., no parties other thanhimself and the owner of the static public key attributed to party U) could employ a DLC
primitive to compute the shared secret Z, from which the key-wrapping key is derived. Party V’s
confidence is commensurate with the specificity of the identifier that is associated with the static
public key attributed to party U, the degree of trust in the association between that identifier andthat static public key, the assurance of validity of the domain parameters and public keys
employed during the transaction, and the evidence available to party V that party U has derived
the correct key-wrapping key (i.e., the one used by party U to wrap the transported keying
material).
On the other hand, if a C(1e, 1s) scheme is employed during the key-agreement portion of thetransaction, party U (the key-transport sender) is only required to provide an ephemeral public
key to party V. Since there is no assumption of a trusted association between an ephemeral
public key and an identifier, the use of a C(1e, 1s) scheme (in and of itself) offers no assurance tothe party V (the key-transport receiver) of the accuracy of any identifier that may be associatedwith party U. Any trusted association desired/required between an identifier and the (purported)
key-transport sender (party U) would have to be provided by methods external to the keyestablishment scheme.
When a C(2e, 2s) scheme is employed during the key-agreement portion of the transaction, thekey-transport receiver (acting as party V) generates an ephemeral key pair that is used in thecomputation of Z. This provides assurance to party V that both the shared secret and the keywrapping key derived from it will vary from one key-transport transaction to the next. Assuranceof the freshness of the key-wrapping key may also be obtained by party V when a C(1e, 2s),C(1e, 1s) or C(0e, 2s) scheme is employed and party V contributes a nonce (see Section 5.4) thatis used in the derivation of the key-wrapping key. The assurance of freshness that party V canobtain in this way is commensurate with the probability that the contributed nonce has not beenpreviously employed in a key derivation process. Assurance that a fresh key-wrapping key isused during each instance of a key-transport transaction provides commensurate assurance toparty V that the confidentiality of the wrapped keying material transported during a particulartransaction with party U will not be threatened by the possibility that the key-wrapping key has
been (or will be) compromised as a result of the use of an identical key in some other transaction
and/or application.
Key-confirmation from party V (the intended key-transport receiver) to party U (the intended
key-transport sender) can be incorporated in a C(2e, 2s), C(1e, 2s), C(1e, 1s) or C(0e, 2s) keyagreement scheme (as specified in Section 6.1.1.5.2, Section 6.2.1.5.2, Section 6.2.2.3, orSection 6.3.3.2, respectively) following the derivation of the key-wrapping key. This enables
party U (the intended key-transport sender) to obtain assurance that party V (the intended keytransport receiver) has derived the correct key-wrapping key. A key-confirmation failure would

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key-transport transaction could be discontinued before the keying material is wrapped and sent.
Key-confirmation from party U (the intended key-transport sender) to party V (the intended keytransport receiver) can be incorporated in a C(2e, 2s), C(1e, 2s), or C(0e, 2s) key-agreement
scheme (as specified in Section 6.1.1.5.1, Section 6.2.1.5.1, or Section 6.3.3.1, respectively) prior
to the key-transport portion of the transaction; in the case of a C(1e, 2s) or C(0e, 2s) scheme,
party V would be required to contribute a nonce that is used as input to the key-derivation
method when the key-wraping key is derived. Key-confirmation provided in this direction (fromparty U to party V) enables party V to obtain assurance that he has derived the same key thatparty U will employ to wrap the transported keying material. A key-confirmation failure may, forexample, prompt party V to discontinue the current key-transport transaction (without attempting
to unwrap any transported keying material) and notify party U that they must try again toestablish a shared key-wrapping key.
As specified in Section 7.2, key-confirmation can also be performed following the transport of
the wrapped keying material, allowing party U (the key-transport sender) to obtain assurance thatparty V (the intended key-transport receiver) has successfully employed the derived keywrapping key to unwrap the transported keying material. Confirming party V’s success inunwrapping the transported keying material also confirms that party V has correctly derived the
key-wrapping key during the key-agreement portion of the transaction. Therefore, at the risk oftransporting keying material that cannot be unwrapped, key-confirmation following the transportof wrapped keying material (as specified in Section 7.2) provides an alternative to incorporating
key-confirmation (from party V to party U) in the key-agreement portion of the transaction.
The use of a C(1e, 2s), C(1e, 1s) or C(0e, 2s) key-agreement scheme to establish the keywrapping key allows for one-pass implementations of key-transport (in cases where keyconfirmation is not required). Assuming that the static public key attributed to party V (theintended key-transport receiver) has been obtained previously, party U (the key-transport sender)
can include the wrapped keying material and all of the data required for party V to derive the
key-wrapping key in a single message. On the other hand, the use of a C(2e, 2s) scheme
necessitates the exchange of two or more messages, since each party must (at least) provide anephemeral public key to the other party in the key-agreement portion of the transaction.
There are additional considerations that apply to the broadcast scenario, in which one sender(acting as party U) transports the same keying material “simultaneously” (or within a shortperiod of time) to multiple receivers (i.e., multiple entities acting as party V) for use following
the key-transport transaction(s).
As noted in Section 7.1, this Recommendation’s general prohibition against the reuse of anephemeral key pair is relaxed in broadcast scenarios, permitting (but not requiring) the keytransport sender (acting as party U in the key-agreement portion of the transaction) to use thesame ephemeral key pair when establishing key-wrapping keys with the multiple key-transportreceivers. However, parties must proceed with caution when engaging in such practices (see, forexample, “On Reusing Ephemeral Keys in Diffie-Hellman Key Agreement Protocols,” by A.
Menezes and B. Ustaoglu; available at http://cacr.uwaterloo.ca/techreports/2008/cacr200824.pdf).

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As part of the proper implementation of this Recommendation, the key-transport sender (acting
as party U) should not reuse an ephemeral public key when establishing key-wrapping keys for
key-transport in a broadcast scenario unless all parties involved and/or agents trusted to act on
their behalf have determined the conditions (including the choice of key-agreement scheme)
under which this practice meets the security requirements of the sender and the various receivers.
If, in a broadcast scenario, the key-transport sender (i.e., party U) requires multiple key-transportreceivers to provide evidence that they have successfully unwrapped the keying material sent tothem using key-confirmation as specified in Section 7.2, it is imperative for the sender totransport a different MAC key to each receiver (as required by this Recommendation). In theabsence of the compromise of any key-wrapping keys, this will deter one receiver frommasquerading as another when returning a key-confirmation MacTag to the sender.

9.

Key Recovery

For some applications, the secret keying material used to protect data may need to be recovered(for example, if the normal reference copy of the secret keying material is lost or corrupted). In
this case, either the secret keying material or sufficient information to reconstruct the secretkeying material needs to be available (for example, the keys, domain parameters and other inputs
to the scheme used to perform the key establishment process).
Keys used during the key establishment process shall be handled in accordance with thefollowing:
1. A static key pair may be saved.
2. An ephemeral public key may be saved.
3. An ephemeral private key shall be destroyed after use and, therefore, shall not berecoverable.
4. A symmetric key may be saved.
Note: This implies that keys derived from schemes where both parties generate ephemeral key
pairs (i.e., the C(2e, 2s) and C(2e, 0s) schemes) cannot be made recoverable by reconstruction of
the secret keying material by parties requiring the ephemeral private key in their calculations.
For those schemes where only party U generates an ephemeral key pair (i.e., the C(1e, 2s) andC(1e, 1s schemes), only party V can recover the secret keying material by reconstruction.
General guidance on key recovery and the protections required for each type of key is provided
in the Recommendation for Key Management [SP 800-57].

10.

Implementation Validation

When the NIST Cryptographic Algorithm Validation Program (CAVP) and the Cryptographic
Module Validation Program (CMVP) have established a validation program for thisRecommendation, a vendor shall have its implementation tested and validated by the CAVP and
CMVP in order to claim conformance to this Recommendation. Information on the CAVP and
the CMVP is available at http://csrc.nist.gov/cryptval/.

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An implementation claiming conformance to this Recommendation shall include one or more of
the following capabilities:•

Domain parameter generation as specified in Section 5.5.1.

•

Explicit domain parameter validation as specified in Section 5.5.2, item 2.

•

Key pair generation as specified in Section 5.6.1; documentation shall include how
assurance of domain parameter validity is expected to be achieved by the key pair owner.

•

Explicit public key validation as specified in Section 5.6.2.3.1 for FFC or as specified in
Sections 5.6.2.3.2 or 5.6.2.3.3 for ECC.

•

A key-agreement scheme from Section 6, together with an approved key derivation
method from Section 5.8 or an application-specific KDF from SP 800-135. Other key
derivation methods may be temporarily allowed for backward compatibility; these otherallowable methods and any restrictions on their use will be specified in FIPS 140-2
Annex D. If key-confirmation is also claimed, the appropriate key-confirmation
technique from Section 5.9 shall be used. Documentation shall include how assurance ofprivate key possession and assurance of domain parameter and public key validity are
expected to be achieved by both the owner and the recipient.

•

A key-transport scheme as specified in Section 7.

An implementer shall also identify the appropriate specifics of the implementation, including:•

The security strength(s) of supported cryptographic algorithms; this will determine theparameter set requirements (see Table 1 and Table 2 in Section 5.5.1),•

The domain parameter generation method (see Section 5.5.1).

•

The hash function (see Section 5.1),•

The MAC key length(s) (see Table 8 and Table 9 in Section 5.9.3),•

The MAC length(s) (see Table 8 and Table 9 in Section 5.9.3).

•

The type of cryptography: FFC or ECC,•

The key-establishment schemes available (see Section 6),•

The key derivation method to be used, including the format of OtherInfo (see Section
5.8),

•

The type of nonces to be generated (see Section 5.4),•

The NIST- Recommended elliptic curve(s) available (if appropriate),•

The key wrap algorithm, and•The key-confirmation scheme (see Section 5.9).

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Appendix A: References
A.1 Normative References[FIPS 140]

FIPS 140-2, Security requirements for Cryptographic Modules, May 25, 2001.

[FIPS 140-2 IG] FIPS 140-2 Implementation Guidance, Available at
http://csrc.nist.gov/groups/STM/cmvp/documents/fips140-2/FIPS1402IG.pdf .
[FIPS 180]

FIPS 180-4, Secure Hash Standard, February 2011.

[FIPS 186]

FIPS 186-3, Digital Signature Standard, June 2009.

[FIPS 197]

FIPS 197, Advanced Encryption Standard, November 2001.

[FIPS 198]

FIPS 198-1, The Keyed-Hash Message Authentication Code (HMAC), July 2008.

[SP 800-38B] NIST SP 800-38B, Recommendation for Block Cipher Modes of Operation: The
CMAC Mode for Authentication, May 2005.
[SP 800-38F] Recommendation for Block Cipher Modes of Operation: Methods for Key
Wrapping, December, 2012.
[SP 800-56C] Recommendation for Key Derivation through Extraction-then-Expansion,
November 2011.
[SP 800-57]

NIST SP 800-57, Part 1: Recommendation for Key Management, August 2005.

[SP 800-90A] Recommendation for Random Number Generation Using Deterministic Random
Bit Generators, January 2012.
[SP 800-108] Recommendation for Key Derivation Using Pseudorandom Functions, October
2009.
[SP 800-131A] Transitions: Recommendation for Transitioning the Use of Cryptographic
Algorithms and Key Lengths, January 2011
[SP 800-135] Recommendation for Existing Application-Specific Key Derivation Functions,
December 2010.
[ANS X9.42] ANS X9.42-2003, Public Key Cryptography for the Financial Services Industry:
Agreement of Symmetric Keys Using Discrete Logarithm Cryptography[ANS X9.62] ANS X9.62-2005, Key Cryptography for the Financial Services Industry: Elliptic
Curve Digital Signature Algorithm (ECDSA).
[ANS X9.63] ANS X9.63-2011, Key Cryptography for the Financial Services Industry: Public
Key Cryptography for the Financial Services Industry: Key Agreement and Key
Transport Using Elliptic Curve Cryptography.

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A.2 Informative References[BM 1998]

S. Blake-Wilson, A. Menezes, Unknown Key-Share Attacks on the Station-toStation (STS) Protocol, Technical Report CORR 98-42, University of Waterloo,
1998. Available at: http://cacr.math.uwaterloo.ca.

[CMU 2009] S. Chatterjee, A. Menezes, and B. Ustaoglu,Reusing Static Keys in Key
Agreement Protocols, INDOCRYPT 2009, LNCS Vol. 5922, pp. 39–56,
Springer-Verlag, 2009. Available at:
http://www.cacr.math.uwaterloo.ca/techreports/2009/cacr2009-36.pdf .
[CBH 2005]

K. R. Choo, C. Boyd, and Y. Hitchcock, On Session Key Construction in
Provably-Secure Key Establishment Protocols, LNCS, Vol. 3715, pp. 116-131,
Springer-Verlag, 2005. Extended version available at:
http://eprint.iacr.org/2005/206.pdf.

[Menezes 2007] A. Menezes, Another look at HMQV. Journal of Mathematical Cryptology,
Vol.1(1), pp. 47-64, 2007
[RBB 2001] P. Rogaway, M. Bellare, D. Boneh, Evaluation of Security Level of
Cryptography: ECMQVS (from SEC 1), Jan. 2001. Available at:
http://www.ipa.go.jp/security/enc/CRYPTREC/fy15/doc/1069_ks-ecmqv.pdf.

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Appendix B: Rationale for Including Identifiers in the KDF Input
It is strongly recommended that identifiers for both parties to a key-agreement transaction be
included among the data input to the key-derivation method – as a simple and efficient means ofbinding those identifiers to the derived keying material. (See Sections 5.8.)
The inclusion of sufficiently-specific identifiers for party U and party V provides assurance thatthe keying material derived by those parties will be different from the keying material that isderived by other parties (or by the same parties acting in opposite roles). As a result, keyagreement schemes gain resilience against unknown-key-share attacks and/or other exploitation
techniques that depend on some type of confusion over the role played by each party (e.g., party
U versus party V). See, for example, references [CBH 2005], [Menezes 2007], [RBB 2001], [BM1998], and [CMU 2009], which all recommend the inclusion of identifiers in the key derivation
method as a means of eliminating certain vulnerabilities.
In addition to identifiers, the inclusion of other context-specific information in the key-derivation
input data can be used to draw finer distinctions between key-agreement transactions, providing
assurance that parties will not derive the same keying material unless they agree on all of the
included information. This can protect against attacks that rely on confusion concerning thecontext in which key-establishment takes place and/or how the derived keying material is to be
used, see [CMU 2009]. Examples of additional context-specific information include (but are notlimited to) the protocol employing the key-derivation method, protocol-defined session numbers,
the key-agreement scheme that was employed to produce the shared secret Z, any ephemeral
public keys and/or nonces exchanged during the key-agreement transaction, the bit length of thederived keying material, and its intended use.
Protocols employing an approved key-agreement scheme may employ alternative methods tobind participant identifiers (and/or other context-specific data) to the derived keying material orotherwise provide assurance that the participants in a key-agreement transaction share the sameview of the context in which the keying material was established (including their respective roles
and identifiers). However, this Recommendation makes no statement as to the adequacy of theseother methods.

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Appendix C: Data Conversions (Normative)
C.1

Integer-to-Byte String Conversion

Input:

A non-negative integer C and the intended length n of the byte string satisfying
28n > C.

Output:A byte string S of length n bytes.

1. Let S1, S2,…, Sn be the bytes of S from leftmost to rightmost, where
Si = (ai1, ai2, ai3, ai4, ai5, ai6, ai7, ai8), for i =1 to n.
2. Assume thatai = ai1 27+ai2 26+ai3 25+ai4 24+ai5 23+ai6 22+ai7 2+ai8, for i =1 to n.
3. The bytes of S shall satisfy:
C =a128(n-1)+a228(n-2)+…+an-128+an.
C.2

Field-Element-to-Byte String Conversion

Input:

An element α in the field Fq.

Output:A byte string S of length n = t / 8 bytes, where t = log2 q.

1. If q is an odd prime, then α must be an integer in the interval [0, q-1]; α shall be
converted to a byte string of length n bytes using the technique specified in Appendix C.1
above.
2. If q = 2m, then we assume that α is (already) represented as a bit string of length m, witheach bit indicating the coefficient (0 or 1) of a specific element of a particular basis forGF(2m) viewed as a vector space over GF(2). Let s1, s2, …, sm be the bits of α fromleftmost to rightmost. Let S1, S2, …, Sn be the bytes of S from leftmost to rightmost. The
rightmost bit sm shall become the rightmost bit of the last byte Sn, and so on through theleftmost bit s1, which shall become the (8n - m + 1)th bit of the first byte S1. The leftmost(8n - m) bits of the first byte S1 shall be zero.
C.3

Field-Element-to-Integer Conversion

Input:

An element α in the field Fq.

Output:An integer x.

1. If q is an odd prime, then x = α (no conversion is required).
2. If q = 2m, then α must be a bit string of length m bits. Let s1, s2, …, sm be the bits of α
from leftmost to rightmost. α shall be converted to an integer x satisfying:x=Σ2(m-i) si

for i = 1 to m.
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Appendix D: Revisions (Informative)
The original version of this document was published in March, 2006. In March, 2007, thefollowing revision was made to allow the dual use of keys during certificate requests:
In Section 5.6.4.2, the second item was originally as follows:“A static key pair may be used in more than one key-establishment scheme. However,
one static public/private key pair shall not be used for different purposes (for example, adigital signature key pair is not to be used for key establishment or vice versa).”
The item was changed to the following, where the changed text is indicated in italics:“A static key pair may be used in more than one key-establishment scheme. However,
one static public/private key pair shall not be used for different purposes (for example, adigital signature key pair is not to be used for key establishment or vice versa) with thefollowing possible exception: when requesting the (initial) certificate for a public static
key-establishment key, the key establishment private key associated with the public key
may be used to sign the certificate request. See SP 800-57, Part 1 on Key Usage forfurther information.”

In May 2013 version, the following revisions were made;•

Abstract – The March 2007 version cites ANS X9.42 and X9.63; this version directly
provides information on the key establishment schemes (DH, MQV) and the underlying
mathematics structure (discrete logs on finite field, elliptic curve).

•

Section 3.1 – Added definitions of assumption, binding, bit string, byte, byte string,
destroy, key-establishment pair, key-wrapping key, trusted association; removed
definitions on assurance of identifier, initiator, responder, (instead initiator and responder,
all the schemes are defined in terms of party U and party V, see revision in Section 4),
extended keying material to derived keying material (derived from the shared secret) andtransported keying material (generated by the sendor in a key-transport scheme.)•

Section 3.2 – The notations, C(ie), C(ie, js), MAC(MacKey, MacData), MacTag,
T_bitlen(X), were introduced; the notation |x | is removed.

•

Section 3.2 – Notations Z, Ze, Zs are used for both FFC and ECC and therefore moved up
as general notations.

•

Section 3.2 – The terms GF(p), GF(p)* were introduced for FFC.

•

Section 4 – Used party U and party V to name the parties, rather than user the initiatorand responder as the parties. Discussions about identifiers vs. identity and binding havebeen moved to Section 4.1.

•

Section 4.1 – Added discussions on the concept of a trusted association;•

Section 5 – Table 1 in March 2007 version has been removed; the information is now
provided in Tables 6 and 7 in Section 5.8.1, and Tables 8 and 9 in Section 5.9.3.

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Section 5.2 – Provided more details on MAC inputs (MacKey and MacData). Added textthat MACs can be used for key derivation, as well as key-confirmation. Added SP 80038B (CMAC) as an approved MAC. Refers to the new Tables 6 and 7.

•

Section 5.2.1 - MacLen now is a parameter, rather than an input variable. Refers to new
Tables 8 and 9, instead of old Tables 1 and 2. Discusses the truncation of the MAC
output.

•Section 5.4 – More discussion has been added about the use of nonces, including new
requirements and recommendations.

•

Section 5.5.1.1 – More information is provided about finite fields. Added therequirement that the leftmost bit of p and q be a 1. Table 1 has been shortened to address
just the values of p and q; information about the hash function is now provided in Tables
6 and 7 of Section 5.8.1, and in Tables 8 and 9 of Section 5.9.3.

•

Section 5.5.1.2 – More information is provided about elliptic curves. More details areprovided on parameter values. Table 2 has been shorted to just address n and h;
information about the hash function is now provided in Tables 6 and 7 of Section 5.8.1,
and in Tables 8 and 9 of Section 5.9.3.

•

Section 5.5.2 – A note about parameters generated by using SHA-1has been removed.
The validation methods are referred to other documents (FIPS 186 and ANS X9.62). It is
not a right place for such statement.

•

Section 5.6 has been reorganized to make it clearer to understand key generation andobtaining the required assurances.

•

Section 5.6.1.1 – FFC key-pair generation has been revised to require a randomly
selected integer in the range in the range [2, q−2], rather than requiring a private key for
FFC key pair generation to be unpredictable and generated by an approved RNG.
Generation in accordance with FIPS 186-3 (as referenced therein) fulfills theserequirements.

•

Section 5.6.1.2 – ECC key-pair generation has been revised to require a randomly
selected integer in the range in the range [2, n−2], rather than requiring a private key for
ECC key pair generation to be unpredictable and generated by an approved RNG.
Generation in accordance with FIPS 186-3 (as referenced therein) fulfills theserequirements.

•

New Section 5.6.2 – Discusses assurances and why they are required. Added Tables 3, 4,
and 5 which summarize types of assurance.

•

New Section 5.6.2.1 – Discusses the assurances required by a key-pair owner about its
own key pair, including owner assurance of correct generation, static and ephemeral
public-key validity, pair-wise consistency and private-key possession.

•

New Section 5.6.2.2 – Discusses the assurances required by a public-key recipient,including static and ephemeral public-key validity, and static and ephemeral private-key
possession.

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New Sections 5.6.3.2 and 5.6.3.3 – Different requirements are included for static andephemeral key pairs. Included a case that an agent may act on behalf of a system user.

•

Section 5.7 – Added requirements to destroy all values if there is an error and to destroy
intermediate calculations have been added for each FFC and ECC primitive. Conversion
calls have been added to convert to a string. Note that this removed such statements forthe action steps for each scheme in Section 6.

•

Section 5.8 – Key derivation has been divided into single-step key derivation methods(Section 5.8.1), an extract-then-expand key-derivation procedure (Section 5.8.2) andapplication-specific key-derivation methods (Section 5.8.3).

•

Section 5.8.1 – Instead of using a hash function, the one-step method is now defined witha function H, which can be a hash function or an HMAC with an approved hash function.
Added tables defining minimum length for the hash functions with regard to eachparameter set; and added more complete discussions about OtherInfo, including theconcatenation and ASN.1 formats included in the previous version. HMAC with anapproved hash function is now approved for key derivation, in addition to the hashfunction specified in the previous version.

•

Section 5.8.1 – Split Table 1 (for FFC) to Table 1 (Section 5.5.1.1), Table 6 (Section
5.8.1) and Table 8 (Section 5.9.3), where Table 1 is for FFC parameter-size sets, Table 6
is for the function H used for key derivation and Table 8 is about the MAC key lengthand MAC tag length. In the new tables, added row on “Maximum security strength
supported”.

•

Section 5.8.1 – In Table 6, changed the minimum output length for function H from 128
to 112 for FFC parameter set.

•

Section 5.8.1 - Split Table 2 (for ECC) to Table 2 (Section 5.5.1.2), Table 7 (Section
5.8.1) and Table 9 (Section 5.9.3), where Table 2 is for ECC parameter-size sets, Table 7
is for the function H used for key derivation, and Table 9 is about the MAC key lengthand MAC tag length. In the new tables, added row on “Maximum security strength
supported”.

•

Section 5.8.2 – Added reference to an approved two-step method – an extraction-thenexpansion method – that is specified in SP 800-56C.

•

Section 5.8.3 – Added reference to the application-specific key derivation methods
provided in SP 800-135.

•

Moved general introduction of key-confirmation to Section 5.9 – Incorporates thematerial from Section 8 (with additional introductory material).

•

New Section 5.9.1.1 – Emphasizes more clearly that a nonce is required if there is noephemeral key; added guidance on what to do if key-confirmation fails.

•

New Section 5.9.2 – Emphasizes that if no ephemeral key is used, then a nonce isrequired.

•

New Section 5.9.3 – Discussions about the security strength of the MacTag are provided,
along with tables on the minimum MacKey length and MacLen values.

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New Section 5.9.3 – Table 8, changed the minimum MacLen, that is, MacTag length to64 bits for all the parameter sets of FFC.

•New Section 5.9.3 – Table 9, changed the minimum MacLen, that is, MacTag length to64 bits for all the parameter sets of ECC.

•

Section 6 – The notation C(ie) replaces C(i), and C(ie, js) replaces C(i, j). If party U does
not contribute a static key, then the requirement for a non-null identifier is now
transaction dependent, rather than required. Rationale for choosing the C(ie, js) schemes
has been moved to a new Section 8, instead of after each class of schemes. Assumptions
are specified for each type of scheme, rather than prerequisites.

•

Section 6.1.1 (and similarly for Sections 6.2.1, 6.2.2 and 6.3) –Added a new assumptionthat if an identifier is used as a label, then the identifier must have a trusted association tothat party’s static key. The discussion on the need for a trusted association has been
added.

•

Section 6.1.1.1 (dhHybrid1) – More guidance is provided about error handling.
Specifically allows the reuse of an ephemeral key pair in a broadcast scenario. This isalso provided in Sections 6.1.1.2, 6.1.1.3 and 6.1.1.4.

•

New Section 6.1.1.5 (and similarly in new Sections 6.1.2.3, 6.2.1.5, 6.2.2.3 and 6.3.3) –Key-confirmation is incorporated to each applied subcategory of schemes. This material
was previously provided in Section 8.4 of the previous version.

•

Section 6.2.1 (C(1e,2s) schemes) – Added additional assumptions which were included in
the previous prerequisites. This includes obtaining assurance of static public key validity
and private keys possession of the key-pair owner.

•

Section 7 – Has been revised to specify DLC-based key-agreement and key-transport in
the same key-establishment transaction, with party U acting as the key-transport sender.
In addition, optional key-confirmation from party V to party U following the keytransport process has been specified.

•

Section 8 – The rationale for choosing each scheme type has been moved from Section 6of the previous version. A new section on the rationale associated with key-transport has
been included.

•

All figures are replaced to reflect the content, text, and terminology changes.

•

Old Appendix A, Summary of Differences between this Recommendation and ANS X9Standards, was removed. Note that X9.42 was withdrawn, while X9.63 has modified tobe consistent with this Recommendation.

•

Appendix B – The requirement of including identifiers as part of the OtherInfo isreplaced with text that. it is strongly recommended that identifiers for both parties to a
key-agreement transaction be included among the data input to a key-derivation method.
A paragraph has been added stating that there may be other ways to bind identifiers toderived keying material, but the recommendation makes no statement on the adequacy of
this.

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The new Appendix A includes all the informative references, which was in Appendix D
in March 2007 version.

•

The old Appendix E becomes Appendix D and the changes on March 2007 version areadded as listed here.

127