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NIST Special Publication 800-56A

Recommendation for Pair-Wise Key Establishment Schemes Using

Discrete Logarithm Cryptography

March 2006

March 2007

SP 800-56A is superseded in its entirety by the publication of

SP 800-56A Revised (March 2007).

NIST Special Publication 800-56A Revised

Recommendation for Pair-Wise Key Establishment Schemes Using

Discrete Logarithm Cryptography (Revised)

Elaine Barker, Don Johnson, Miles Smid

March 2007

http://dx.doi.org/10.6028/NIST.SP.800-56ar

Computer Security Division (Information Technology Lab)

SP 800-56A Revision 2 (as of August 7, 2015)

http://csrc.nist.gov/groups/ST/toolkit/key_management.html

N/A
C O M P U T E R S E C U R I T Y 

Recommendation for Pair-Wise 

Key Establishment Schemes 

Using Discrete Logarithm 

Cryptography

Elaine Barker, Don Johnson, and Miles Smid 

NIST Special Publication 800-56A 

March, 2006 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

2

Abstract 

This Recommendation specifies key establishment schemes using discrete logarithm 

cryptography, based on standards 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). 

KEY WORDS: assurances; Diffie-Hellman; elliptic curve cryptography; finite field 

cryptography; key agreement; key confirmation; key derivation; key establishment; key 

management; key recovery; key transport; MQV. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

3

Acknowledgements 

The National Institute of Standards and Technology (NIST) gratefully acknowledges and 

appreciates contributions by Rich Davis, Mike Hopper and Laurie Law from the National 

Security Agency concerning the many security issues associated with this Recommendation. 

NIST also thanks the many contributions by the public and private sectors whose thoughtful and 

constructive comments improved the quality and usefulness of this publication.
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

4

Authority 

This document has been developed by the National Institute of Standards and Technology 

(NIST) in furtherance of its statutory responsibilities under the Federal Information Security 

Management Act (FISMA) of 2002, Public Law 107-347. 

NIST is responsible for developing standards and guidelines, including minimum requirements, 

for providing adequate information security for all agency operations and assets, but such 

standards and guidelines shall not apply to national security 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 A-130, Appendix IV: Analysis of 

Key Sections. Supplemental information is provided in A-130, Appendix III. 

This Recommendation has been prepared for use by federal agencies. It may be used by 

nongovernmental organizations on a voluntary basis and is not subject to copyright. (Attribution 

would be appreciated by NIST.) 

Nothing in this document should be taken to contradict 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 of Commerce, Director of the OMB, or any other federal official. 

Conformance testing for implementations of key establishment schemes, as specified in this 

Recommendation, will be conducted within the framework of the Cryptographic Module 

Validation Program (CMVP), a joint effort of NIST and the Communications Security 

Establishment of the Government of Canada. An implementation of a key establishment scheme 

must adhere to the requirements in this Recommendation in order to be validated under the 

CMVP. The requirements of this Recommendation are indicated by the word “shall.” 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

5

Table of Contents 

1. Introduction............................................................................................................ 10

2. Scope and Purpose ............................................................................................... 11

3. Definitions, Symbols and Abbreviations ............................................................. 12

3.1 Definitions...................................................................................................................... 12 

3.2 Symbols and Abbreviations ........................................................................................... 16 

4. Key Establishment Schemes Overview ............................................................... 20

4.1 Key Agreement Preparations by an Owner ................................................................... 21 

4.2 Key Agreement Process................................................................................................. 24 

4.3 DLC-based Key Transport Process................................................................................ 25 

5. Cryptographic Elements ....................................................................................... 26

5.1 Cryptographic Hash Functions ...................................................................................... 26 

5.2 Message Authentication Code (MAC) Algorithm......................................................... 26 

5.2.1 MacTag Computation ........................................................................................ 26 

5.2.2 MacTag Checking.............................................................................................. 27 

5.2.3 Implementation Validation Message ................................................................. 27 

5.3 Random Number Generation ......................................................................................... 27 

5.4 Nonces........................................................................................................................... 27 

5.5 Domain Parameters........................................................................................................ 28

5.5.1 Domain Parameter Generation........................................................................... 28 

5.5.1.1 FFC Domain Parameter Generation.................................................... 28 

5.5.1.2 ECC Domain Parameter Generation................................................... 29 

5.5.2 Assurances of Domain Parameter Validity........................................................ 30 

5.5.3 Domain Parameter Management........................................................................ 31 

5.6 Private and Public Keys................................................................................................. 31 

5.6.1 Private/Public Key Pair Generation................................................................... 31 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

6

5.6.1.1 FFC Key Pair Generation.................................................................... 31 

5.6.1.2 ECC Key Pair Generation................................................................... 32 

5.6.2 Assurances of the Arithmetic Validity of a Public Key..................................... 32 

5.6.2.1 Owner Assurances of Static Public Key Validity............................... 32 

5.6.2.2 Recipient Assurances of Static Public Key Validity........................... 33 

5.6.2.3 Recipient Assurances of Ephemeral Public Key Validity .................. 33 

5.6.2.4 FFC Full Public Key Validation Routine............................................ 34 

5.6.2.5 ECC Full Public Key Validation Routine........................................... 34 

5.6.2.6 ECC Partial Public Key Validation Routine....................................... 35 

5.6.3 Assurances of the Possession of a Static Private Key........................................ 36 

5.6.3.1 Owner Assurances of Possession of a Static Private Key................... 37 

5.6.3.2 Recipient Assurance of Owner’s Possession of a Static Private Key. 38 

5.6.3.2.1 Recipient Obtains Assurance through a Trusted Third Party ............. 38 

5.6.3.2.2 Recipient Obtains Assurance Directly from the Claimed Owner....... 38 

5.6.4 Key Pair Management........................................................................................ 39 

5.6.4.1 Common Requirements on Static and Ephemeral Key Pairs.............. 39 

5.6.4.2 Specific Requirements on Static Key Pairs ........................................ 40 

5.6.4.3 Specific Requirements on Ephemeral Key Pairs ................................ 40 

5.7 DLC Primitives .............................................................................................................. 41 

5.7.1 Diffie-Hellman Primitives ................................................................................. 41 

5.7.1.1 Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive........ 41 

5.7.1.2 Elliptic Curve Cryptography Cofactor Diffie-Hellman (ECC CDH) 

Primitive 42 

5.7.2 MQV Primitives................................................................................................. 42 

5.7.2.1 Finite Field Cryptography MQV (FFC MQV) Primitive ................... 42 

5.7.2.1.1 MQV2 Form of the FFC MQV Primitive............................ 43 

5.7.2.1.2 MQV1 Form of the FFC MQV Primitive............................ 43 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

7

5.7.2.2 ECC MQV Associate Value Function ................................................ 44 

5.7.2.3 Elliptic Curve Cryptography MQV (ECC MQV) Primitive............... 44 

5.7.2.3.1 Full MQV Form of the ECC MQV Primitive...................... 45 

5.7.2.3.2 One-Pass Form of the ECC MQV Primitive........................ 45 

5.8 Key Derivation Functions for Key Agreement Schemes............................................... 46 

5.8.1 Concatenation Key Derivation Function (Approved Alternative 1).................. 46 

5.8.2 ASN.1 Key Derivation Function (Approved Alternative 2).............................. 49 

6. Key Agreement....................................................................................................... 51

6.1 Schemes Using Two Ephemeral Key Pairs, C(2) .......................................................... 54 

6.1.1 Each Party Has a Static Key Pair and Generates an Ephemeral Key Pair, C(2, 2)

............................................................................................................................ 54 

6.1.1.1 dhHybrid1, C(2, 2, FFC DH).............................................................. 56 

6.1.1.2 Full Unified Model, C(2, 2, ECC CDH)............................................. 58 

6.1.1.3 MQV2, C(2, 2, FFC MQV) ................................................................ 59 

6.1.1.4 Full MQV, C(2, 2, ECC MQV) .......................................................... 61 

6.1.1.5 Rationale for Choosing a C(2, 2) Scheme .......................................... 63 

6.1.2 Each Party Generates an Ephemeral Key Pair; No Static Keys are Used, C(2, 0)

............................................................................................................................ 63 

6.1.2.1 dhEphem, C(2, 0, FFC DH)................................................................ 64 

6.1.2.2 Ephemeral Unified Model, C(2, 0, ECC CDH).................................. 66 

6.1.2.3 Rationale for Choosing a C(2, 0) Scheme .......................................... 68 

6.2 Schemes Using One Ephemeral Key Pair, C(1) ............................................................ 68 

6.2.1 Initiator Has a Static Key Pair and Generates an Ephemeral Key Pair; 

Responder Has a Static Key Pair, C(1, 2).......................................................... 68 

6.2.1.1 dhHybridOneFlow, C(1, 2, FFC DH) ................................................. 70 

6.2.1.2 One-Pass Unified Model, C(1, 2, ECC CDH) .................................... 72 

6.2.1.3 MQV1, C(1, 2, FFC MQV) ................................................................ 75 

6.2.1.4 One-Pass MQV, C(1, 2, ECC MQV).................................................. 77 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

8

6.2.1.5 Rationale for Choosing a C(1, 2) Scheme .......................................... 79 

6.2.2 Initiator Generates Only an Ephemeral Key Pair; Responder Has Only a Static 

Key Pair, C(1, 1) ................................................................................................ 80 

6.2.2.1 dhOneFlow, C(1, 1, FFC DH) ............................................................ 81 

6.2.2.2 One-Pass Diffie-Hellman, C(1, 1, ECC CDH) ................................... 83 

6.2.2.3 Rationale in Choosing a C(1, 1) Scheme............................................ 85 

6.3 Scheme Using No Ephemeral Key Pairs, C(0, 2) .......................................................... 86 

6.3.1 dhStatic, C(0, 2, FFC DH) ................................................................................. 87 

6.3.2 Static Unified Model, C(0, 2, ECC CDH) ......................................................... 89 

6.3.3 Rationale in Choosing a C(0, 2) Scheme........................................................... 91 

7. DLC based Key Transport..................................................................................... 91

8. Key Confirmation................................................................................................... 93

8.1 Assurance of Possession Considerations when using Key Confirmation...................... 94 

8.2 Unilateral Key Confirmation for Key Agreement Schemes .......................................... 95 

8.3 Bilateral Key Confirmation for Key Agreement Schemes ............................................ 97 

8.4 Incorporating Key Confirmation into a Key Agreement Scheme ................................. 97 

8.4.1 C(2, 2) Scheme with Unilateral Key Confirmation Provided by U to V........... 97 

8.4.2 C(2, 2) Scheme with Unilateral Key Confirmation Provided by V to U........... 99 

8.4.3 C(2, 2) Scheme with Bilateral Key Confirmation ............................................. 99 

8.4.4 C(1, 2) Scheme with Unilateral Key Confirmation Provided by U to V......... 100 

8.4.5 C(1, 2) Scheme with Unilateral Key Confirmation Provided by V to U......... 101 

8.4.6 C(1, 2) Scheme with Bilateral Key Confirmation ........................................... 102 

8.4.7 C(1, 1) Scheme with Unilateral Key Confirmation Provided by V to U......... 103 

8.4.8 C(0, 2) Scheme with Unilateral Key Confirmation Provided by U to V......... 104 

8.4.9 C(0, 2) Scheme with Unilateral Key Confirmation Provided by V to U......... 105 

8.4.10 C(0, 2) Scheme with Bilateral Key Confirmation ........................................... 106 

9. Key Recovery ....................................................................................................... 107
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

9

10.Implementation Validation .................................................................................. 108

Appendix A: Summary of Differences between this Recommendation and ANS X9 

Standards (Informative)....................................................................................... 110

Appendix B: Rationale for Including Identifiers in the KDF Input......................... 113

Appendix C: Data Conversions (Normative)........................................................... 114

C.1 Integer-to-Byte String Conversion............................................................................... 114 

C.2 Field-Element-to-Byte String Conversion ................................................................... 114 

C.3 Field-Element-to-Integer Conversion .......................................................................... 114 

Appendix D: References (Informative) .................................................................... 115

Figures 

Figure 1: Owner Key Establishment Preparations.........................................................................23 

Figure 2: Key Agreement Process .................................................................................................24 

Figure 3: Key Transport Process....................................................................................................25 

Figure 4: General Protocol when Each Party Generates Both Static and Ephemeral Key 

Pairs ................................................................................................................................55 

Figure 5: General Protocol when Each Party Generates Ephemeral Key Pairs; No Static Keys are 

Used................................................................................................................................64 

Figure 6: General Protocol when the Initiator has both Static and Ephemeral Key Pairs, and the 

Responder has only a Static Key Pair.............................................................................69 

Figure 7: General Protocol when the Initiator has Only an Ephemeral Key Pair, and the 

Responder has Only a Static Key Pair............................................................................80 

Figure 8: General Protocol when Each Party has only a Static Key Pair ......................................86 

Figure 9: C(2, 2) Scheme with Unilateral Key Confirmation from Party U to Party V ................98 

Figure 10: C(2, 2) Scheme with Unilateral Key Confirmation from Party V to Party U ..............99 

Figure 11: C(2, 2) Scheme with Bilateral Key Confirmation......................................................100 

Figure 12: C(1, 2) Scheme with Unilateral Key Confirmation from Party U to Party V ............101 

Figure 13: C(1, 2) Scheme with Unilateral Key Confirmation from Party V to Party U ............102 

Figure 14: C(1, 2) Scheme with Bilateral Key Confirmation......................................................103 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

10

Figure 15: C(1, 1) Scheme with Unilateral Key Confirmation from Party V to Party U ............104 

Figure 16: C(0, 2) Scheme with Unilateral Key Confirmation from Party U to Party V ............105 

Figure 17: C(0, 2) Scheme with Unilateral Key Confirmation from Party V to Party U ............106 

Figure 18: C(0, 2) Scheme with Bilateral Key Confirmation......................................................107 

Tables 

Table 1: FFC Parameter Size Sets .................................................................................................29 

Table 2: ECC Parameter Size Sets.................................................................................................30 

Table 3: Key Agreement Scheme Categories ................................................................................51 

Table 4: Key Agreement Scheme Subcategories...........................................................................52 

Table 5: Key Agreement Schemes.................................................................................................52 

Table 6: dhHybrid1 Key Agreement Scheme Summary ...............................................................57 

Table 7: Full Unified Model Key Agreement Scheme Summary..................................................59 

Table 8: MQV2 Key Agreement Scheme Summary .....................................................................61 

Table 9: Full MQV Key Agreement Scheme Summary................................................................62 

Table 10: dhEphem Key Agreement Scheme Summary ...............................................................66 

Table 11: Ephemeral Unified Model Key Agreement Scheme .....................................................67 

Table 12: dhHybridOneFlow Key Agreement Scheme Summary ................................................72 

Table 13: One-Pass Unified Model Key Agreement Scheme Summary.......................................74 

Table 14: MQV1 Key Agreement Scheme Summary ...................................................................77 

Table 15: One-Pass MQV Model Key Agreement Scheme Summary..........................................79 

Table 16: dhOneFlow Key Agreement Scheme Summary............................................................83 

Table 17: One-Pass Diffie-Hellman Key Agreement Scheme Summary......................................85 

Table 18: dhStatic Key Agreement Scheme Summary..................................................................88 

Table 19: Static Unified Model Key Agreement Scheme Summary.............................................91 

Table 20: Key Agreement Schemes Using Unilateral and Bilateral Key Confirmation ...............93 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

11

1. Introduction 

Many U.S. Government Information Technology (IT) systems need to employ well-established 

cryptographic 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) 800-

67, 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 shared secret keying material 

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 key establishment 

schemes. 

2. Scope and Purpose 

This Recommendation provides the specifications of key establishment schemes that are 

appropriate for use by the U.S. Federal Government, based on standards 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. A key establishment scheme can be characterized as either a 

key agreement scheme or a key transport scheme. The asymmetric-key-based key agreement 

schemes in this Recommendation are based on the Diffie-Hellman (DH) and Menezes-QuVanstone (MQV) algorithms. In addition, an asymmetric-key-based key transport scheme is 

specified. It is intended that an adjunct key establishment schemes Recommendation will contain 

key transport scheme(s) from ANS X9.44 Key Agreement and Key Transport using FactoringBased Cryptography, when they become available. 

This Recommendation provides a description of selected schemes from ANS X9 standards. 

When there are differences between this Recommendation and the referenced ANS X9 

standards, this key establishment schemes Recommendation shall have precedence for U.S. 

Government applications. 

This Recommendation is intended for use in conjunction with NIST Special Publication 800-57, 

Recommendation for Key Management [7]. This key establishment schemes Recommendation, 

the Recommendation for Key Management [7], and the referenced ANS X9 standards are 

intended to provide sufficient information for a vendor to implement secure key establishment 

using asymmetric algorithms in FIPS 140-2 [1] 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, are 

not specified in this Recommendation. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

12

3. Definitions, Symbols and Abbreviations 

3.1 Definitions 

Approved FIPS approved or NIST Recommended. An algorithm or technique that is 

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. 

Assurance of 

identifier 

Confidence that identifying information (such as a name) is correctly 

associated with an entity. 

Assurance of 

possession of a 

private key 

Confidence that an entity possesses a private key associated with a public 

key. 

Assurance of 

validity 

Confidence that either a key or a set of domain parameters is 

arithmetically correct. 

Bit length The length in bits of a bit string. 

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 the 

generator point specified in the domain parameters. 

Domain parameters The parameters used with a cryptographic algorithm that are common to a 

domain of users. 

Entity An individual (person), organization, device, or process. “Party” is a 

synonym. 

Ephemeral key A key that is intended for a very short period of use. The key is ordinarily 

used in exactly one transaction of a cryptographic scheme; an exception 

to this is when the ephemeral key is used in multiple transactions for a 

key transport broadcast. Contrast with static key. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

13

Hash function A function that maps a bit string of arbitrary length to a fixed length bit 

string. Approved hash functions satisfy the following properties: 

1. (One-way) It is computationally infeasible to find any input that 

maps 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. 

Approved hash functions are specified in FIPS 180-2 [2]. 

Identifier A bit string that is associated with a person, device or organization. It 

may be an identifying name, or may be something more abstract (for 

example, a string consisting of an IP address and timestamp). 

If a party owns a static key pair that is used in a key agreement 

transaction, then the identifier assigned to that party is one that is bound 

to that static key pair. If the party does not contribute a static public key 

as part of a key agreement transaction, then the identifier of that party is a 

non-null identifier selected in accordance with the protocol utilizing the 

scheme. 

Initiator The party that begins a key agreement transaction. Contrast with 

responder. 

Key agreement A key establishment procedure where the resultant secret keying material 

is a function of information contributed by two participants, so that no 

party can predetermine the value of the secret keying material 

independently from the contributions of the other parties. Contrast with 

key transport. 

Key agreement 

transaction 

The instance that results in shared secret keying material among different 

parties using 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) actually 

possesses the correct secret keying material and/or shared secret. 

Key derivation The process by which keying material is derived from a shared secret and 

other information. 

Key establishment The procedure that results in shared secret keying material among 

different parties. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

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Key establishment 

transaction 

An instance of establishing secret keying material using a key 

establishment scheme. 

Key transport A key establishment procedure whereby one party (the sender) selects a 

value for the secret keying material and then securely distributes that 

value to another party (the receiver). Contrast with key agreement. 

Key transport 

transaction 

The instance that results in shared secret keying material between 

different parties using a key transport scheme. 

Key wrap A method of encrypting keying material (along with associated integrity 

information) that provides both confidentiality and integrity protection 

using a symmetric key algorithm. 

Keying material The data that is necessary to establish and maintain a cryptographic 

keying relationship. Some keying material may be secret, while other 

keying material may be public. As used in this Recommendation, secret 

keying material may include keys, secret initialization vectors or other 

secret information; public keying material includes any non-secret data 

needed to establish a relationship. 

MacTag Data that allows an entity to verify the integrity of the information. Other 

documents sometimes refer to this data as a MAC. 

Message 

Authentication Code 

(MAC) algorithm 

Defines a family of one-way cryptographic functions that is 

parameterized by a symmetric key and produces a MacTag on arbitrary 

data. A MAC algorithm can be used to provide data origin authentication 

as well as data integrity. In this Recommendation, a MAC algorithm is 

used for key confirmation and validation testing purposes. 

Nonce A time-varying value that has at most a negligible chance of repeating, 

for example, a random value that is generated anew for each use, a 

timestamp, a sequence number, or some combination of these. 

Owner For a static key pair, the owner is the entity that is authorized to use the 

static private key associated with a 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, the owner is the entity that 

generated the key pair. 

Party An individual (person), organization, device, or process. “Entity” is a 

synonym for party. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

15

Provider The party during key confirmation that provides assurance to the other 

party (the recipient) that the two parties have indeed established a shared 

secret. 

Public key 

certificate 

A set of data that contains an entity’s identifier(s), the entity's public key 

(including an indication of the associated set of domain parameters, if 

any) and possibly other information, and is digitally signed by a trusted 

party, thereby binding the public key to the included identifier(s). 

Receiver The party that receives secret keying material via a key transport 

transaction. Contrast with sender. 

Recipient A party that receives (1) keying material: such as a static public key (e.g., 

in a certificate) or an ephemeral public key; (2) assurance: such as an 

assurance of the validity of a candidate public key or assurance of 

possession of the private key associated with a public key; or (3) key 

confirmation. Contrast with provider. 

Responder The party that does not begin a key agreement transaction. Contrast with 

initiator. 

Scheme A (cryptographic) scheme consists of an unambiguous specification of a 

set of 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. 

Security strength 

(Also “Bits of 

security”) 

A number associated with the amount of work (that is, the number of 

operations) that is required to break a cryptographic algorithm or system. 

Security properties The security features (e.g., entity authentication, playback protection, or 

key confirmation) that a cryptographic scheme may, or may not, provide. 

Sender The party that sends secret keying material to the receiver using a key 

transport transaction. 

Shall This term is used to indicate a requirement of a Federal Information 

processing Standard (FIPS) or a requirement that needs to be fulfilled to 

claim conformance to this Recommendation. Note that shall may be 

coupled with not to become shall not. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

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Shared secret keying 

material 

The secret keying material that is either (1) derived by applying the key 

derivation function to the shared secret and other shared information 

during a key agreement process, or (2) is transported during a key 

transport process. 

Shared secret A secret value that has been computed using a key agreement scheme and 

is used as input to a key derivation function. 

Should This term is used to indicate an important recommendation. Ignoring the 

recommendation could result in undesirable results. Note that should may 

be coupled with not to become should not. 

Static key A key that is intended for use for a relatively long period of time and is 

typically intended for use in many instances of a cryptographic key 

establishment scheme. Contrast with an ephemeral key. 

Symmetric key 

algorithm 

A cryptographic algorithm that uses one secret key that is shared between 

authorized parties. 

Trusted party A trusted party is a party that is trusted by an entity to faithfully perform 

certain services for that entity. An entity may choose to act as 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, the initiator and responder in a scheme are 

considered to be the first and second parties in a key establishment 

transaction.) 

3.2 Symbols and Abbreviations 

General: 

AES Advanced Encryption Standard (as specified in FIPS 197 [4]). 

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

ANS American National Standard. 

ASN.1 Abstract Syntax Notation One. 

CA Certification Authority. 
NIST SP 800-56A, Recommendation for Pair-Wise Key Establishment Schemes 

Using Discrete Logarithm Cryptography 

 March, 2006 

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CDH The cofactor Diffie-Hellman key agreement primitive. 

DH The (non-cofactor) 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 an elliptic 

curve. For example, see ANS X9.63 [12]. 

FF Finite Field. 

FFC Finite Field Cryptography, the public key cryptographic methods using a finite field. 

For example, see ANS X9.42 [10]. 

HMAC Keyed-hash Message Authentication Code (as specified in FIPS 198 [5]). 

ID The bit string denoting the identifier associated with an entity. 

H An Approved hash function. 

KC Key Confirmation. 

KDF Key Derivation Function. 

MAC Message Authentication Code. 

MQV The Menezes-Qu-Vanstone key agreement primitive. 

Null The empty bit string 

SHA Secure Hash Algorithm. 

TTP A Trusted Third Party. 

U The initiator of a key establishment process. 

V The responder in a key establishment process. 

{X} Indicates that the inclusion of X is optional. 
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X || Y Concatenation of two strings X and Y. 

|x| The length of x in bits. 

[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. 

The following notations for FFC and ECC are consistent with those used in the ANS X9.42 and 

ANS X9.63 standards; however, it should be recognized that the notation between the standards 

is inconsistent (for example, x and y are used as the private and public keys in ANS X9.42, 

whereas x and y are used as the coordinates of a point in ANS X9.63). 

FFC (ANS X9.42): 

g An FFC domain parameter; the generator of the subgroup of order q. 

mod p The reduction modulo p of an integer value.

p An FFC domain parameter; the (large) prime field order. 

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 arbitrarily. 

q An FFC domain parameter; the (small) prime multiplicative subgroup order.

rU, rV Party U or Party V’s ephemeral private key. These are integers in the range 

[1, q-1].

tU, tV Party U or Party V’s ephemeral public key. These are integers in the range 

[2, p-2], representing elements in the finite field of size p.

SEED An FFC domain parameter; an initialization value that is used during 

domain parameter generation that can also be used to provide assurance at a 

later time that the resulting domain parameters were generated arbitrarily. 

xU, xV Party U or Party V’s static private key. These are integers in the range 

[1, q-1]. 

yU, yV Party U or Party V’s static public key. These are integers in the range 

[2, p-2], representing elements in the finite field of size p.
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Z A shared secret that is used to derive secret keying material using a key 

derivation function. 

Ze An ephemeral shared secret that is computed using the Diffie-Hellman 

primitive.

Zs A static shared secret that is computed using the Diffie-Hellman primitive.

ECC (ANS X9.63): 

a, b An ECC domain parameter; two field elements that define the equation of an 

elliptic curve. 

avf(Q) The associate value of the elliptic curve point Q. 

de,U, de,V Party U’s and Party V’s ephemeral private keys. These are integers in the range 

[1, n-1]. 

ds,U, ds,V Party U’s and Party V’s static private keys. These are integers in the range 

[1, n-1]. 

FR Field Representation indicator. An indication of the basis 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 is the reduction polynomial (a trinomial or a pentanomial). See 

[12] for details. 

G An ECC domain parameter, which is a distinguished point on an elliptic curve 

that generates the subgroup of order n. 

h An ECC domain parameter, the cofactor, which is the order of the elliptic curve 

divided by the order of the point G. 

n An ECC domain parameter; the order of the point G. 

O The point at infinity; a special point in an elliptic curve group that serves as the 

(additive) identity. 

q An ECC domain parameter; the field size. 

Qe,U, Qe,V Party U’s and Party V’s ephemeral public keys. These are points on the elliptic 

curve defined by the domain parameters. 
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Qs,U, Qs,V Party U’s and Party V’s static public keys. These are points on the elliptic curve 

defined by the domain parameters. 

SEED An ECC domain parameter; an initialization value that is used during domain 

parameter generation that can also be used to provide assurance at a later time 

that the resulting domain parameters were generated arbitrarily. 

xP, yP Elements of the finite field of size q, representing the x and y coordinates, 

respectively, of a point P. These are integers in the interval [0, p-1] in the case 

that q is an odd prime p, or are bit strings of length m bits in the case that q = 2m. 

Z A shared secret that is used to derive secret keying material using a key 

derivation function. 

Ze An ephemeral shared secret that is computed using the Diffie-Hellman primitive. 

Zs A static shared secret that is computed using the Diffie-Hellman primitive. 

4. Key Establishment Schemes Overview 

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 key agreement (where both parties contribute to the shared secret and, therefore, the 

derived secret keying material), the secret keying material to be established is not sent directly; 

rather, information is exchanged between both parties that allows each party to derive the secret 

keying material. Key agreement schemes may use either symmetric key or asymmetric key 

(public key) techniques. The key agreement schemes described in this Recommendation use 

public key techniques. The party that begins a key agreement scheme is called the initiator, and 

the other party is called the responder. 

During key transport (where one party selects the secret keying material), wrapped (that is, 

encrypted) secret keying material is transported from the sender to the receiver. Key transport 

schemes may use either symmetric key or public key techniques; only key transport schemes 

based on Discrete Logarithm Cryptography (DLC) cryptography are described in this 

Recommendation. The party that sends the secret keying material is called the sender, and the 

other party is called the receiver. 

The security of the DLC schemes in this Recommendation is based on the intractability of the 

discrete logarithm problem. The schemes calculated over a finite field (FF) are based on ANS 

X9.42. The schemes calculated using elliptic curves (EC) are based on ANS X9.63. 
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This Recommendation specifies several processes that are associated with key establishment 

(including processes for generating domain parameters and for deriving secret keying material 

from a shared secret). In each case, equivalent processes may be used. Two processes 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. Some processes are used 

to provide assurance (for example, 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 other party is called the recipient 

(of the assurance). 

Note that the terms initiator, responder, sender, receiver, provider and recipient have specific 

meanings in this Recommendation. 

A number of steps are performed to establish secret keying material as described in Sections 4.1 

and 4.2. 

4.1 Key Agreement Preparations by an Owner 

The owner of a private/public key pair is the entity that is authorized to use the private key of 

that key pair. Figure 1 depicts the steps that may be required of that entity when preparing for a 

key agreement process. 

The first step is to obtain appropriate domain parameters that are generated as specified in 

Section 5.5.1; either the entity itself generates the domain parameters, or the entity obtains 

domain parameters that another entity has generated. Having obtained the domain parameters, 

the entity obtains assurance of the validity of those domain parameters; approved methods for 

obtaining this assurance are provided in Section 5.5.2. 

If the entity will be using a key establishment scheme that requires that the entity have a static 

key pair, the entity obtains this key pair. Either the entity generates the key pair as specified in 

Section 5.6.1 or a trusted party generates the key pair as specified in Section 5.6.1 and provides it 

to the entity. The entity (i.e., the owner) obtains assurance of the validity of its static public key 

and also obtains assurance that it actually possesses the (correct) static private key. Approved 

methods for obtaining assurance of public key validity by the owner are addressed in Section 

5.6.2.1; approved methods for an owner to obtain assurance of the actual possession of the 

private key are provided in Section 5.6.3.1. 

An identifier (see Section 3.1) is used to label the entity that is authorized to use the static private 

key corresponding to a particular static public key (i.e., the identifier labels the key pair’s 

owner). This label may uniquely distinguish the entity from all others, in which case it could 

rightfully be considered 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. A key pair’s owner is responsible for ensuring that the identifier 

associated with its static public key is appropriate for the applications in which it will be used. 
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This Recommendation assumes that there is a trustworthy binding of each entity’s identifier to 

the entity’s static public key. The binding of an identifier to a static public key may be 

accomplished by a trusted authority (i.e., a binding authority; for example, a registration 

authority working with a CA who creates a certificate containing both the static public key and 

the identifier). The binding authority verifies the identifier chosen for the owner. The binding 

authority is also responsible for obtaining assurance of: the validity of the domain parameters 

associated with the owner’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, for 

example, Section 5.5.2, Section 5.6.2.2 [method 1], and Section 5.6.3.2.2, where the binding 

authority acts as the recipient of the static public key.) Binding Authorities shall obtain 

assurance of possession either by using one of the methods specified in Section 5.6.3.2.2 or by 

using an Approved alternative. 
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Owner obtains

Assurance of Possession

of the 

Static Private Key

(5.6.3.1)

Owner obtains

Assurance of

Public Key Validity

(5.6.2.1)

Obtain

Static Key Pair

(5.6.1)

Obtain Assurance of

Domain Parameter 

Validity

(5.5.2)

Obtain

Domain Parameters

(5.5.1)

Owner Ready for Key Establishment

Entity itself

generates

Another entity

generates

Owner

generates

TTP

generates

Depending

Provide

Assurance of Possession 

and Identifier to a

Binding Authority

Figure 1: Owner Key Establishment Preparations 

After the above steps have been performed, the entity (i.e., the static key pair owner) is ready to 

enter into a key establishment process with another compatibly prepared entity. 
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4.2 Key Agreement Process 

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 this 

Recommendation; however, some discrepancies in order may occur, depending on the 

communication protocol in which the key agreement process is performed. Depending on the key 

agreement scheme and the available keys, either entity could be the key agreement initiator. Note 

that some of the shown actions may not be a part of some schemes. For example, key 

confirmation is optional (see Section 8). The specifications of this recommendation indicate 

when a particular action is required. 

Figure 2: Key Agreement Process 

Each entity obtains the identifier associated with the other entity, and verifies that the identifier 

of the other entity corresponds to the entity with whom the participant wishes to establish secret 

keying material. 

Each entity that requires the other entity’s static public key for use in the key establishment 

scheme obtains that public key and obtains assurance of its validity. Approved methods for 

obtaining assurance of the validity of a static public key are provided in Section 5.6.2.2. 
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Each entity that requires the other entity’s ephemeral public key for use in the key establishment 

scheme obtains that public key and obtains assurance of its validity. Ephemeral key pairs are 

generated as specified in Section 5.6.1; the ephemeral private key is not provided to the other 

entity. Approved methods for obtaining assurance of the validity of an ephemeral public key are 

provided in Section 5.6.2.3. 

If the key agreement scheme requires that an entity provide a nonce, the nonce is generated as 

specified in Section 5.4 and provided to the other entity. 

Each participant in the key agreement process uses the appropriate public and private keys to 

generate a shared secret as specified in Section 6 for each key agreement scheme. Using the 

shared secret, each participant then derives secret keying material from the shared secret and 

other information as specified in Section 5.8. 

If one or both of the participants wish to confirm that the other entity has computed the same 

shared secret or the same secret keying material as part of the key agreement process, key 

confirmation is performed as specified in Section 8.4. 

Assurance of static private key possession is obtained prior to using the derived keying material 

for purposes beyond those of the key agreement transaction itself (see Section 5.6.3.2). 

4.3 DLC-based Key Transport Process 

Figure 3: Key Transport Process 

Figure 3 depicts the steps that are performed when transporting secret keying material from one 

entity to another using a key transport scheme. Depending on the available keys, either entity 

could be the key transport sender. Prior to performing key transport, a key-wrapping key is 

established by using a key agreement process as specified in Section 7. Key confirmation may be 

performed to obtain assurance that both parties possess the same key-wrapping key. The sender 

selects secret keying material to be sent to the other entity, wraps the keying material using the 

Establish Key-wrapping Key Select the Keying Material Obtain Wrapped Keying 

Material Establish Key-wrapping Key

Wrap Keying Material Unwrap Keying material

Transport Wrapped Keying 

Material

Key Transport Sender Key Transport Receiver

Key Transport Complete
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key-wrapping key and sends the wrapped keying material to the other entity. The receiving entity 

receives the wrapped keying material and unwraps it using the previously established keywrapping key. 

5. Cryptographic Elements 

This section describes the basic computations that are performed and the assurances that need to 

be obtained when performing DLC based key establishment. The schemes described in Section 6 

are based upon the correct implementation of these computations and assurances. 

Tables 1 and 2 of Section 5.5 list parameter size sets to be used in the selection of cryptographic 

elements. All cryptographic elements used together shall be selected in accordance with the 

same parameter size set. 

5.1 Cryptographic Hash Functions 

An Approved hash function shall be used when a hash function is required (for example, for the 

key derivation function or to compute a MAC when HMAC, as specified in FIPS 198, is used). 

FIPS 180-2 [2] specifies Approved hash functions. The hash function shall be selected in 

accordance with the parameter lists in Tables 1 and 2 of Section 5.5. 

5.2 Message Authentication Code (MAC) Algorithm 

A Message Authentication Code (MAC) algorithm defines a family of one-way (MAC) functions 

that is parameterized by a symmetric key. In key establishment schemes, an entity is sometimes 

required to compute a MacTag on received or derived data using the MAC function determined 

by a symmetric key derived from a shared secret. The MacTag is sent to another entity in order 

to confirm that the shared secret was correctly computed. An Approved MAC algorithm shall be 

used to compute a MacTag, for example, HMAC [5]. 

The MAC algorithm is used to provide key confirmation as specified in this Recommendation 

when key confirmation is desired, and is used to validate implementations of the key 

establishment schemes specified in this Recommendation (see Section 5.2.3). MacTag

computation and checking are defined in Sections 5.2.1 and 5.2.2 of this Recommendation. 

5.2.1 MacTag Computation 

The computation of the MacTag is represented as follows: 

MacTag = MAC(MacKey, MacLen, MacData). 

The MacTag computation shall be performed using an Approved MAC algorithm. In the above 

equation, MAC represents an Approved MAC algorithm; MacKey represents a symmetric key 

obtained from the DerivedKeyingMaterial (see Section 5.8); MacLen represents the length of 

MacTag; and MacData represents the data on which the MacTag is computed. The minimum for 
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MacLen is specified in Tables 1 and 2 of Section 5.5. The minimum size for MacKey is also 

specified in Tables 1 and 2. See [5] and [6]. 

5.2.2 MacTag Checking 

To check a received MacTag (e.g., received during key confirmation and/or implementation 

validation), a new MacTag is computed—using the values of MacKey, MacLen, and MacData

possessed by the recipient (as specified in Section 5.2.1). The new MacTag is compared with the 

received MacTag. If their values are equal, then it may be inferred that the same MacKey,

MacLen, and MacData values were used in the two MacTag computations. 

5.2.3 Implementation Validation Message 

For purposes of validating an implementation of the schemes in this Recommendation during an 

implementation validation test (under the NIST Cryptographic Validation Program), the value of 

MacData shall be the string “Standard Test Message”, followed by a 16-byte field for a nonce. 

The default value for this field is all binary zeros. Different values for this field will be specified 

during testing. This is for the purposes of testing when no key confirmation capability exists (see 

Section 10). 

Note: ANS X9.42 defines MacData as “ANSI X9.42 Testing Message”. ANS X9.63 does not 

address implementation validation at this level of detail. The implementation test message used 

for NIST validation is a different text string from the implementation test message for ANS 

X9.42 validation. 

5.3 Random Number Generation 

Whenever this Recommendation requires the use of a randomly generated value (for example, 

for keys or nonces), the values shall be generated using an Approved random bit generator 

(RBG) providing an appropriate security strength. 

5.4 Nonces 

A nonce is a time-varying value that has (at most) a negligible chance of repeating. For example, 

a nonce may be composed of one (or more) of the following components: 

1. A random value that is generated anew for each nonce, using an Approved random bit 

generator. The security strength of the random bit generator and the entropy of the nonce 

shall be at least one half of the minimum required bit length of the subgroup order (as 

specified in Tables 1 and 2 of Section 5.5). 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 
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4. A combination of a timestamp and a monotonically increasing sequence number such that 

the sequence number is reset only when the timestamp changes. (For 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.) 

Nonces are used, for example, in implementation validation testing (Section 5.2.3), in C(0, 2) 

schemes (Section 6.3), and in key confirmation (Section 8). 

When using a nonce, a random nonce should be used. 

5.5 Domain Parameters 

Discrete Logarithm Cryptography (DLC), which includes Finite Field Cryptography (FFC) and 

Elliptic 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. A 

user of a candidate set of domain parameters (for example, either an initiator or a responder) 

shall have assurance of domain parameter validity prior to using them. 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. 

Some schemes in ANS X9.42 and X9.63 allow the set of domain parameters used and associated 

with static keys to be different from the set of domain parameters used and associated with 

ephemeral keys. For this Recommendation, however, only one set of domain parameters shall be 

used during any key establishment transaction using a given run of a scheme (that is, the statickey domain parameters and the ephemeral-key domain parameters used in one scheme shall be 

the same).

5.5.1 Domain Parameter Generation 

5.5.1.1 FFC Domain Parameter Generation 

Domain parameters for FFC schemes are of the form (p, q, g{, SEED, pgenCounter}), where p is 

the (larger) prime field order, q is the (smaller) prime (multiplicative) subgroup order, g is a 

generator of the q-order cyclic subgroup of GF(p)*, and SEED and pgenCounter are optional 

values used in the canonical process of generating and validating p and q, and possibly g,

depending on the method of generation. FFC Domain parameters shall be generated using a 

method specified in FIPS 186-3 [3] based on a parameter size set selected from Table 1. 
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Table 1: FFC Parameter Size Sets 

FFC Parameter Set Name FA FB FC 

Bit length of field order p (i.e., ⎡ ⎤ log ) 2 p 1024 2048 20481

Bit length of subgroup order q (i.e., ⎡log ) 2 q⎤ 160 224 256 

Minimum bit length of the hash function output 160 224 256 

Minimum MAC key size (for use in key confirmation) 80 112 128 

Minimum MacLen (for use in key confirmation) 80 112 128 

As shown in Table 1, there are three parameter size sets (named FA through FC) for FFC; all the 

parameters of a particular set shall be used together. For U.S. government applications, one or 

more sets shall be selected based on the solution requirements. See the comparable security table 

in the Recommendation for Key Management [7] to assess the comparable security of any 

particular parameter size set. The Recommendation for Key Management [7] provides guidance 

on selecting an appropriate security strength and an appropriate FFC parameter set. If the 

appropriate security strength does not have an FFC parameter set, then Elliptic Curve 

Cryptography should be used (see Section 5.5.1.2). 

For this Recommendation, the size of p (public key size) is a multiple of 1024 bits; the exact 

length depends on the FFC parameter set selected. For this Recommendation, the size of q is a 

specific bit length depending on the FFC parameter set selected. 

5.5.1.2 ECC Domain Parameter Generation 

Domain parameters for ECC schemes are of the form (q, FR, a, b{, SEED}, G, n, h), where q is 

the field size; FR is an indication of the basis used; a and b are two field elements that define the 

equation of the curve; SEED is an optional bit string that is included if the elliptic curve was 

randomly generated in a verifiable fashion; G is a generating point (possibly generated from the 

SEED) consisting of (xG, yG) of prime order on the curve; n is the order of the point G; and h is 

the cofactor (which is equal to the order of the curve divided by n). Note that the field size q may 

be either an odd prime p or 2m, where m is a prime. 

 

1

 Parameter size set FC is included with the same field order length 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. 
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Table 2: ECC Parameter Size Sets 

ECC Parameter Set Name EA EB EC ED EE 

Bit length of ECC subgroup order n 

(i.e., ⎡ ⎤ log ) 2 n

160-

223 

224-

255 

256-

383 

384-

511 

512+ 

Maximum bit length of ECC cofactor h 10 14 16 24 32 

Minimum bit length of the hash function 

output 

160 224 256 384 512 

Minimum MAC key size (for use in key 

confirmation) 

80 112 128 192 256 

Minimum MacLen (for use in key 

confirmation) 

80 112 128 192 256 

As shown in Table 2, there are five parameter size sets (named EA, EB, EC, ED and EE) for 

ECC; all the members of a particular set shall be used together. For U.S. government 

applications, one or more sets shall be selected based on the solution requirements. See the 

comparable security table in the Recommendation for Key Management [7] to assess the 

comparable security of any particular parameter size set. The Recommendation for Key 

Management [7] provides guidance on selecting the appropriate security strength and an 

appropriate ECC key size. 

The five different cofactor maximums each ensure that the subgroup of order n is unique and that 

cofactor multiplication is reasonably efficient. The ECC domain parameters shall either be 

generated as specified in ANS X9.62 [13] or selected from the recommended elliptic curve 

domain parameters specified in FIPS 186-3 [3]. (Note: ANS X9.62, rather than ANS X9.63, 

specifies the most current method of generating ECC domain parameters at the time of writing 

this Recommendation.) 

5.5.2 Assurances of Domain Parameter Validity 

Secure key establishment depends on the arithmetic validity of the set of domain parameters used 

by 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 according to the requirements 

specified in Section 5.5.1. 

2. The party performs an explicit domain parameter validation as specified in: 
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a. FIPS 186-3 for FFC based on a parameter size set selected from Table 1. 

b. ANS X9.62-2 for ECC. 

3. The party has received assurance from a trusted third party (for example, a CA or NIST2

) 

that the set of domain parameters was valid at the time that they were generated by reason 

of either method 1 or 2 above. 

Note: Some domain parameters have been generated using SHA-1, and SHA-1 will be 

required during their validation. At some time in the future, it is expected that SHA-1 will 

no longer be an Approved hash function. However, if a set of domain parameters was 

successfully validated with SHA-1 while it was still an Approved hash function, then 

those domain parameters will continue to qualify as valid even after the use of SHA-1 is 

no longer Approved. In particular, this is true of the NIST Recommended Elliptic Curves. 

The application performing the key establishment on behalf of the party should determine 

whether or not to allow key establishment based upon the method(s) of assurance that was used. 

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 

A particular set of domain parameters 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

be correctly associated with its specific set of domain parameters. 

5.6 Private and Public Keys 

This section specifies requirements for the generation of key pairs, assurances of public key 

validity, assurances of private key possession, and key pair management. 

5.6.1 Private/Public Key Pair Generation 

5.6.1.1 FFC Key Pair Generation 

For the FFC schemes, each static and ephemeral private key and public key shall be generated 

using an Approved method and the selected valid domain parameters (p, q, g{, SEED, 

pgenCounter}) (see Appendix B of FIPS 186-3). Each private key shall be unpredictable and 

shall be generated in the range [1, q-1] using an Approved random bit generator. The static 

public key y is computed from the static private key x by using the following formula: y = gx

 mod 

p. Similarly the ephemeral public key t is computed from the ephemeral private key r by using 

the following formula: t = gr mod p. 

 

2

 If using an elliptic curve from the list of NIST recommended curves in FIPS 186-3 [3]. 
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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 and the selected domain parameters (q, FR, a, b{, SEED}, 

G, n, h) (see Appendix B of FIPS 186-3). Each private key, d, shall be unpredictable and shall 

be generated in the range [1, n-1] using an Approved random bit generator. The public key Q is 

computed by using the following formula: Q = (xQ, yQ) = dG. 

5.6.2 Assurances of the Arithmetic Validity of a Public Key

Secure key establishment depends on the arithmetic validity of the public key. To explain the 

assurance requirements, some terminology needs to be defined. 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 entity that generated the key as part of a key establishment transaction. The 

recipient of an ephemeral public key is the entity that receives the key during a key establishment 

transaction with the owner. 

Both the owner and a recipient of a candidate public key shall have assurance of its arithmetic 

validity before using it, as specified below. The application performing the key establishment on 

behalf of the owner and recipient should determine whether or not to allow key establishment 

based upon the method(s) of assurance that was used. Such knowledge may be explicitly 

provided to the application in some manner, or may be implicitly provided by the operation of 

the application itself. Prior to obtaining this assurance of arithmetic validity, the owner and 

recipient of the public key shall have assurance of the validity of the domain parameters. The 

procedures presented for obtaining public key validity assume that the domain parameters have 

been validated. 

5.6.2.1 Owner Assurances of Static Public Key Validity 

The owner of a static public key shall obtain assurance of its validity in one or more of the 

following ways: 

1. Owner Full Validation - The owner performs a successful full public key validation (see 

Sections 5.6.2.4 and 5.6.2.5). For example, a key generation routine may perform full 

public key validation as part of its processing. 

2. TTP Full Validation – The owner receives assurance that a trusted third party (trusted by 

the owner) has performed a successful full public key validation (see Sections 5.6.2.4 and 

5.6.2.5). 

3. Owner Generation – The owner has generated the public key from the private key (see 

Section 5.6.1). 
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4. TTP Generation – The owner has received assurance that a trusted third party (trusted by 

the owner) has generated the public/private key pair and has provided the key pair to the 

owner (see Section 5.6.1). 

The application performing the key establishment on behalf of the owner should determine 

whether or not to allow key establishment based upon the method(s) of assurance that was used. 

Such knowledge may be explicitly provided to the application in some manner, or may be 

implicitly provided by the operation of the application itself. Note that the use of a TTP to 

generate a key pair for an owner means that the TTP is trusted (by both the owner and any 

recipient) to not use the owner’s private key to masquerade as the owner. 

5.6.2.2 Recipient Assurances of Static Public Key Validity 

The recipient of a static public key shall obtain assurance of its validity in one or more of the 

following ways: 

1. Recipient Full Validation - The recipient performs a successful full public key validation 

(see Sections 5.6.2.4 and 5.6.2.5). 

2. TTP Full Validation – The recipient receives assurance that a trusted third party (trusted 

by the recipient) has performed a successful full public key validation (see Sections 

5.6.2.4 and 5.6.2.5). 

3. TTP Generation – The recipient receives assurance that a trusted third party (trusted by 

the recipient) has generated the public/private key pair in accordance with Section 5.6.1 

and has provided the key pair to the owner. 

The application performing the key establishment on behalf of the recipient should determine 

whether or not to allow key establishment based upon the method(s) of assurance that was used. 

Such knowledge may be explicitly provided to the application in some manner, or may be 

implicitly provided by the operation of the application itself. Note that the use of a TTP to 

generate a key means that the TTP is trusted (by both the recipient and the owner) to not use the 

owner’s private key to masquerade as the owner. 

5.6.2.3 Recipient Assurances of Ephemeral Public Key Validity 

The recipient of an ephemeral public key shall obtain assurance of its validity in one or more of 

the following ways: 

1. Recipient Full Validation - The recipient performs a successful full public key validation 

(see Sections 5.6.2.4 and 5.6.2.5). 

2. TTP Full Validation – The recipient receives assurance that a trusted third party (trusted 

by the recipient) has performed a successful full public key validation (see Sections 

5.6.2.4 and 5.6.2.5). For example, a trusted processor may only forward an ephemeral 

public key to the recipient if the public key passes a full public key validation. 
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3. Recipient ECC Partial Validation - If using an ECC method (only), the recipient performs 

a successful partial public key validation (see Section 5.6.2.6). 

4. TTP ECC Partial Validation – If using an ECC method (only), the recipient receives 

assurance that a trusted third party (trusted by the recipient) has performed a successful 

partial public key validation (see Section 5.6.2.6). For example, a trusted processor may 

only forward an ECC ephemeral public key to the recipient if it passes a partial public 

key validation. 

The application performing the key establishment on behalf of the recipient should determine 

whether or not to allow key establishment based upon the method(s) of assurance that was used. 

Such knowledge may be explicitly provided to the application in some manner, or may be 

implicitly provided by the operation of the application itself. 

5.6.2.4 FFC Full Public Key Validation Routine 

FFC full public key validation refers to the process of checking all the arithmetic properties of a 

candidate 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 method 

shall be used with static and ephemeral FFC public keys when assurance of the validity of the 

keys is obtained by method 1 or method 2 of Sections 5.6.2.1, 5.6.2.2, and 5.6.2.3. 

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. 

 (Ensures that the key has the unique correct representation and range in the field.) 

2. Verify that y

q

 = 1 (mod p). 

 (Ensures that the key has the correct order and is in the correct subgroup.) 

Output: If any of the above checks fail, then output an error indicator. Otherwise, output an 

indication of successful full validation. 

5.6.2.5 ECC Full Public Key Validation Routine 

ECC full public key validation refers to the process of checking all the arithmetic properties of a 

candidate 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 the 
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associated private key and so may be done at any time by anyone. This method may be used for a 

static ECC public key, or an ephemeral ECC public key, when assurance of the validity of the 

key is obtained by method 1 or method 2 of Sections 5.6.2.1, 5.6.2.2, and 5.6.2.3.

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 O. This can be done by inspection if the point is 

entered in the standard affine representation. 

 (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 prime 

p, 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 of 

an element in the underlying field.) 

3. If q is an odd prime p, verify that (yQ)

2 ≡ (xQ)3 + axQ + b (mod p). 

 If q = 2m, verify that (yQ)

2 + xQ yQ = (xQ)3 + a(xQ)2 + b in the finite field of size 2m. 

 (Ensures that the public key is on the correct elliptic curve.) 

4. Verify that nQ = O. 

 (Ensures that the public key has the correct order. Along with check 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.) 

Output: If any of the above checks fail, then output an error indicator. Otherwise, output an 

indication of successful validation. 

5.6.2.6 ECC Partial Public Key Validation Routine 

ECC partial public key validation refers to the process of checking some (but not all) of the 

arithmetic properties of a candidate ECC public key to ensure that it is in the correct group (but 

not necessarily the correct subgroup) specified by the associated ECC domain parameters. ECC 

Partial Public Key Validation omits the validation of subgroup membership, and therefore is 

usually 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. This 

method may only be used for an ephemeral ECC public key when assurance of the validity of the 

key is obtained by method 3 or 4 of Section 5.6.2.3. 
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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 O. This can be done by inspection if the point is 

entered in the standard affine representation. 

 (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 prime 

p, 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 of 

an element in the underlying field.) 

3. If q is an odd prime p, verify that (yQ)

 2 ≡ (xQ)3 + axQ + b (mod p). 

 If q = 2m, verify that (yQ)

2 + xQ yQ = (xQ)3 + a(xQ)2 + b in the finite field of size 2m. 

 (Ensures 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 checks fail, then output an error indicator. Otherwise, output an 

indication of validation success. 

5.6.3 Assurances of the Possession of a Static Private Key 

The security of key agreement schemes that use static key pairs depends on limiting knowledge 

of the static private keys to those who have been authorized to use them (i.e., their respective 

owners). In addition to preventing unauthorized entities from gaining access to private keys, it is 

also important to obtain assurance that authorized users do have access to their correct static 

private keys. 

Assurance of possession requirements for the owner of a static private key are specified in 

Section 5.6.3.1. Assurance of possession requirements for recipients of a static public key are 

specified in Section 5.6.3.2. 

When assurance of possession of a static private key is obtained, the assurance of the validity of 

the associated public key shall be obtained either prior to or concurrently with obtaining 

assurance of possession. Note that as time passes, an owner may lose possession of the 

associated private key, either by choice or due to an error; for this reason, current assurance of 

possession can be of more value for some applications. See Section 5.6.3.2.2 and Section 8.1 for 

ways to obtain more current assurance of possession. 
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5.6.3.1 Owner Assurances of Possession of a Static Private Key 

The owner of a static public key shall have assurance that the owner actually possesses the 

correct associated private key in one or more of the following ways: 

1. Owner Receives Assurance via Explicit Key Confirmation – The owner employs the static 

key pair to successfully engage another party in a key agreement transaction incorporating 

explicit key confirmation. The key confirmation shall be performed with the owner as key 

confirmation recipient in order to obtain assurance that the private key functions correctly. 

See Section 8 for further explanation. 

2. Owner Receives Assurance via Use of an Encrypted Certificate - The owner uses the static 

private key while engaging in a key agreement transaction with a Certificate Authority 

(trusted by the owner), providing the CA with the corresponding static public key. As part 

of this transaction, the CA generates a certificate containing the owner’s static public key 

and encrypts the certificate using a symmetric key derived from the shared secret they have 

(allegedly) established. Only the encrypted form of the certificate is provided to the owner. 

By successfully decrypting the certificate, the owner obtains assurance of possession of the 

correct private key (at the time of the key agreement transaction). 

3. Owner Receives Assurance via Key Regeneration – The owner regenerates a public key 

from the static private key and verifies that the regenerated public key is equal to the 

original static public key. Note that this method may be useful if the static private key has 

been generated by a party other than the owner or as an integrity check on a key pair that 

has been stored for a long period of time. 

4. Owner Receives Assurance via Trusted Provision - A trusted party (trusted by the owner) 

provides the static private key and static public key to the owner using a trusted 

distribution method. Reliance upon this method assumes (1) that the trusted party will 

provide a private key that is consistent with the public key and (2) that the trusted party 

will not use the private key to masquerade as the owner. 

5. Owner Receives Assurance via Key Generation - The act of generating a key pair, with the 

public key being computed from the private key, is a way for the owner to obtain 

assurance of possession of the correct private key. This method allows an owner who 

protects his/her own keys to have assurance of possession without additional computation. 

Note that this method may not detect algorithm implementation errors, hardware errors, 

random bit flips, etc. Further assurance may be obtained through the use of one or more of 

the above methods. 

The owner of a static public key (or agents trusted to act on the owner’s behalf) should

determine that the method used for obtaining assurance of the owner’s possession of the correct 

static private key is sufficient and appropriate to meet the security requirements of the owner’s 

intended application(s). 
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5.6.3.2 Recipient Assurance of Owner’s Possession of a Static Private Key 

At the time of binding an identifier to the owner’s static public key, the binding authority shall

obtain assurance that the owner is in possession of the correct static private key. This assurance 

shall either be obtained using one of the methods specified in Section 5.6.3.2.2 or by using an 

Approved alternative. 

Recipients other than binding authorities shall obtain this assurance – either through a trusted 

third party (see Section 5.6.3.2.1) or directly – before using the derived keying material for 

purposes beyond those required during the key agreement transaction itself. If the recipient 

chooses to obtain this assurance directly, then to comply with this Recommendation the parties 

shall use one of the methods specified in Section 5.6.3.2.2. 

5.6.3.2.1 Recipient Obtains Assurance through a Trusted Third Party 

The recipient of a static public key may receive assurance that its owner is in possession of the 

correct static private key from a trusted third party, either before or during a key agreement 

transaction that makes use of that static public key. The methods used by a third party trusted by 

the recipient to obtain that assurance are beyond the scope of this Recommendation (see 

however, Section 8.1.5.1.1.2 of SP 800-57 [7]). The recipient of a static public key (or agents 

trusted to act on behalf of the recipient) should know the method(s) used by the third party, in 

order to determine that the assurance obtained on behalf of the recipient is sufficient and 

appropriate to meet the security requirements of the recipient’s intended application(s). 

5.6.3.2.2 Recipient Obtains Assurance Directly from the Claimed Owner 

When two parties engage in a key agreement transaction, there is (at least) an implicit claim of 

ownership made whenever a static public key is provided on behalf of a particular party. That 

party is considered to be a claimed owner of the corresponding static key pair – as opposed to 

being a true owner – until adequate assurance can be provided that the party is actually the one 

authorized to use the static private key. 

The recipient of a static public key can directly obtain assurance of the claimed owner’s current 

possession of the corresponding private key by successfully completing a key agreement 

transaction that explicitly incorporates key confirmation, with the claimed owner serving as the 

key confirmation provider (see Section 8). Note that the recipient of the static public key in 

question is also the key confirmation recipient. When assurance of possession is obtained 

through key confirmation performed in compliance with this Recommendation, the underlying 

key agreement scheme used shall be one of the following, and the recipient seeking assurance 

shall serve as the key agreement initiator: 

• dhHybridOneFlow or (Cofactor) One-Pass Unified Model, 

• MQV1 or One-Pass MQV, 

• dhOneFlow or (Cofactor) One-Pass Diffie-Hellman. 
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(See Sections 6 and 8 for details.) The recipient of a static public key (or agents trusted to act on 

the recipient’s behalf) shall determine that the key agreement scheme employed to obtain 

assurance of possession in this manner is sufficient and appropriate to meet the security 

requirements of the recipient’s intended application(s). 

In each of the permitted scenarios, the key confirmation recipient contributes (at least) an 

ephemeral public key that must be arithmetically combined with the static private key claimed by 

the key confirmation provider during the provider's computation of the shared secret. Successful 

key confirmation (performed in the context described here) demonstrates that the correct static 

private key has been used in the key confirmation provider’s calculations, and thus also provides 

assurance that the claimed owner is the true owner (see Section 8.1). 

Key confirmation may be requested by the recipient even when the recipient has obtained 

independent assurance that the claimed owner of a static public key is indeed its true owner. This 

is appropriate in situations where the recipient desires assurance that the owner is currently in 

possession of the correct static private key (and that the owner is currently able to use it 

correctly), or in situations where there is no access to a trusted party who can provide assurance 

of the owner’s (previous) demonstration of private key possession. 

Note that the recommendation that assurance of possession be obtained before using derived 

keying material for purposes beyond those of the key agreement transaction itself does not 

prohibit the parties to a key agreement transaction from using derived keying material during the 

transaction, for purposes required by the key agreement scheme. For example, in a transaction 

involving a key agreement scheme that incorporates key confirmation, the parties establish a 

(purportedly) shared secret, derive keying material, and — as part of that same transaction — use 

a portion of the derived keying material as the MacKey in their key confirmation computations. 

5.6.4 Key Pair Management 

5.6.4.1 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 [7]): 

1. A public/private key pair shall be correctly associated with its corresponding specific set 

of domain parameters. Each key pair shall not be used with more than one set of domain 

parameters. 

2. Each DLC private key shall be unpredictable and created using an Approved key 

generation method as specified in Section 5.6.1. 

3. Private keys shall be protected from unauthorized access, disclosure, modification and 

substitution. 

4. Public keys shall be protected from unauthorized modification and substitution. This is 

often accomplished by using public key certificates that have been signed by a 

Certification Authority (CA). 
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5.6.4.2 Specific Requirements on Static Key Pairs 

The specific requirements for static key pairs are as follows: 

1. A recipient of a static public key shall be assured of the data integrity and correct 

association of (a) the public key, (b) the set of domain parameters for that key, and (c) the 

identifier of the entity that owns the key pair (that is, the party with whom the recipient 

intends to establish a key). This assurance is often provided by verifying a public-key 

certificate that was signed by a trusted third party (for example, a CA), but may be 

provided by direct distribution of the keying material from the owner, provided that the 

recipient trusts the owner to do this. 

2. 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, a 

digital signature key pair is not to be used for key establishment or vice versa). 

3. An owner and a recipient of a static public key shall have 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. The application performing the key establishment 

on behalf of the recipient should determine whether or not to allow key establishment 

based upon the method(s) of assurance that was used. Such knowledge may be explicitly 

provided to the application in some manner, or may be implicitly provided by the 

operation of the application itself. 

4. An owner and a recipient of a static public key shall have assurance of the owner’s 

possession of the associated private key (see Section 5.6.3). The owner shall know the 

method used to obtain assurance of possession of the owner’s private key. 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. 

5.6.4.3 Specific Requirements on Ephemeral Key Pairs 

The 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 

Transport transactions that are transporting identical secret keying material 

simultaneously (or within a short period of time; see Section 7). After its use, an 

ephemeral private key shall be zeroized. 

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. 
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3. A recipient of an ephemeral public key shall have assurance of the validity of the public 

key (see Section 5.6.2). The application performing the key establishment on behalf of 

the recipient should determine whether or not to allow key establishment based upon the 

method(s) of assurance that was used. Such knowledge may be explicitly provided to the 

application in some manner, or may be implicitly provided by the operation of the 

application itself. 

5.7 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 the 

Modified 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 shall

be used by the MQV2 and MQV1 schemes, which are based on finite field cryptography 

and the MQV algorithm. 

4. The ECC MQV primitive (Section 5.7.2.2 of this Recommendation): This primitive shall

be 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 function 

(see Section 5.8). 

5.7.1 Diffie-Hellman Primitives 

5.7.1.1 Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive 

The shared secret Z is computed using the domain parameters (p, q, g{, SEED, pgenCounter}), 

the other party’s public key and one’s own private key. This primitive is used in Section 6 by the 

dhHybrid1, 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 the initiator U or the responder V. 

Input: 

1. (p, q, g{, SEED, pgenCounter}): Domain parameters, 
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2. xA : One’s own private key, and 

3. yB : The other party’s public key. 

Process: 

1. Z y p Ax = B mod

2. If Z=1, output an error indicator. 

3. Else, output Z. 

Output: The shared secret Z or an error indicator. 

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

The shared secret Z is computed using the domain parameters (q, FR, a, b{, SEED}, G, n, h), the 

other 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 the initiator U or the 

responder 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 = O, output an error indicator. 

3. Z = xP where xP is the x-coordinate of P. 

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 

The shared secret Z is computed using the domain parameters (p, q, g{, SEED, pgenCounter}), 

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 be 

either the initiator U or the responder V. 

Input: 
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1. (p, q, g{, SEED, pgenCounter}): Domain parameters, 

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,3

5. tA : One’s own second public key, and 

6. tB : The other party’s second public key. 

Process: 

1.w = ⎡q / 2⎤. 

2. w w

A A T = (t mod 2 ) + 2 . 

3. S r T x q A = ( A + A A )mod . 

4. w w

B B T = (t mod 2 ) + 2 . 

5. Z t y p B S A T = (( B ( B )) )mod . 

6. If Z = 1, output an error indicator. Else, 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 the initiator U or the responder V. 

In this form, one’s own second private and public keys (input 4 and 5 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 

(input 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 

This form of invoking the FFC MQV primitive is used in Section 6.2.1.3 by the MQV1 scheme. 

In this form, the initiator uses a static key pair and an ephemeral key pair, but the responder uses 

 

3

 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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only a static key pair. One-Pass MQV is done using the MQV primitive by using the responder’s 

static key pair as the responder’s second key pair (as the responder has no ephemeral key pair). 

The initiator uses the responder’s static public key for the responder’s second public key; that is, 

when the initiator uses the algorithm in Section 5.7.2.1, input 6 becomes the other party’s static 

public key (yB). 

The responder uses his/her static private key for his second private key; that is, when the 

responder uses the algorithm in Section 5.7.2.1, input 4 becomes the responder’s static private 

key xA, and input 5 becomes the responder’s 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 to 

compute 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, a point on the subgroup not equal to the point at infinity). 

Process: 

1. Convert xQ to an integer xqi using the convention specified in Appendix C.3. 

2. Calculate 

xqm = xqi mod ⎡ ⎤ / 2 2 f (where f = ⎡log ). 2 n⎤

3. Calculate the associate value function 

avf(Q) = xqm + ⎡ ⎤ / 2 2 f . (See footnote4

). 

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. The ECC version of 

MQV uses the cofactor h in its calculations. Assume that the party performing the computation is 

party A, and the other party is party B. Note that party A could be either the scheme initiator U 

or the scheme responder V. 

 

4

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

1. (q, FR, a, b{, SEED}, G, n, h): Domain parameters,

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,5

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 = O, output an error indicator. 

4. Z=xP, where xP is the x-coordinate of P.

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 that 

party A could be either the initiator U or the responder V. 

In this form, one’s own second private and public keys (input 4 and 5 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 (input 6 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, the initiator has a static key pair and an ephemeral key pair, but the 

responder has only a static key pair. One-Pass MQV is done using the MQV primitive with the 

responder’s static key pair as the responder’s second key pair (as the responder has no ephemeral 

keys). 

 

5

 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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The initiator uses the responder’s static public key as the responder’s second public key. When 

the initiator uses the algorithm in Section 5.7.2.3, input 6 becomes the other party’s static public 

key (Qs,B). 

The responder uses his static private key as his second private key. When the responder uses the 

algorithm in Section 5.7.2.3, input 4 becomes the responder’s static private key ds,A, and input 5 

becomes the responder’s static public key (Qs,A). 

5.8 Key Derivation Functions for Key Agreement Schemes 

An Approved key derivation function (KDF) shall be used to derive secret keying material from 

a shared secret. The output from a KDF shall only be used for secret keying material, such as a 

symmetric key used for data encryption or message integrity, a secret initialization vector, or a 

master key that will be used to generate other keys (possibly using a different process). Nonsecret keying material (such as a non-secret initialization vector) shall not be generated using the 

shared secret. 

Each call to the KDF requires a freshly computed shared secret, and this shared secret shall be 

zeroized immediately following its use. The derived secret keying material shall be computed in 

its entirety before outputting any portion of it. In schemes using only static keys, the freshly 

computed shared secret may be the same as the previous shared secret. In these cases, the 

initiator supplied nonce (NonceU, see Section 6.3) used in the scheme will vary so the same 

keying material is not regenerated. 

The derived secret keying material may be parsed into one or more keys or other secret 

cryptographic keying material (for example, secret initialization vectors). If Key Confirmation 

(KC) or implementation validation testing are to be performed as specified in Section 8 or 

Section 5.2.3, respectively, then the MAC key shall be formed from the first bits of the KDF 

output and zeroized after its use (i.e., the MAC key shall not be used for purposes other than key 

confirmation or implementation validation testing). 

This section specifies two Approved KDFs for use in key establishment. The Approved methods 

are provided in Section 5.8.1 and 5.8.2. 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 140-2 Annex D. Any hash function used in a KDF shall be 

Approved (see Section 5.1) and shall also meet the selection requirements specified herein (see 

Section 5.5.1). 

5.8.1 Concatenation Key Derivation Function (Approved Alternative 1) 

 This section specifies an Approved key derivation function, based on concatenation. 

The Concatenation KDF is as follows: 

Function call: kdf (Z, OtherInput), 

where OtherInput is keydatalen and OtherInfo. 
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Fixed Values (implementation dependent):

1. hashlen: an integer that indicates the length (in bits) of the output of the hash function 

used to derive blocks of secret keying material. 

2. max_hash_inputlen: an integer that indicates the maximum length (in bits) of the bit 

string(s) input to the hash function. 

Auxiliary Function: 

1. H: an Approved hash function. 

Input: 

1. Z: a byte string that is the shared secret. 

2. keydatalen: An integer that indicates the length (in bits) of the secret keying material to 

be generated; keydatalen shall be less than or equal to hashlen × (232 –1). 

3. OtherInfo: A bit string equal to the following concatenation: 

 AlgorithmID || PartyUInfo || PartyVInfo {|| SuppPubInfo }{|| SuppPrivInfo } 

 where the subfields are defined as follows: 

3.1 AlgorithmID: A bit string 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 1-80 are to be used as an 80-bit 

HMAC key and that bits 81-208 are to be used as a 128-bit AES key. 

3.2 PartyUInfo: A bit string containing public information that is required by the 

application using this KDF to be contributed by party U to the key derivation 

process. At a minimum, PartyUInfo shall include IDU, the identifier of party U. See 

the notes below. 

3.3 PartyVInfo: A bit string containing public information that is required by the 

application using this KDF to be contributed by party V to the key derivation 

process. At a minimum, PartyVInfo shall include IDV, the identifier of party V. See 

the notes below. 

3.4 (Optional) SuppPubInfo: A bit string containing additional, mutually-known public 

information. 

3.5 (Optional) SuppPrivInfo: A bit string containing additional, mutually-known private 

information (for example, a shared secret symmetric key that has been 

communicated through a separate channel). 

Each of the three subfields AlgorithmID, PartyUInfo, and PartyVInfo shall be the 

concatenation of an application-specific, fixed-length sequence of substrings of 

information. Each substring representing a separate unit of information shall have one of 

these two formats: Either it is a fixed-length bit string, or it has the form Datalen || Data, 
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where Data is a variable-length string of zero or more bytes, and Datalen is a fixedlength, big-endian counter that indicates the length (in bytes) of Data. (In this variablelength format, a null string of data shall be represented by using Datalen to indicate that 

Data has length zero.) An application using this KDF shall specify the ordering and 

number of the separate information substrings used in each of the subfields AlgorithmID, 

PartyUInfo, and PartyVInfo, and shall also specify which of the two formats (fixedlength or variable-length) is used for each substring. The application shall specify the 

lengths for all fixed-length quantities, including the Datalen counters. 

The subfields SuppPrivInfo and SuppPubInfo (when allowed by the application) shall be 

formed by the concatenation of an application-specific, fixed-length sequence of 

substrings of additional information that may be used in key derivation upon mutual 

agreement of parties U and V. Each substring representing a separate unit 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 (in bytes) of Data. The information substrings 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 an application allows the use of the OtherInfo subfield 

SuppPrivInfo and/or the subfield SuppPubInfo, then the application shall specify the 

ordering and the number of additional information substrings that may be used in the 

allowed subfield(s) and shall specify the fixed-length of the Datalen counters. 

Process: 

1. reps = ⎡ keydatalen / hashlen⎤.

2. If reps > (232 −1), then ABORT: output an error indicator and stop.

3. Initialize a 32-bit, big-endian bit string counter as 0000000116. 

4. If counter || Z || OtherInfo is more than max_hash_inputlen bits long, 

then ABORT: output an error indicator and stop. 

5. For i = 1 to reps by 1, do the following: 

5.1 Compute Hashi = H(counter || Z || OtherInfo). 

5.2 Increment counter (modulo 232), treating it as an unsigned 32-bit integer.

6. Let Hhash be set to Hashreps if (keydatalen / hashlen) is an integer; otherwise, let Hhash

be set to the (keydatalen mod hashlen) leftmost bits of Hashreps. 

7. Set DerivedKeyingMaterial = Hash1 || Hash2 || ... || Hashreps-1 || Hhash. 

Output: 

The bit string DerivedKeyingMaterial of length keydatalen bits (or an error indicator). 

Any scheme attempting to call this key derivation function with keydatalen greater than 
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or equal to hashlen × (232 −1) shall output an error indicator and stop without outputting 

DerivedKeyingMaterial. Any call to the key derivation function involving an attempt to 

hash a bit string that is greater than max_hash_inputlen bits long shall cause the KDF to 

output an error indicator and stop without outputting DerivedKeyingMaterial. 

Notes: 

1. IDU and IDV shall be represented in OtherInfo as separate units of information, using 

either the fixed-length format or the variable-length format described above – according 

to the requirements of the application using this KDF. The rationale for including the 

identifiers in the KDF input is provided in Appendix B. 

2. Party U shall be the initiator, and party V shall be the responder, as assigned by the 

protocol employing the key agreement scheme used to determine the shared secret Z. 

5.8.2 ASN.1 Key Derivation Function (Approved Alternative 2) 

This section specifies an Approved key derivation function utilizing ASN.1 DER encoding of 

OtherInfo. In all other respects, it is the same as the key derivation function specified in Section 

5.8.1. 

The ASN.1 KDF is as follows: 

Function call: kdf (Z, OtherInput) 

where OtherInput is keydatalen and OtherInfo. 

Fixed Values (implementation dependent):

1. hashlen: an integer that indicates the length (in bits) of the output of the hash function 

used to derive blocks of secret keying material. 

2. max_hash_inputlen: an integer that indicates the maximum length (in bits) of the bit 

string(s) input to the hash function. 

Auxiliary Function: 

1. H: an Approved hash function.

Input: 

1. Z: a byte string that is the shared secret. 

2. keydatalen: An integer that indicates the length (in bits) of the secret keying material to 

be generated; keydatalen shall be less than or equal to hashlen × (232 –1). 

3. OtherInfo: A bit string specified in ASN.1 DER encoding, which consists of the 

following information: 

3.1 AlgorithmID: A bit string that indicates how the derived keying material will be 

parsed and for which algorithm(s) the derived secret keying material will be used. 
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For example, AlgorithmID might indicate that bits 1-80 are to be used as an 80-bit 

HMAC key and that bits 81-208 are to be used as a 128-bit AES key. 

3.2 PartyUInfo: A bit string containing public information that is required by the 

application using this KDF to be contributed by party U to the key derivation 

process. At a minimum, PartyUInfo shall include IDU, the identifier of party U. See 

the notes below. 

3.3 PartyVInfo: A bit string containing public information that is required by the 

application using this KDF to be contributed by party V to the key derivation 

process. At a minimum, PartyVInfo shall include IDV, the identifier of party V. See 

the notes below. 

3.4 (Optional) SuppPubInfo: A bit string containing additional, mutually-known public 

information. 

3.5 (Optional) SuppPrivInfo: A bit string containing additional, mutually-known private 

information (for example, a shared secret symmetric key that has been 

communicated through a separate channel). 

Process: 

1. reps = ⎡keydatalen / hashlen⎤.

2. If reps > (232 −1), then ABORT: output an error indicator and stop.

3. Initialize a 32-bit, big-endian bit string counter as 0000000116. 

4. If counter || Z || OtherInfo is more than max_hash_inputlen bits long, 

then ABORT: output an error indicator and stop. 

5. For i = 1 to reps by 1, do the following: 

5.1 Compute Hashi = H(counter || Z || OtherInfo). 

5.2 Increment counter (modulo 232), treating it as an unsigned 32-bit integer.

6. Let Hhash be set to Hashreps if (keydatalen / hashlen) is an integer; otherwise, let Hhash

be set to the (keydatalen mod hashlen) leftmost bits of Hashreps. 

7. Set DerivedKeyingMaterial = Hash1 || Hash2 || ... || Hashreps-1 || Hhash. 

Output: 

The DerivedKeyingMaterial as a bit string of length keydatalen bits (or an error 

indicator). The ASN.1 KDF produces secret keying material that is at most hashlen × (232 

–1) bits in length. Any call to this key derivation function using a keydatalen value that is 

greater than hashlen × (232–1) shall cause the KDF to output an error indicator and stop 

without outputting DerivedKeyingMaterial. Any call to the key derivation function 

involving an attempt to hash a bit string that is greater than max_hash_inputlen bits long 
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shall cause the KDF to output an error indicator and stop without outputting 

DerivedKeyingMaterial. 

Notes: 

1. IDU and IDV shall be represented in OtherInfo as separate units of information. The 

rationale for including the identifiers in the KDF input is provided in Appendix B. 

2. Party U shall be the initiator, and party V shall be the responder, as assigned by the 

protocol employing the key agreement scheme used to determine the shared secret Z. 

6. Key Agreement 

This Recommendation provides three categories of key agreement schemes (see Table 3). 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(i), parties U and V have a total of i 

ephemeral key pairs. The first category, C(2), consists of schemes requiring the generation of 

ephemeral key pairs by both parties; a C(2) scheme is suitable for an interactive key 

establishment protocol. The second category, C(1), consists of schemes requiring the generation 

of an ephemeral key pair by only one party; a C(1) scheme is suitable for a store and forward 

scenario, but may also be used in an interactive key establishment protocol. The third category, 

C(0), consists of schemes that do not use ephemeral keys. 

Key confirmation may be added to many of these schemes to provide assurance that the 

participants share the same keying material; see Section 8 for details on key confirmation. Each 

party should have such assurance. Although other methods are often used to provide this 

assurance, this Recommendation makes no statement as to the adequacy of these other methods. 

Table 3: Key Agreement Scheme Categories 

Category Comment 

C(2): Two ephemeral key pairs Each party generates an ephemeral key pair. 

C(1): One ephemeral key pair Only the initiator generates an ephemeral key pair. 

C(0): 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 4). In subcategory C(i, j), parties U and V have a total of i 

ephemeral key pairs and j static key pairs. 
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Table 4: Key Agreement Scheme Subcategories 

Category Subcategory 

C(2, 2): Each party generates an ephemeral key pair and has a 

static key pair. 

C(2): Two ephemeral key 

pairs 

C(2, 0): Each party generates an ephemeral key pair; no static 

key pairs are used. 

C(1, 2): The initiator generates an ephemeral key pair and has a 

static key pair; the responder has only a static key pair. 

C(1): One ephemeral key pair 

C(1, 1): The initiator generates an ephemeral key pair, but has no 

static key pair; the responder has only a static key pair. 

C(0): Zero ephemeral key 

pairs 

C(0, 2): Each party has 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, C(2, 2, FFC DH) completely classifies the dhHybrid1 

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 5). 

Table 5: Key Agreement Schemes 

Category Subcategory Primitive Scheme Full Classification 

C(2) C(2, 2) FFC DH dhHybrid1 C(2, 2, FFC DH) 

C(2) C(2, 2) ECC CDH (Cofactor) Full Unified Model C(2, 2, ECC CDH) 

C(2) C(2, 2) FFC MQV MQV2 C(2, 2, FFC MQV) 

C(2) C(2, 2) ECC MQV Full MQV C(2, 2, ECC MQV) 

C(2) C(2, 0) FFC DH dhEphem C(2, 0, FFC DH) 

C(2) C(2, 0) ECC CDH (Cofactor) Ephemeral Unified 

Model 

C(2, 0, ECC CDH) 

C(1) C(1, 2) FFC DH dhHybridOneFlow C(1, 2, FFC DH) 
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Category Subcategory Primitive Scheme Full Classification 

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

Model 

C(1, 2, ECC CDH) 

C(1) C(1, 2) FFC MQV MQV1 C(1, 2, FFC MQV) 

C(1) C(1, 2) ECC MQV One-Pass MQV C(1, 2, ECC MQV) 

C(1) C(1, 1) FFC DH dhOneFlow C(1, 1, FFC DH) 

C(1) C(1, 1) ECC CDH (Cofactor) One-Pass DiffieHellman C(1, 1, ECC CDH) 

C(0) C(0, 2) FFC DH dhStatic C(0, 2, FFC DH) 

C(0) C(0, 2) ECC CDH Cofactor Static Unified Model C(0, 2, ECC CDH) 

Each party in a key agreement process shall use the same set of valid domain parameters. These 

parameters shall be established and validated prior to the initiation of the key agreement process. 

See Section 5.5 for a discussion of domain parameters. 

Party U shall have an identifier IDU. If Party U owns a static key pair that is used in a given key 

agreement transaction, then IDU shall be the identifier that is bound to that key pair. If Party U 

does not contribute a static public key as part of a given key agreement transaction, then IDU

shall be a non-null identifier selected in accordance with the protocol utilizing the scheme. 

Similar rules apply for the selection of Party V’s identifier, IDV. 

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 key 

confirmation process. Note that the flow diagrams in this Recommendation omit explicit mention 

of various validation checks that are required. The flow diagrams and descriptions in this 

Recommendation assume a successful completion of the key establishment process. 

Rationale for selecting schemes for each subcategory, C(i, j), is included. These rationale 

sections will provide the user or developer with additional information to help make a choice as 

to which key establishment scheme to use. The rationale includes discussions of the security 

properties for the schemes. In general, the security properties for each scheme within a 

subcategory are the same; when this is not the case, the exceptions are identified. See Section 

6.1.1.5 specifically. These rationale sections do not contain an in-depth discussion of all possible 

security properties of all schemes. For further discussion, see ANS X9.42 and ANS X9.63. Note 

that the specific security properties achieved depend on whether a static key is used, whether an 
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ephemeral key is used, the specific method of calculating the shared secret, and the key 

confirmation method used, if any. 

It is important that a scheme not be chosen based solely on the number of security properties it 

possesses. Rather, a scheme should be selected based on how well the scheme fulfills the system 

requirements. For instance, in a bandwidth-constrained system, a scheme with fewer passes per 

exchange might be preferable to a scheme with more passes and more security properties. 

It is also important to understand that 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, are not specified in this Recommendation. 

6.1 Schemes Using Two Ephemeral Key Pairs, C(2) 

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 of 

static keys by the parties. In the first subcategory, each party has both static and ephemeral keys 

(see Section 6.1.1), while in the second subcategory, each party has only ephemeral keys (see 

Section 6.1.2). 

6.1.1 Each Party Has a Static Key Pair and Generates an Ephemeral Key Pair, 

C(2, 2) 

For these schemes, each party (U and V) has 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 shared secret keying material is derived from the shared 

secret (see Figure 4). 
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Figure 4: General Protocol when Each Party Generates Both Static and Ephemeral Key 

Pairs 

Prerequisites: The following are prerequisites for the use of all C(2, 2) schemes. 

1. Each party shall have an authentic copy of the same set of domain parameters, D. These 

parameters shall 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 shall have assurance of the validity of these domain parameters 

as specified in Section 5.5.2.

2. Each party shall have 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, the 

static key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Each party shall

obtain assurance of the validity of its own static public key as specified in Section 

5.6.2.1. Each party shall obtain assurance of its possession of the correct value for its 

own private key as specified in Section 5.6.3.1. 

3. The parties shall have agreed upon an Approved key derivation function (see Section 5.8) 

as well as an Approved hash function appropriate for use with the key derivation function 

and associated parameters (see Section 5.5). If key confirmation is used, the parties shall

have agreed upon an Approved MAC and associated parameters (see Tables 1 and 2). 

4. Prior to or during the key agreement process, each party shall obtain the identifier 

associated with the other party during the key agreement scheme and the static public key 

that is bound to that identifier. These static public keys shall be obtained in a trusted 

manner (e.g., from a certificate signed by a trusted CA). Each party shall obtain 

U U’s Ephemeral Public Key V

V’s Ephemeral Public Key

U’s Static Public Key

V’s Static Public Key

1.V uses its static and ephemeral 

private keys and U’s static and 

ephemeral public keys to form a 

shared secret.

2.V invokes the Key Derivation 

Function using the shared secret. 

1. U uses its static and ephemeral 

private keys and V’s static and 

ephemeral public keys to form a 

shared secret.

2. U invokes the Key Derivation 

Function using the shared secret.
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assurance of the validity of the other party’s static public key as specified in Section 

5.6.2.2 

The recipient of a static public key shall obtain assurance that its (claimed) owner is (or was) in 

possession of the corresponding static private key, as specified in Section 5.6.3.2. 

6.1.1.1 dhHybrid1, C(2, 2, FFC DH) 

This section describes the dhHybrid1 scheme from ANS X9.42. The prerequisites for this 

scheme shall be satisfied as specified in Section 6.1.1. In particular, party U shall obtain the 

static public key yV of party V, and party V shall obtain the static public key yU of party U. 

With the exception of key derivation, dhHybrid1 is “symmetric” in the actions of parties U (the 

initiator) and V (the responder). 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 symbol “U” by “V” (and vice versa) in the description of the key agreement transformation. 

Note, however, that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key tU to V. Receive an ephemeral public key tV

(purportedly) from V. If tV is not received, output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Zs – an integer in 

the range [2, p-2] – from the set of domain parameters D, U’s static private key xU, and 

V’s static public key yV. Convert Zs to a byte string (which is also denoted by Zs) using 

the Integer-to-Byte-String Conversion specified in Appendix C.1, and then zeroize the 

results of all intermediate calculations used in the computation of Zs. If the call to the 

FFC DH primitive outputs an error indicator, zeroize the results of all intermediate 

calculations used in the attempted computation of Zs, output an error indicator, and stop. 

4. Use the FCC DH primitive to derive a shared secret Ze – another integer in the range [2, 

p-2] – from the set of domain parameters D, U’s ephemeral private key rU, and V’s 

ephemeral public key tV. Convert Ze to a byte string (which is also denoted by Ze) using 

the Integer-to-Byte-String Conversion specified in Appendix C.1, and then zeroize the 

results of all intermediate calculations used in the computation of Ze. If this call to the 

FFC DH primitive outputs an error indicator, zeroize Zs and the results of all intermediate 

calculations used in the attempted computation of Ze, output an error indicator, and stop. 
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5. Compute the shared secret Z = Ze || Zs. Zeroize the results of all intermediate calculations 

used in the computation of Z (including Ze and Zs). 

6. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

7. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

dhHybrid1 is summarized in Table 6. 

Table 6: dhHybrid1 Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key xU

2. Static public key yU

1. Static private key xV

2. Static public key yV

Ephemeral Data 1. Ephemeral private key rU

2. Ephemeral public key tU

1. Ephemeral private key rV

2. Ephemeral public key tV

Computation Compute Zs by calling FFC DH 

using xU and yV

Compute Ze by calling FFC DH 

using rU and tV

Compute Z = Ze || Zs

Compute Zs by calling FFC DH 

using xV and yU

Compute Ze by calling FFC DH 

using rV and tU

Compute Z = Ze || Zs

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) 

Zeroize Z

Compute kdf(Z,OtherInput)

Zeroize Z
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6.1.1.2 Full Unified Model, C(2, 2, ECC CDH) 

This section describes the Full Unified Model scheme from ANS X9.63. The prerequisites for 

this scheme shall be satisfied as specified in Section 6.1.1. In particular, party U shall obtain the 

static public key Qs,V of party V, and party V shall obtain the static public key Qs,U of party U. 

With the exception of key derivation, Full Unified Model is “symmetric” in the actions of parties 

U (the initiator) and V (the responder). 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 symbol “U” by “V” (and vice versa) in the description of the key agreement transformation. 

Note, however, that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key Qe,U to V. Receive an ephemeral public key Qe,V

(purportedly) from V. If Qe,V is not received, output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Zs – an element 

of the finite field of size q – from the set of domain parameters D, U’s static private key 

ds,U, and V’s static public key Qs,V. Convert Zs to a byte string (which is also denoted by 

Zs) using the Field-element-to-Byte-String Conversion specified in Appendix C.2, and 

then zeroize the results of all intermediate calculations used in the computation of Zs. If 

the call to the ECC CDH primitive outputs an error indicator, zeroize the results of all 

intermediate calculations used in the attempted computation of Zs, output an error 

indicator, and stop. 

4. Use the ECC CDH primitive to derive a shared secret Ze – another element of the finite 

field of size q – from the set of domain parameters D, U’s ephemeral private key de,U, and 

V’s ephemeral public key Qe,V. Convert Ze to a byte string (which is also denoted by Ze) 

using the Field-element-to-Byte-String Conversion specified in Appendix C.2, and then 

zeroize the results of all intermediate calculations used in the computation of Ze. If this 

call to the ECC CDH primitive outputs an error indicator, zeroize Zs and the results of all 

intermediate calculations used in the attempted computation of Ze, output an error 

indicator, and stop. 

5. Compute the shared secret Z = Ze || Zs. Zeroize the results of all intermediate calculations 

used in the computation of Z (including Ze and Zs). 

6. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 
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OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

7. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

The Full Unified Model is summarized in Table 7. 

Table 7: Full Unified Model Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

2. Static public key Qs,U

1. Static private key ds,V

2. Static public key Qs,V

Ephemeral Data 1. Ephemeral private key de,U

2. Ephemeral public key Qe,U

1. Ephemeral private key de,V

2. Ephemeral public key Qe,V

Computation Compute Zs by calling ECC CDH 

using ds,U and Qs,V

Compute Ze by calling ECC CDH 

using de,U and Qe,V

Compute Z = Ze || Zs

Compute Zs by calling ECC CDH 

using ds,V and Qs,U

Compute Ze by calling ECC CDH 

using de,V and Qe,U

Compute Z = Ze || Zs

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) 

Zeroize Z

Compute kdf(Z,OtherInput) 

Zeroize Z

6.1.1.3 MQV2, C(2, 2, FFC MQV) 

This section describes the MQV2 scheme from ANS X9.42. The prerequisites for this scheme 

shall be satisfied as specified in Section 6.1.1. In particular, party U shall obtain the static public 

key yV of party V, and party V shall obtain the static public key yU of party U. 

With the exception of key derivation, MQV2 is “symmetric” in the actions of parties U (the 

initiator) and V (the responder). Only the actions performed by party U are specified here; a 
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specification of the actions performed by party V may be obtained by systematically replacing 

the symbol “U” by “V” (and vice versa) in the description of the key agreement transformation. 

Note, however, that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Receive an ephemeral public key tV

(purportedly) from V. If tV is not received, output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the MQV2 form of the FFC MQV primitive in Section 5.7.2.1 to derive a shared 

secret Z – an integer in the range [2, p-2] – from the set of domain parameters D, U’s 

static private key xU, V’s static public key yV, U’s ephemeral private key rU, U’s 

ephemeral public key tU, and V’s ephemeral public key tV. If the call to the FFC MQV 

primitive outputs an error indicator, zeroize the results of all intermediate calculations 

used in the attempted computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

MQV2 is summarized in Table 8. 
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Table 8: MQV2 Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key xU

2. Static public key yU

1. Static private key xV

2. Static public key yV

Ephemeral Data 1. Ephemeral private key rU

2. Ephemeral public key tU

1. Ephemeral private key rV

2. Ephemeral public key tV

Computation Compute Z by calling FFC MQV 

using xU, yV, rU, tU, and tV

Compute Z by calling FFC MQV 

using xV, yU, rV, tV, and tU

Derive Secret 

Keying Material 

Compute kdf(Z, OtherInput) 

Zeroize Z

Compute kdf(Z, OtherInput) 

Zeroize Z

6.1.1.4 Full MQV, C(2, 2, ECC MQV) 

This section describes the Full MQV scheme from ANS X9.63. The prerequisites for this scheme 

shall be satisfied as specified in Section 6.1.1. In particular, party U shall obtain the static public 

key Qs,V of party V, and party V shall obtain the static public key Qs,U of party U. 

With the exception of key derivation, the Full MQV scheme is “symmetric” in the actions of 

parties U (the initiator) and V (the responder). 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 symbol “U” by “V” (and vice versa) in the description of the key 

agreement transformation. Note, however, that 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 transformation to a) establish a shared secret value Z with 

party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Receive an ephemeral public key Qe,V

(purportedly) from V. If Qe,V is not received, output an error indicator and stop. 

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, output an error indicator and stop. 
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3. Use the Full MQV form of the ECC MQV primitive in Section 5.7.2.3.1 to derive a 

shared secret value Z – an element of the finite field of size q – from the set of domain 

parameters D, U’s static private key ds,U, V’s static public key Qs,V, U’s ephemeral private 

key de,U, U’s ephemeral public key Qe,U, and V’s ephemeral public key Qe,V. If the call to 

the ECC MQV primitive outputs an error indicator, zeroize the results of all intermediate 

calculations used in the attempted computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-Byte 

String Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

The Full MQV is summarized in Table 9. 

Table 9: Full MQV Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

2. Static public key Qs,U

1. Static private key ds,V

2. Static public key Qs,V

Ephemeral Data 1. Ephemeral private key de,U

2. Ephemeral public key Qe,U

1. Ephemeral private key de,V

2. Ephemeral public key Qe,V

Computation Compute Z by calling ECC MQV 

using ds,U, Qs,V, de,U, Qe,U, and Qe,V

Compute Z by calling ECC MQV 

using ds,V, Qs,U, de,V, Qe,V, and Qe,U

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) 

Zeroize Z

Compute kdf(Z,OtherInput) 

Zeroize Z
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6.1.1.5 Rationale for Choosing a C(2, 2) Scheme 

Since these schemes use two static keys, each party has assurance that no unintended party can 

compute the shared secret without the compromise of a static private key. 

Since these schemes use two ephemeral keys, each party has assurance that the shared secret 

varies from one key establishment transaction to the next. Even if both static and ephemeral 

private keys of one party from one transaction are compromised, the shared secrets from other 

legitimate C(2, 2) transactions (that is, between honest parties) are still protected by the use of 

different ephemeral private keys. 

Key confirmation can be provided in either or both directions for these schemes using the 

methods as specified in Sections 8.4.1, 8.4.2, and 8.4.3. Upon completion of a Key Confirmation 

as in Section 8, the recipient of the confirmation has assurance of the identifier of the key 

confirmation provider (through the identifier bound to the static key), as well as confirmation of 

the active participation of the provider. 

For a given set of FFC domain parameters, MQV2 is expected to have better performance than 

dhHybrid1. 

For a given set of ECC domain parameters, Full MQV is expected to have better performance 

than the Full Unified Model. 

The MQV schemes (MQV2 and Full MQV) provide assurance to each party that if a malicious 

party compromises their static private key, the malicious party cannot masquerade as a third 

party to the party whose key was compromised. In other words, if a malicious party, E, 

compromises party A’s static private key, then E cannot masquerade as any other party to A. The 

dhHybrid1 and Full Unified Model do not provide this assurance to either party. (Of course, for 

any scheme, if a static private key is compromised by an adversary, then that adversary can 

masquerade as the owner of that static private key to any other entity.) 

6.1.2 Each Party Generates an Ephemeral Key Pair; No Static Keys are Used, 

C(2, 0) 

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 

and then compute the shared secret. The secret keying material is derived using the shared secret 

(see Figure 5). 
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Figure 5: General Protocol when Each Party Generates Ephemeral Key Pairs; No Static 

Keys are Used 

Prerequisites: The following are prerequisites for the use of all C(2, 0) schemes:

1. Each party shall have an authentic copy of the same set of domain parameters, D. These 

parameters shall 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 shall have assurance of the validity of these domain parameters 

as specified in Section 5.5.2.

2. Each party shall have agreed upon an Approved key derivation function (see Section 5.8) 

as well as an Approved hash function appropriate for use with the key derivation function 

(see Section 5.5) with associated parameters. 

3. Prior to or during the key agreement process, each party shall obtain the identifier 

associated with the other party during the key agreement scheme. 

6.1.2.1 dhEphem, C(2, 0, FFC DH) 

This section describes the dhEphem scheme from ANS X9.42. The prerequisites for this scheme 

shall be satisfied as specified in Section 6.1.2. 

With the exception of key derivation, dhEphem is “symmetric” in the actions of parties U (the 

initiator) and V (the responder). 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 symbol “U” by “V” (and vice versa) in the description of the key agreement transformation. 

Note, however, that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive secret keying material as follows: 

U V

U’s Ephemeral Public Key

V’s Ephemeral Public Key

1. U uses its ephemeral private key 

and V’s ephemeral public key to 

form a shared secret.

2. U invokes the Key Derivation 

Function using the shared secret.

1. V uses its ephemeral private key 

and U’s ephemeral public key to 

form a shared secret.

2. V invokes the Key Derivation 

Function using the shared secret.
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1. Generate an ephemeral key pair (rU, tU) from the domain parameters D as specified in 

Section 5.6.1. Send the public key tU to V. Receive an ephemeral public key tV

(purportedly) from V. If tV is not received, output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the FCC DH primitive in Section 5.7.1.1 to derive a shared secret Z – an integer in 

the range [2, p-2] – from the set of domain parameters D, U’s ephemeral private key rU,

and V’s ephemeral public key tV. If the call to the FFC DH primitive outputs an error 

indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

dhEphem is summarized in Table 10. 

 
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Table 10: dhEphem Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data N/A N/A 

Ephemeral Data 1. Ephemeral private key rU

2. Ephemeral public key tU

1. Ephemeral private key rV

2. Ephemeral public key tV

Computation Compute Z by calling FFC DH using 

rU and tV 

Compute Z by calling FFC DH using 

rV and tU

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput)

Zeroize Z

Compute kdf(Z,OtherInput)

Zeroize Z

6.1.2.2 Ephemeral Unified Model, C(2, 0, ECC CDH) 

This section describes the Ephemeral Unified Model scheme from ANS X9.63. The prerequisites 

for this scheme shall be satisfied as specified in Section 6.1.2. 

With the exception of key derivation, Ephemeral Unified Model is “symmetric” in the actions of 

parties U (the initiator) and V (the responder). 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 symbol “U” by “V” (and vice versa) in the description of the key 

agreement transformation. Note, however, that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key Qe,U to V. Receive an ephemeral public key Qe,V

(purportedly) from V. If Qe,V is not received, output an error indicator and stop. 

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, output an error indicator and stop. 
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3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z – an element of 

the finite field of size q – from the set of domain parameters D, U’s ephemeral private 

key de,U, and V’s ephemeral public key Qe,V. If the call to the ECC CDH primitive outputs 

an error indicator, zeroize the results of all intermediate calculations used in the 

attempted computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-ByteString Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator.. 

The Ephemeral Unified Model is summarized in Table 11. 

Table 11: Ephemeral Unified Model Key Agreement Scheme 

Party U Party V 

Domain 

Parameters 

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

Static Data N/A N/A 

Ephemeral Data 1. Ephemeral private key de,U

2. Ephemeral public key Qe,U

1. Ephemeral private key de,V

2. Ephemeral public key Qe,V

Computation Compute Z by calling ECC CDH 

using de,U and Qe,V

Compute Z by calling ECC CDH 

using de,V and Qe,U

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput)

Zeroize Z

Compute kdf(Z,OtherInput)

Zeroize Z
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6.1.2.3 Rationale for Choosing a C(2, 0) Scheme 

These schemes offer no assurance to either party of the identifier of the entity with whom they 

have established a shared secret, since there is no binding between an identifier and an ephemeral 

public key. 

These schemes offer assurance to both parties that the current shared secret is isolated from prior 

and future compromises of shared secrets and private keys because all cryptographic material 

used in the computation of the shared secret is ephemeral and is zeroized immediately after use. 

Despite the fact that these schemes offer very few assurances, they may be used as a component 

in a carefully constructed protocol. These schemes may be appropriate for applications where, 

for one reason or another, there is no need to know the identifier of the party with whom one is 

establishing a shared secret, or where the identifier is bound to the public key through some other 

method. 

These schemes have the property of being relatively fast to compute, due to the lack of any 

certificate validation. They also require no support from a Certification Authority. These 

schemes are also often used as building blocks in larger protocols where other parts of the 

protocol add additional security properties. 

This Recommendation does not specify how to add key confirmation to these schemes. 

6.2 Schemes Using One Ephemeral Key Pair, C(1) 

This category consists of two subcategories that are determined by the possession (or nonpossession) of a static key pair by each of the parties. Let party U serve as the initiator, and party 

V serve as the responder. Only the initiator (party U) generates an ephemeral key pair. In the first 

subcategory, both the initiator and the responder have a static key pair, and the initiator also 

generates an ephemeral key pair (see Section 6.2.1). In the second subcategory, the initiator 

generates an ephemeral key pair, but has no static key pair; the responder has only a static key 

pair (see Section 6.2.2). 

6.2.1 Initiator Has a Static Key Pair and Generates an Ephemeral Key Pair; 

Responder Has a Static Key Pair, C(1, 2) 

For these schemes, party U (the initiator) uses both static and ephemeral private/public key pairs. 

Party V (the responder) 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 shared secret keying material is derived using the shared secret (see Figure 6). 
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Figure 6: General Protocol when the Initiator has both Static and Ephemeral Key Pairs, 

and the Responder has only a Static Key Pair 

Prerequisites: The following are prerequisites for the use of all C(1, 2) schemes. 

1. Each party shall have an authentic copy of the same set of domain parameters, D. These 

parameters shall 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 shall have assurance of the validity of these domain parameters 

as specified in Section 5.5.2.

2. Each party shall have 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, the 

static key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Each party shall

obtain assurance of the validity of its own static public key as specified in Section 

5.6.2.1. Each party shall obtain assurance of its possession of the correct value for its 

own private key as specified in Section 5.6.3.1. 

3. The parties shall have agreed upon an Approved key derivation function (see Section 5.8) 

as well as an Approved hash function appropriate for use with the key derivation function 

and associated parameters (see Section 5.5). If key confirmation is used, the parties shall

have agreed upon an Approved MAC and associated parameters (see Tables 1 and 2). 

4. Prior to or during the key agreement process, each party shall obtain the identifier 

associated with the other party during the key agreement scheme and the static public key 

that is bound to that identifier. These static public keys shall be obtained in a trusted 

manner (e.g., from a certificate signed by a trusted CA). Each party shall obtain 

assurance of the validity of the other party’s static public key as specified in Section 

5.6.2.2. 

U V

U’s Ephemeral Public Key

U’s Static Public Key

V’s Static Public Key

1. U uses its static and 

ephemeral private keys and 

V’s static public key to form 

a shared secret.

2. U invokes the Key 

Derivation Function using 

the shared secret.

1. V uses its static private key 

and U’s static and ephemeral 

public keys to form a shared 

secret.

2. V invokes the Key Derivation 

Function using the shared 

secret.
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The recipient of a static public key shall obtain assurance that its (claimed) owner is (or was) in 

possession of the corresponding static private key, as specified in Section 5.6.3.2. 

6.2.1.1 dhHybridOneFlow, C(1, 2, FFC DH) 

This section describes the dhHybridOneFlow scheme from ANS X9.42. The prerequisites for 

this scheme shall be satisfied as specified in Section 6.2.1. In particular, party U shall obtain the 

static public key yV of party V, and party V shall obtain the static public key yU of party U. 

Note that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key tU to V. 

2. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Zs – an integer in 

the range [2, p-2] – from the set of domain parameters D, U’s static private key xU, and 

V’s static public key yV. Convert Zs to a byte string (which is also denoted by Zs) using 

the Integer-to-Byte-String Conversion specified in Appendix C.1, and then zeroize the 

results of all intermediate calculations used in the computation of Zs. If the call to the 

FFC DH primitive outputs an error indicator, zeroize the results of all intermediate 

calculations used in the attempted computation of Zs, output an error indicator, and stop. 

3. Use the FCC DH primitive to derive a shared secret Ze – another integer in the range [2, 

p-2] – from the set of domain parameters D, U’s ephemeral private key rU, and V’s static 

public key yV. Convert Ze to a byte string (which is also denoted by Ze) using the Integerto-Byte-String Conversion specified in Appendix C.1, and then zeroize the results of all 

intermediate calculations used in the computation of Ze. If this call to the FFC DH 

primitive outputs an error indicator, zeroize Zs and the results of all intermediate 

calculations used in the attempted computation of Ze, output an error indicator, and stop. 

4. Compute the shared secret Z = Ze || Zs. Zeroize the results of all intermediate calculations 

used in the computation of Z (including Ze and Zs). 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 
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Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Receive an ephemeral public key tU (purportedly) from U. If tU is not received, output an 

error indicator and stop. 

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, output an error indicator and stop. 

3. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Zs – an integer in 

the range [2, p-2] – from the set of domain parameters D, V’s static private key xV, and 

U’s static public key yU. Convert Zs to a byte string (which is also denoted by Zs) using 

the Integer-to-Byte-String Conversion specified in Appendix C.1, and then zeroize the 

results of all intermediate calculations used in the computation of Zs. If the call to the 

FFC DH primitive outputs an error indicator, zeroize the results of all intermediate 

calculations used in the attempted computation of Zs, output an error indicator, and stop. 

4. Use the FCC DH primitive to derive a shared secret Ze – another integer in the range [2, 

p-2] – from the set of domain parameters D, V’s static private key xV, and U’s ephemeral 

public key tU. Convert Ze to a byte string (which is also denoted by Ze) using the Integerto-Byte-String Conversion specified in Appendix C.1, and then zeroize the results of all 

intermediate calculations used in the computation of Ze. If this call to the FFC DH 

primitive outputs an error indicator, zeroize Zs and the results of all intermediate 

calculations used in the attempted computation of Ze, output an error indicator, and stop. 

5. Compute the shared secret Z = Ze || Zs. Zeroize the results of all intermediate calculations 

used in the computation of Z (including Ze and Zs). 

6. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

7. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

dhHybridOneFlow is summarized in Table 12. 
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Table 12: dhHybridOneFlow Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key xU

2. Static public key yU

1. Static private key xV

2. Static public key yV

Ephemeral Data 1. Ephemeral private key rU

2. Ephemeral public key tU

N/A 

Computation Compute Zs by calling FFC DH 

using xU and yV

Compute Ze by calling FFC DH 

using rU and yV 

Compute Z = Ze || Zs

Compute Zs by calling FFC DH 

using xV and yU 

Compute Ze by calling FFC DH 

using xV and tU 

Compute Z = Ze || Zs

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput)

Zeroize Z

Compute kdf(Z,OtherInput)

Zeroize Z

6.2.1.2 One-Pass Unified Model, C(1, 2, ECC CDH) 

This section describes the One-Pass Unified Model scheme from ANS X9.63. The prerequisites 

for this scheme shall be satisfied as specified in Section 6.2.1. In particular, party U shall obtain 

the static public key Qs,V of party V, and party V shall obtain the static public key Qs,U of party 

U. 

Note that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. 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 – an element 

of the finite field of size q – from the set of domain parameters D, U’s static private key 
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ds,U, and V’s static public key Qs,V. Convert Zs to a byte string (which is also denoted by 

Zs) using the Field-element-to-Byte-String Conversion specified in Appendix C.2, and 

then zeroize the results of all intermediate calculations used in the computation of Zs. If 

the call to the ECC CDH primitive outputs an error indicator, zeroize the results of all 

intermediate calculations used in the attempted computation of Zs, output an error 

indicator, and stop. 

3. Use the ECC CDH primitive to derive a shared secret Ze – another element of the finite 

field of size q – from the set of domain parameters D, U’s ephemeral private key de,U, and 

V’s static public key Qs,V. Convert Ze to a byte string (which is also denoted by Ze) using 

the Field-element-to-Byte-String Conversion specified in Appendix C.2, and then zeroize 

the results of all intermediate calculations used in the computation of Ze. If this call to the 

ECC CDH primitive outputs an error indicator, zeroize Zs and the results of all 

intermediate calculations used in the attempted computation of Ze, output an error 

indicator, and stop. 

4. Compute the shared secret Z = Ze || Zs. Zeroize the results of all intermediate calculations 

used in the computation of Z (including Ze and Zs). 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Receive an ephemeral public key Qe,U (purportedly) from U. If Qe,U is not received, 

output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Zs – an element 

of the finite field of size q – from the set of domain parameters D, V’s static private key 

ds,V, and U’s static public key Qs,U. Convert Zs to a byte string (which is also denoted by 

Zs) using the Field-element-to-Byte-String Conversion specified in Appendix C.2, and 

then zeroize the results of all intermediate calculations used in the computation of Zs. If 

the call to the ECC CDH primitive outputs an error indicator, zeroize the results of all 

intermediate calculations used in the attempted computation of Zs, output an error 

indicator, and stop. 
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4. Use the ECC CDH primitive to derive a shared secret Ze – another element of the finite 

field of size q – from the set of domain parameters D, V’s static private key ds,V, and U’s 

ephemeral public key Qe,U. Convert Ze to a byte string (which is also denoted by Ze) using 

the Field-element-to-Byte-String Conversion specified in Appendix C.2, and then zeroize 

the results of all intermediate calculations used in the computation of Ze. If this call to the 

ECC CDH primitive outputs an error indicator, zeroize Zs and the results of all 

intermediate calculations used in the attempted computation of Ze, output an error 

indicator, and stop. 

5. Compute the shared secret Z = Ze || Zs. Zeroize the results of all intermediate calculations 

used in the computation of Z (including Ze and Zs). 

6. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

7. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

The One-Pass Unified Model is summarized in Table 13. 

Table 13: One-Pass Unified Model Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key ds,U

2. Static public key Qs,U

1. Static private key ds,V

2. Static public key Qs,V

Ephemeral Data 1. Ephemeral private key de,U

2. Ephemeral public key Qe,U

N/A 
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Party U Party V 

Computation Compute Zs by calling ECC CDH 

using ds,U and Qs,V 

Compute Ze by calling ECC CDH 

using de,U and Qs,V

Compute Z = Ze || Zs

Compute Zs by calling ECC DH 

using ds,V and Qs,U

Compute Ze by calling ECC DH 

using ds,V and Qe,U 

Compute Z = Ze || Zs

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput)

Zeroize Z

Compute kdf(Z,OtherInput)

Zeroize Z

6.2.1.3 MQV1, C(1, 2, FFC MQV) 

This section describes the MQV1 scheme from ANS X9.42. The prerequisites for this scheme 

shall be satisfied as specified in Section 6.2.1. In particular, party U shall obtain the static public 

key yV of party V, and party V shall obtain the static public key yU of party U. 

Note that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key tU to V. 

2. Use the MQV1 form of the FFC MQV primitive in Section 5.7.2.1 to derive a shared 

secret Z – an integer in the range [2, p-2] – from the set of domain parameters D, U’s 

static private key xU, V’s static public key yV, U’s ephemeral private key rU, U’s 

ephemeral public key tU, and (for a second time) V’s static public key yV. If the call to the 

FFC MQV primitive outputs an error indicator, zeroize the results of all intermediate 

calculations used in the attempted computation of Z, output an error indicator, and stop. 

3. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 
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5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Receive an ephemeral public key tU (purportedly) from U. If tU is not received, output an 

error indicator and stop. 

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, output an error indicator and stop. 

3. Use the MQV1 form of the FFC MQV primitive in Section 5.7.2.1 to derive a shared 

secret Z – an integer in the range [2, p-2] – from the set of domain parameters D, V’s 

static private key xV, U’s static public key yU, V’s static private key xV (for a second time), 

V’s static public key yV, and U’s ephemeral public key tU. If the call to the FFC MQV 

primitive outputs an error indicator, zeroize the results of all intermediate calculations 

used in the attempted computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

MQV1 is summarized in Table 14. 
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Table 14: MQV1 Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key xU

2. Static public key yU

1. Static private key xV

2. Static public key yV

Ephemeral Data 1. Ephemeral private key rU

2. Ephemeral public key tU

N/A 

Computation C 

Compute Z by calling FFC MQV 

using xU, yV, rU, tU, and yV (again) 

Compute Z by calling FFC MQV 

using xV, yU, xV (again), yV, and tU

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) 

Zeroize Z

Compute kdf(Z,OtherInput) 

Zeroize Z

6.2.1.4 One-Pass MQV, C(1, 2, ECC MQV) 

This section describes the 1-Pass MQV scheme from ANS X9.63. The prerequisites for this 

scheme shall be satisfied as specified in Section 6.2.1. In particular, party U shall obtain the 

static public key Qs,V of party V, and party V shall obtain the static public key Qs,U of party U. 

Note that 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 transformation to a) establish a shared secret value Z with 

party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key Qe,U to V. 

2. Use the One-Pass MQV form of the ECC MQV primitive in Section 5.7.2.3.2 to derive a 

shared secret value Z – an element of the finite field of size q – from the set of domain 

parameters D, U’s static private key ds,U, V’s static public key Qs,V, U’s ephemeral private 

key de,U, U’s ephemeral public key Qe,U, and (for a second time) V’s static public key 
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Qs,V. If the call to the ECC MQV primitive outputs an error indicator, zeroize the results 

of all intermediate calculations used in the attempted computation of Z, output an error 

indicator, and stop. 

3. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-Byte 

String Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z. 

4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator.

Party V shall execute the following transformation to a) establish a shared secret value Z with 

party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Receive an ephemeral public key Qe,U (purportedly) from U. If Qe,U is not received, 

output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the One-Pass MQV form of the ECC MQV primitive in Section 5.7.2.3 to derive a 

shared secret value Z – an element of the finite field of size q – from the set of domain 

parameters D, V’s static private key ds,V, U’s static public key Qs,U, V’s static private key 

ds,V (for a second time), V’s static public key Qs,V, and U’s ephemeral public key Qe,U. If 

the call to the ECC MQV primitive outputs an error indicator, zeroize the results of all 

intermediate calculations used in the attempted computation of Z, output an error 

indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-Byte 

String Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 
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Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

The Full One-Pass MQV is summarized in Table 15. 

Table 15: One-Pass MQV Model Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key ds,U

2. Static public key Qs,U

1. Static private key ds,V

2. Static public key Qs,V

Ephemeral Data 1. Ephemeral private key de,U

2. Ephemeral public key Qe,U

N/A 

Computation Compute Z by calling ECC MQV 

using ds,U, Qs,V, de,U, Qe,U, and Qs,V

(again) 

Compute Z by calling ECC MQV 

using ds,V, Qs,U, ds,V (again), Qs,V, and 

Qe,U

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput)

Zeroize Z 

Compute kdf(Z,OtherInput)

Zeroize Z

6.2.1.5 Rationale for Choosing a C(1, 2) Scheme 

These schemes offer different assurances to the different parties participating in the exchange. 

One party (the initiator) has both static and ephemeral keys. The other party (the responder) has 

only a static key. 

Both parties are assured that only they and the other intended party can compute the shared 

secret. The initiator, by virtue of its ephemeral contribution, has assurance that previous derived 

secret keying material will not be reused. 

A compromise of the static private key of the initiator does not, by itself, compromise prior or 

future shared secrets (and therefore, secret keying material) of legitimate C(1, 2) transactions, 

nor does the compromise of only the initiator’s ephemeral private key. However, the 

compromise of the static private key of the responder leads to the compromise of all future 

shared secrets where this party acts as a responder using the same static key. Additionally, any 
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previous shared secrets (and therefore, secret keying material) that are computed with this party 

acting as responder become compromised if a malicious party stored the initiator’s ephemeral 

public key. 

For a given set of FFC domain parameters, MQV1 is expected to have better performance than 

the dhHybridOneFlow. For a given set of ECC domain parameters, One-Pass MQV is expected 

to have better performance than the One-Pass Unified Model. 

Key confirmation can be provided in either or both directions using the methods specified in 

Sections 8.4.4, 8.4.5, and 8.4.6. Upon completion of a Key Confirmation as specified in Section 

8, the recipient of the confirmation has assurance of the identifier of the key confirmation 

provider (through the identifier bound to the static key), as well as confirmation of the active 

participation of the provider 

6.2.2 Initiator Generates Only an Ephemeral Key Pair; Responder Has Only a 

Static Key Pair, C(1, 1) 

For these schemes, Party U generates an ephemeral key pair, but has 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) 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 7). 

Figure 7: General Protocol when the Initiator has Only an Ephemeral Key Pair, and the 

Responder has Only a Static Key Pair 

Prerequisites: The following are prerequisites for the use of all C(1, 1) schemes. 

1. Each party shall have an authentic copy of the same set of domain parameters, D. These 

parameters shall 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 shall have assurance of the validity of these domain parameters 

as specified in Section 5.5.2.

U V

V’s Static Public Key

U’s Ephemeral Public Key

1. U uses its ephemeral private key 

and V’s static public key to form 

a shared secret.

2. U invokes the Key Derivation 

Function using the shared secret.

1. V uses its static private key and 

U’s ephemeral public key to form 

a shared secret.

2. V invokes the Key Derivation 

Function using the shared secret.
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2. The responder shall have 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, the static key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). The 

responder shall obtain assurance of the validity of its own static public key as specified in 

Section 5.6.2.1. The responder shall obtain assurance of its possession of the correct 

value of its own private key as specified in Section 5.6.3.1. 

3. The parties shall have agreed upon an Approved key derivation function (see Section 5.8) 

as well as an Approved hash function appropriate for use with the key derivation function 

and associated parameters (see Section 5.5). If key confirmation is used, the parties shall

have agreed upon an Approved MAC and associated parameters (see Tables 1 and 2). 

4. Prior to or during the key agreement process, each party shall obtain the identifier 

associated with the other party during the key agreement scheme. The initiator shall

obtain the static public key that is bound to the responder’s identifier. This static public 

key shall be obtained in a trusted manner (e.g., from a certificate signed by a trusted CA). 

The initiator shall obtain assurance of the validity of the responder’s static public key as 

specified in Section 5.6.2.2 

The following is a prerequisite for using the derived keying material for purposes beyond the 

C(1,1) scheme itself. 

The initiator shall obtain assurance that the responder is (or was) in possession of the 

appropriate static private key, as specified in Section 5.6.3.2. 

6.2.2.1 dhOneFlow, C(1, 1, FFC DH) 

This section describes the dhOneFlow scheme from ANS X9.42. The prerequisites for this 

scheme shall be satisfied as specified in Section 6.2.2. In particular, party U shall obtain the 

static public key yV of party V. 

Note that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. Send the public key tU to V. 

2. Use the FCC DH primitive in Section 5.7.1.1 to derive a shared secret Z – an integer in 

the range [2, p-2] – from the set of domain parameters D, U’s ephemeral private key rU,

and V’s static public key yV. If the call to the FFC DH primitive outputs an error 

indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 
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3. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Receive an ephemeral public key tU (purportedly) from U. If tU is not received, output an 

error indicator and stop. 

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, output an error indicator and stop. 

3. Use the FCC DH primitive in Section 5.7.1.1 to derive a shared secret Z – an integer in 

the range [2, p-2] – from the set of domain parameters D, V’s static private key xV, and 

U’s ephemeral public key tU. If the call to the FFC DH primitive outputs an error 

indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 
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dhOneFlow is summarized in Table 16. 

 Table 16: dhOneFlow Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data N/A 1. Static private key xV

2. Static public key yV

Ephemeral Data 1. Ephemeral private key rU

2. Ephemeral public key tU

N/A 

Computation Compute Z by calling FFC DH using 

rU and yV 

Compute Z by calling FFC DH using 

xV and tU 

Derive Secret 

Material 

Compute kdf(Z,OtherInput) 

Zeroize Z

Compute kdf(Z,OtherInput) 

Zeroize Z

6.2.2.2 One-Pass Diffie-Hellman, C(1, 1, ECC CDH) 

This section describes the One-Pass Diffie-Hellman scheme from ANS X9.63. The prerequisites 

for this scheme shall be satisfied as specified in Section 6.2.2. In particular, party U shall obtain 

the static public key Qs,V of party V. 

Note that 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 in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive 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. 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 Z – an element of 

the finite field of size q – from the set of domain parameters D, U’s ephemeral private 

key de,U, and V’s static public key Qs,V. If this call to the ECC CDH primitive outputs an 

error indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 
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3. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-ByteString Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z. 

4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Receive an ephemeral public key Qe,U (purportedly) from U. If Qe,U is not received, 

output an error indicator and stop. 

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, output an error indicator and stop. 

3. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z – an element of 

the finite field of size q – from the set of domain parameters D, V’s static private key ds,V,

and U’s ephemeral public key Qe,U. If this call to the ECC CDH primitive outputs an error 

indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 

4. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-ByteString Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z. 

5. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV). (See Section 5.8.) If the key 

derivation function outputs an error indicator, zeroize all copies of Z, output an error 

indicator, and stop. 

6. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

The One-Pass Diffie-Hellman is summarized in Table 17. 
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Table 17: One-Pass Diffie-Hellman Key Agreement Scheme Summary 

Party U Party V 

Domain 

Parameters 

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

Static Data N/A 1. Static private key ds,V

2. Static public key Qs,V

Ephemeral Data 1. Ephemeral private key de,U

2. Ephemeral public key Qe,U

N/A 

Computation Compute Z by calling ECC CDH 

using de,U and Qs,V

Compute Z by calling ECC CDH 

using ds,V and Qe,U 

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) 

Zeroize Z

Compute kdf(Z,OtherInput) 

Zeroize Z 

6.2.2.3 Rationale in Choosing a C(1, 1) Scheme 

In these schemes, one party (the initiator) has only an ephemeral key, while the other party (the 

responder) has only a static key. Different assurances are given to the different parties in the key 

establishment transaction. 

Due to the use of a static key by the responder, the initiator (only) has assurance that no 

unintended party can compute the shared secret without the compromise of private material. The 

responder has no such assurance, since the responder has no assurance about who is providing 

the ephemeral key (that is, unless there are additional appropriate elements in the protocol using 

this scheme). 

Due to the use of an ephemeral key by the initiator, the initiator has assurance that a previous 

shared secret will not be reused. 

There is no assurance to either party that the security of the shared secret is isolated from 

compromises of private keys from prior or future C(1, 1) transactions. A compromise of the 

initiator’s ephemeral private key compromises the shared secret for that individual transaction 

only. However, a compromise of the responder’s static private key compromises all shared 

secrets resulting from future C(1, 1) transactions in which that party is a responder, as well as 

any shared secrets resulting from prior C(1, 1) transactions for which a malicious party stored the 

ephemeral public keys. 

The responder does not have any assurances as to the identifier of the initiator. 
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The responder may provide key confirmation to the initiator as specified in Section 8.4.7, giving 

the initiator assurance as to the identifier and active participation of the responder. 

6.3 Scheme Using No Ephemeral Key Pairs, C(0, 2) 

In this category, the parties use only static key pairs that have been generated using the same 

domain parameters. Each party obtains the other party’s static public keys. A nonce shall be sent 

by party U (the initiator) to party V (the scheme responder) to ensure that the derived keying 

material is different for each key establishment transaction. 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 function, the shared secret, and the nonce (see Figure 

8). 

Figure 8: General Protocol when Each Party has only a Static Key Pair 

Prerequisites: The following are prerequisites for the use of all C(0, 2) schemes. 

1. Each party shall have an authentic copy of the same set of domain parameters, D. These 

parameters shall 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 shall have assurance of the validity of these domain parameters 

as specified in Section 5.5.2.

2. Each party shall have 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, 
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the static key pair is (x, y); for ECC schemes, the static key pair is (ds, Qs). Each party 

shall obtain assurance of the validity of its own static public key as specified in Section 

5.6.2.1. Each party shall obtain assurance of its possession of the correct value for its 

own private key as specified in Section 5.6.3.1. 

3. The parties shall have agreed upon an Approved key derivation function (see Section 5.8) 

as well as an Approved hash function appropriate for use with the key derivation function 

and associated parameters (see Section 5.5). In addition the parties shall agree on the 

form of the nonce (see Section 5.4). If key confirmation is used, the parties shall have 

agreed upon the form of the an Approved MAC and associated parameters (see Tables 1 

and 2). 

4. Prior to or during the key agreement process, each party shall obtain the identifier 

associated with the other party during the key agreement scheme and the static public key 

that is bound to that identifier. These static public keys shall be obtained in a trusted 

manner (e.g., from a certificate signed by a trusted CA). Each party shall obtain 

assurance of the validity of the other party’s static public key as specified in Section 

5.6.2.2. 

The recipient of a static public key shall obtain assurance that its (claimed) owner is (or was) in 

possession of the corresponding static private key, as specified in Section 5.6.3.2. 

6.3.1 dhStatic, C(0, 2, FFC DH) 

This section describes the dhStatic scheme from ANS X9.42. The prerequisites for this scheme 

shall be satisfied as specified in Section 6.3. In particular, party U shall obtain the static public 

key yV of party V, and party V shall obtain the static public key yU of party U. 

Note that U and V must use identical orderings of the bit strings that are input to the key 

derivation function. In particular, this requirement applies to the placement of NonceU in the

PartyUInfo subfield of OtherInfo.

Party U shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive secret keying material as follows: 

1. Obtain a nonce, NonceU (see Section 5.4). Send NonceU to V. 

2. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Z – an integer in 

the range [2, p-2] – from the set of domain parameters D, U’s static private key xU, and 

V’s static public key yV. If the call to the FFC DH primitive outputs an error indicator, 

zeroize the results of all intermediate calculations used in the attempted computation of Z, 

output an error indicator, and stop. 

3. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 
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4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDu and IDv, and NonceU.). NonceU shall be in the 

PartyUInfo subfield of OtherInfo. If the key derivation function outputs an error 

indicator, zeroize all copies of Z, output an error indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party U, and b) derive shared secret keying material from Z. 

Actions: V shall derive secret keying material as follows: 

1. Obtain U’s nonce, NonceU, from U. If NonceU is not available, output an error indicator 

and stop. 

2. Use the FFC DH primitive in Section 5.7.1.1 to derive a shared secret Z – an integer in 

the range [2, p-2] – from the set of domain parameters D, V’s static private key xV, and 

U’s static public key yU. If the call to the FFC DH primitive outputs an error indicator, 

zeroize the results of all intermediate calculations used in the attempted computation of Z, 

output an error indicator, and stop. 

3. Convert Z to a byte string (which is also denoted by Z) using the Integer-to-Byte-String 

Conversion specified in Appendix C.1, and then zeroize the results of all intermediate 

calculations used in the computation of Z. 

4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV, and NonceU). NonceU shall be in the 

PartyUInfo subfield of OtherInfo. If the key derivation function outputs an error 

indicator, zeroize all copies of Z, output an error indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

dhStatic is summarized in Table 18. 

Table 18: dhStatic Key Agreement Scheme Summary 

Party U Party V 
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Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key xU

2. Static public key yU

1. Static private key xV

2. Static public key yV

Ephemeral Data NonceU 

Computation Compute Z by calling FFC DH 

using xU, and yV

Compute Z by calling FFC DH using 

xV, and yU

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) using 

NonceU 

Zeroize Z

Compute kdf(Z,OtherInput) using 

NonceU

Zeroize Z

6.3.2 Static Unified Model, C(0, 2, ECC CDH) 

This section describes the Static Unified Model scheme from ANS X9.63. The prerequisites for 

this scheme shall be satisfied as specified in Section 6.3. In particular, party U shall obtain the 

static public key Qs,V of party V, and party V shall obtain the static public key Qs,U of party U. 

Note that U and V must use identical orderings of the bit strings that are input to the key 

derivation function. The same requirement applies to the placement of NonceU in the PartyUInfo

subfield of OtherInfo (see item 4 below).

Party U shall execute the following key agreement transformation in order to a) establish a 

shared secret value Z with party V, and b) derive shared secret keying material from Z. 

Actions: U shall derive secret keying material as follows: 

1. Obtain a nonce, NonceU (see Section 5.4). Send NonceU to V. 

2. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z – an element of 

the finite field of size q – from the set of domain parameters D, U’s static private key ds,U,

and V’s static public key Qs,V. If the call to the ECC CDH primitive outputs an error 

indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 

3. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-ByteString Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Z.
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4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDU and IDV, and NonceU). NonceU shall be in the 

PartyUInfo subfield of OtherInfo. If the key derivation function outputs an error 

indicator, zeroize all copies of Z, output an error indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Party V shall execute the following key agreement transformation in order to a) establish a 

shared secret value, Z, with party U, and b) derive shared secret keying material from Z. 

1. Obtain U’s nonce, NonceU, from U. If NonceU is not if available, output an error indicator 

and stop. 

2. Use the ECC CDH primitive in Section 5.7.1.2 to derive a shared secret Z – an element of 

the finite field of size q – from the set of domain parameters D, V’s static private key ds,V,

and U’s static public key Qs,U. If the call to the ECC CDH primitive outputs an error 

indicator, zeroize the results of all intermediate calculations used in the attempted 

computation of Z, output an error indicator, and stop. 

3. Convert Z to a byte string (which is also denoted by Z) using the Field-element-to-ByteString Conversion specified in Appendix C.2, and then zeroize the results of all 

intermediate calculations used in the computation of Zs.

4. Use the agreed-upon key derivation function to derive secret keying material 

DerivedKeyingMaterial of length keydatalen bits from the shared secret value Z and 

OtherInput (including the identifiers IDu and IDv, and NonceU.). NonceU shall be in the 

PartyUInfo subfield of OtherInfo. If the key derivation function outputs an error 

indicator, zeroize all copies of Z, output an error indicator, and stop. 

5. Zeroize all copies of the shared secret Z and output DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length keydatalen bits or an error indicator. 

Note: If key confirmation is to be incorporated into this scheme, additional input may be 

required, and additional steps must be taken by U and V beyond the computation of 

DerivedKeyingMaterial. See Section 8 for details. 

Static Unified Model is summarized in Table 19. 
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Table 19: Static Unified Model Key Agreement Scheme Summary

Party U Party V 

Domain 

Parameters 

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

Static Data 1. Static private key ds,U

2. Static public key Qs,U

1. Static private key ds,V

2. Static public key Qs,V

Ephemeral Data NonceU 

Computation Compute Z by calling ECC CDH 

using ds,U, and Qs,V

Compute Z by calling ECC CDH 

using ds,V, and Qs,U

Derive Secret 

Keying Material 

Compute kdf(Z,OtherInput) using 

NonceU

Zeroize Z 

Compute kdf(Z,OtherInput) using 

NonceU

Zeroize Z 

6.3.3 Rationale in Choosing a C(0, 2) Scheme 

As identifiers are bound to the static public keys that are used, each party has assurance that the 

intended party and no other party can compute the shared secret, without the compromise of a 

private key. If an entity’s private key is compromised, then all shared secrets of prior and future 

C(0, 2) transactions involving that party are compromised. 

Both parties are assured that only they and the other intended party can compute the shared 

secret. The initiator, by virtue of NonceU, has assurance that previous derived secret keying 

material will not be reused. 

Key confirmation can be provided in either or both directions for these schemes as specified in 

Sections 8.4.8, 8.4.9, and 8.4.10. Upon completion of a Key Confirmation, the recipient of the 

confirmation has assurance of the identifier of the key confirmation provider (through the 

identifier bound to the static key), as well as confirmation of the active participation of the 

provider. 

7. DLC based Key Transport 

The FFC and ECC key agreement schemes in this Recommendation that employ a receiver’s 

static key6

 may be followed by a key transport scheme using an Approved key-wrapping 

 

6

 To prevent receiver identifier spoofing, since the sender would know the identifier of the intended receiver. 
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algorithm, such as the AES key-wrap algorithm [8]. In order to meet this requirement, DLCbased key transport shall be used with one of the following types of schemes: 

1. C(2, 2) schemes, 

2. C(1, 2) schemes, 

3. C(1, 1), schemes with the receiver serving as the scheme responder, and 

4. C(0, 2) schemes. 

DLC-based key transport shall not be used with C(2, 0) schemes, or C(1, 1) schemes with the 

receiver serving as the scheme initiator. 

The DLC-based key transport scheme is as follows: 

1. A key agreement scheme is used to establish a shared secret and derive keying material 

between the sender and the receiver. Key confirmation (as specified in Section 8) may 

optionally be used to provide assurance that the shared secret is the same for both the 

initiator and responder. 

2. The sender obtains a KeyWrappingKey from the DerivedKeyingMaterial. 

3. The sender selects secret keying material, KeyingMaterial, to transport to the receiver. 

4. The sender calculates WrappedKey = KeyWrap(KeyWrappingKey, KeyingMaterial) 

using KeyWrap( ), an Approved key wrapping algorithm. 

5. The sender sends WrappedKey to the receiver. 

6. The receiver receives WrappedKey from the sender. 

7. The receiver obtains a KeyWrappingKey from the DerivedKeyingMaterial that is 

computed by applying the key derivation function to the shared secret. 

8. The receiver calculates KeyingMaterial = KeyUnwrap(KeyWrappingKey, WrappedKey) 

using KeyUnwrap( ), the corresponding key unwrapping algorithm. 

Note that if the key agreement scheme used in Step 1 is such that the receiver (acting as the 

responder) does not contribute an ephemeral key pair to the calculation of the shared secret (that 

is, either a C(1, 2), C(1, 1), or C(0, 2) scheme has been used), then Steps 1 through 5 can be 

performed by the sender (acting as the initiator) without direct involvement of the receiver. This 

can be useful in a store-and-forward environment, such as e-mail. 

A default “rule” of this Recommendation is that ephemeral keys shall not be reused (see Section 

5.6.4.3). An exception to this rule is that the sender may use the same ephemeral key pair in step 

1 above in multiple DLC-based Key Transport transactions if the same secret keying material is 

being transported in each transaction and if all these transactions occur “simultaneously” (or 
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within a short period of time). However, the security properties of the key establishment scheme 

may be affected by reusing the ephemeral key in this manner. 

8. Key Confirmation 

The term key confirmation refers to the use of explicit messages (such as messages containing 

the information defined in this Recommendation) to provide assurance to one party (the key 

confirmation recipient) that another party (the key confirmation provider) actually possesses the 

correct shared secret and/or derived keying material (from the key confirmation recipient’s 

perspective). Key agreement, accompanied by key confirmation (as described in this 

Recommendation), can also be used to provide the recipient with assurance of either the 

provider’s current or prior possession of the static private key that is associated with a particular 

static public key (see Section 8.1). A key establishment scheme is said to provide “unilateral 

key confirmation” when it provides this assurance to only one of the participants, and the 

scheme is said to provide “bilateral key confirmation” when this assurance is provided to both 

participants (that is, unilateral key confirmation is provided in both directions). 

Oftentimes, key confirmation is provided implicitly by a means outside of the key establishment 

scheme (for example, by decrypting an encrypted message from the other party using a key 

derived from the shared secret), but this is not always the case. In some cases, it may be 

appropriate to include the exchange of explicit key confirmation information within the key 

establishment process itself. Key confirmation may enhance the security properties that are 

achieved by a key establishment scheme. For key confirmation to comply with this 

Recommendation, key confirmation shall be incorporated into key establishment schemes as 

specified in this section. 

Unilateral key confirmation may be incorporated into any scheme that uses a static key pair 

associated with the provider. Successful key confirmation will provide assurance to the recipient 

that the provider has correctly derived keying material. Bilateral key confirmation may be added 

to any key agreement scheme in which each party possesses a static key pair. 

Table 20 provides a summary of the scheme classes for which unilateral or bilateral key 

confirmation is specified. Note that key confirmation for the C(2, 0) key agreement schemes is 

not specified, since neither party has a static key pair; if needed, key confirmation would have to 

be provided by some other means. 

Table 20: Key Agreement Schemes Using Unilateral and Bilateral Key Confirmation 

Scheme Class Unilateral Bilateral 

C(2, 2) U to V, or V to U Yes 

C(2, 0) No No 
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Scheme Class Unilateral Bilateral 

C(1, 2) U to V, or V to U Yes 

C(1, 1) V to U No 

C(0, 2) U to V, or V to U Yes 

If key confirmation is incorporated into a scheme in which a recipient does not provide an 

ephemeral public key, a nonce shall be provided for the key confirmation process. 

The process used to provide key confirmation requires string representations of the ephemeral 

public keys. The same notation will be used to represent these keys for schemes based on Finite 

Field cryptography (FFC) and elliptic curve cryptography (ECC): 

EphemPubKeyi = the byte string representation of a participant i’s ephemeral public key. 

For FFC schemes, an ephemeral public key, ti, is converted from a field element in Fp to a byte 

string by representing the field element as an integer in the interval [2, p-2], and then converting 

the integer to a byte string as specified in Appendix C.1. 

For ECC, the coordinates of the ephemeral public key, Qe, i, are converted from field elements to 

byte strings as specified in Appendix C.2 and concatenated (with x first) to form a single byte 

string. 

8.1 Assurance of Possession Considerations when using Key Confirmation 

Key agreement, accompanied by key confirmation (as described in this Recommendation), can 

be used to provide the recipient with assurance of either the provider’s current or prior 

possession of the static private key that is associated with a particular static public key. Current 

possession and prior possession are defined relative to the time of a particular key agreement 

transaction (including the delivery of the key confirmation message to the recipient). 

As has been observed previously (see Section 5.6.3.2), an adequate demonstration of current 

possession of a static private key can be used to provide assurance that the claimed owner of a 

static key pair is the true owner. Note also that as time passes, even the true owner of a key pair 

may lose possession of the associated private key, either deliberately or due to an error. For these 

reasons, assurance of current possession can be of value for some applications. When it is 

desired (or required), assurance of current possession shall be obtained as specified below. 

While assurance of prior possession may be sufficient for some purposes, if both assurance of 

current and prior possession are feasible to obtain, then assurance of current possession is 

preferred. 

As stated in Section 5.6.3.2, in order for the recipient of a successful key confirmation to obtain 

assurance of the key confirmation provider’s current possession of the static private key 
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corresponding to the static public key claimed by that provider (and used in the key agreement 

transaction), the underlying key agreement scheme used shall be one of the following, and the 

recipient seeking assurance shall serve as the key agreement initiator: 

• dhHybridOneFlow or (Cofactor) One-Pass Unified Model 

• MQV1 or One-Pass MQV. 

• dhOneFlow or (Cofactor) One-Pass Diffie-Hellman. 

The key agreement scheme (including the key confirmation) shall be performed as described in 

this Recommendation. 

In each of the schemes specified above, the key confirmation recipient contributes an ephemeral 

public key that is arithmetically combined with the static private key claimed by the provider as 

part of the provider’s computation of the shared secret. The unpredictability of the ephemeral 

public key contributed by the recipient ensures that the provider’s calculations are performed 

contemporaneously, and so a successful computation of the shared secret (as indicated through 

key confirmation) offers assurance that the provider has current possession of that static private 

key. 

If the underlying key agreement scheme is not one of those indicated above (but is one of those 

described in this Recommendation), it is possible that some of the calculations required of the 

key confirmation provider (in particular, those portions depending on the provider’s knowledge 

of a static private key) were completed at a time prior to the current key agreement transaction. 

(For example, if the key agreement scheme were either dhHybrid1or the Full Unified Model, the 

static shared secret Zs could be computed in advance or recovered from a previous transaction.) 

The same is true if the key agreement scheme employed is one of those listed above, but the key 

confirmation recipient does not serve as the key agreement initiator. In such cases, the recipient 

of a successful key confirmation might obtain assurance that the key confirmation provider was 

— at some point in time — in possession of the static private key corresponding to the static 

public key claimed by that provider. However, the recipient would not obtain assurance that the 

provider actually had possession of the static private key during the current transaction. 

8.2 Unilateral Key Confirmation for Key Agreement Schemes 

Unilateral key confirmation occurs when one participant in a key establishment scheme (the 

“provider”) provides assurance to the other participant (the “recipient”) that the same shared 

secret has actually been generated by both the provider and the recipient, and thus provides 

assurance that shared keying material has been established between the provider and the 

recipient. This is an optional feature for any scheme in which the provider possesses a static key 

pair. If the intended key confirmation recipient does not contribute an ephemeral public key or 

nonce during the key establishment process, then a nonce shall be provided for key confirmation. 

Unilateral key confirmation may be added in either direction to the C(2, 2), C(1, 2) and C(0, 2) 

schemes; it may also be added to the C(1, 1) schemes, but in one direction only: when the 
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scheme Responder (V) is the key confirmation provider, and the scheme Initiator (U) is the key 

confirmation recipient (see Table 20 in Section 8). 

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. Note that the provider may be either 

the scheme initiator (party U) or the scheme responder (party V), as long as the provider has a 

static key pair, and the recipient is the other party. 

1. If the recipient does not have an ephemeral key pair and has not already provided a nonce 

as part of the scheme, then the recipient shall provide a nonce to be used in its place (see 

Section 5.4). 

2. The provider computes 

MacDataP = message_stringP || IDP || IDR || EphemDataP || EphemDataR {|| Text}

where 

message_stringP is a six byte string with a value of “KC_1_U” or “KC_1_V” depending 

on whether U or V is providing the MacTag. Note that these values will differ for 

bilateral key confirmation in Section 8.3.

IDP is the identifier of the provider. 

IDR is the identifier of the recipient. 

EphemDataP = EphemPubKeyP (if used in the key agreement scheme),

NonceP (if there is no EphemPubKeyP, and NonceP is required to 

derive secret keying material, for example, when P is the 

initiator in a C(0, 2) scheme), or

 Null (otherwise), where Null is the empty byte string.

EphemDataR = EphemPubKeyR (if used in the key agreement scheme) 

NonceR (otherwise). 

Text1 is an optional bit string that may be used during key confirmation and that is known 

by the parties establishing the secret keying material. 

3. After computing the shared secret and applying the key derivation function to obtain 

DerivedKeyingMaterial (see Section 5.8), the provider parses DerivedKeyingMaterial

into two parts, MacKey and KeyData: 

MacKey || KeyData = DerivedKeyingMaterial.

4. The provider computes MacTagP (see Section 5.2.1) and sends it to the recipient: 

MacTagP = MAC (MacKey, MacLen, MacDataP). 
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5. The recipient computes MacDataP, MacKey, KeyData and MacTagP in the same manner 

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 is assured 

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) from which 

MacKey and KeyData are derived (see Section 5.8). Thus, the recipient also has 

assurance that the provider could compute KeyData correctly. 

6. Zeroize the MacKey. 

8.3 Bilateral Key Confirmation for Key Agreement Schemes 

Bilateral key confirmation is obtained by performing unilateral key confirmation in both 

directions: from a provider U to a recipient V, and from a provider V to a recipient U. U and V 

may use different values for the optional Text in their MacTag computations, provided that both 

parties are aware of the value(s) used. message_stringP is a six byte string with a value of 

“KC_2_U” or “KC_2_V”, depending on whether U or V is providing the MacTag. Bilateral key 

confirmation may be added to the C(2, 2), C(1, 2) and C(0, 2) schemes, as shown in the relevant 

subsections of Section 8.4. 

8.4 Incorporating Key Confirmation into a Key Agreement Scheme 

This section illustrates how to incorporate key confirmation (as described in Section 8.2 and 

Section 8.3 above) into specific key agreement schemes of Section 6. 

The flow depictions separate the scheme flow from the key confirmation. The depictions and 

accompanying discussions assume that the prerequisites of the scheme have been satisfied, that 

the key agreement transaction has proceeded successfully (as described in Section 6) through the 

stage of key derivation, and that the received MacTags are successfully verified as specified in 

Section 5.2.2. If the MacTags do not verify, then key confirmation has not been obtained, and the 

key agreement transaction shall be discontinued. 

8.4.1 C(2, 2) Scheme with Unilateral Key Confirmation Provided by U to V 

Figure 9 depicts a typical flow for a C(2, 2) scheme with unilateral key confirmation from U to 

V. In this situation, party U, the scheme initiator, and party V, the scheme responder, 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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Figure 9: C(2, 2) Scheme with Unilateral Key Confirmation from Party U to Party V 

To provide (and receive) key confirmation (as described in Section 8.2), U (and V) set 

EphemDataU = EphemPubKeyU, and EphemDataV = EphemPubKeyV: 

Party U provides MacTagU to V (as specified in Section 8.2, 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 {|| Text}. 

The recipient V 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 U. 
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8.4.2 C(2, 2) Scheme with Unilateral Key Confirmation Provided by V to U 

Figure 10 depicts a typical flow for a C(2, 2) scheme with unilateral key confirmation from V to 

U. In this situation, party V, the scheme responder, and party U, the scheme initiator, 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 that party V has actively participated in the process. 

Figure 10: C(2, 2) Scheme with Unilateral Key Confirmation from Party V to Party U 

To provide (and receive) key confirmation (as described in Section 8.2), V (and U) set 

EphemDataV = EphemPubKeyV, and EphemDataU = EphemPubKeyU: 

Party V provides MacTagV to U (as specified in Section 8.2, 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 {|| Text}. 

The recipient U 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 V. 

Note that in Figure 10, 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. 

8.4.3 C(2, 2) Scheme with Bilateral Key Confirmation 

Figure 11 depicts a typical flow for a C(2, 2) scheme with bilateral key confirmation. In this 

method, party U, the scheme initiator, and party V, the scheme responder, assume the roles of 

both the provider and the recipient in order to obtain bilateral key confirmation. The successful 

Party U

(Initiator

and KC

Recipient)

Party V

(Responder

and KC

Provider)

EphemPubKeyU

EphemPubKeyV, MacTagV

Provider’s Static Public Key

Recipient’s Static Public Key

C(2,2) Scheme

Key Confirmation

Party U

(Initiator

and KC

Recipient)

Party V

(Responder

and KC

Provider)

EphemPubKeyU

EphemPubKeyV, MacTagV

Provider’s Static Public Key

Recipient’s Static Public Key

C(2,2) Scheme

Key Confirmation
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completion of the key confirmation process provides each party with assurance that the other 

party has derived the same secret Z value, and also provides each party with assurance that the 

other party has actively participated in the process. 

Figure 11: C(2, 2) Scheme with Bilateral Key Confirmation 

To provide bilateral key confirmation (as described in Section 8.3), U and V exchange and verify 

MacTags that have been computed (as specified in Sections 5.2.1) using EphemDataU = 

EphemPubKeyU, and EphemDataV = EphemPubKeyV: 

Party V provides MacTagV to U (as specified in Section 8.2, with P = V and R = U); MacTagV is 

computed by V (and verified by U) using 

 MacDataV = “KC_2_V” || IDV || IDU || EphemPubKeyV || EphemPubKeyU {|| Text1}. 

Party U provides MacTagU to V (as specified in Section 8.2, with P = U and R = V); MacTagU is 

computed by U (and verified by V) using 

 MacDataU = “KC_2_U” || IDU || IDV || EphemPubKeyU || EphemPubKeyV {|| Text2}. 

Note that in Figure 11, 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. 

8.4.4 C(1, 2) Scheme with Unilateral Key Confirmation Provided by U to V 

Figure 12 depicts a typical flow for a C(1, 2) scheme with unilateral key confirmation from U to 

V. In this situation, party U, the scheme initiator, and party V, the scheme responder, assume the 

roles of key confirmation provider and recipient, respectively. Since V does not contribute an 

MacTagU

Party U Party V

EphemPubKeyU

EphemPubKeyV, MacTagV

Party U’s Static Public Key

Party V’s Static Public Key

C(2,2) Scheme

Key Confirmation

(Initator) (Responder)

(KC Recipient)

(KC Provider)

(KC Provider)

(KC Recipient)
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ephemeral public key during the key agreement process, a nonce (NonceV) shall be provided to 

U prior to the computation of the MacTag and used as the 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 (see 

Section 5.4), then party V also obtains assurance that party U has actively participated in the 

process. 

Figure 12: C(1, 2) Scheme with Unilateral Key Confirmation from Party U to Party V 

To provide (and receive) key confirmation (as described in Section 8.2), U (and V) set 

EphemDataU = EphemPubKeyU, and EphemDataV = NonceV: 

Party U provides MacTagU to V (as specified in Section 8.2, 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 {|| Text}. 

The recipient V 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 U. 

8.4.5 C(1, 2) Scheme with Unilateral Key Confirmation Provided by V to U 

Figure 13 depicts a typical flow for a C(1, 2) scheme with unilateral key confirmation from V to U. 

In this situation, party V, the scheme responder, and party U, the scheme initiator, 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 that party V has actively participated in the process. 

To provide (and receive) key confirmation (as described in Section 8.2), both parties set 

EphemDataV = Null, and EphemDataU = EphemPubKeyU: 

Party U

(Initiator 

and KC

Provider)

Party V

(Responder

and KC

Recipient) NonceV

EphemPubKeyU

Party U’s Static Public Key

Party V’s Static Public Key

MacTagU

C(1,2) Scheme

Key Confirmation
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Party V provides MacTagV to U (as specified in Section 8.2, 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 {|| Text}. 

The recipient U 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. 

Figure 13: C(1, 2) Scheme with Unilateral Key Confirmation from Party V to Party U 

 8.4.6 C(1, 2) Scheme with Bilateral Key Confirmation 

Figure 14 depicts a typical flow for a C(1, 2) scheme with bilateral key confirmation. In this 

method, party U, the scheme initiator, and party V, the scheme responder, assume the roles of 

both the provider and the recipient in order to obtain bilateral key confirmation. V shall

contribute a nonce (NonceV ) prior to U’s computation of the MacTagU. The successful 

completion of the key confirmation process provides each party with assurance that the other 

party 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 (see Section 5.4), then party V also 

obtains assurance that party U has actively participated in the process. 

C(1,2) Scheme

Party U

(Initiator

and KC

Recipient)

Party V

(Responder

and KC

Provider)

EphemPubKeyU

MacTagV

Party U’s Static Public Key

Party V’s Static Public Key

Key Confirmation
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Figure 14: C(1, 2) Scheme with Bilateral Key Confirmation 

To provide bilateral key confirmation (as described in Section 8.3), U and 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 U (as specified in Section 8.2, with P = V and R = U); MacTagV is 

computed by V (and verified by U) using 

 MacDataV = “KC_2_V” || IDV || IDU || NonceV || EphemPubKeyU {|| Text1}. 

Party U provides MacTagU to V (as specified in Section 8.2, with P = U and R = V); MacTagU is 

computed by U (and verified by V) using 

 MacDataU = “KC_2_U” || IDU || IDV || EphemPubKeyU || NonceV {|| Text2}. 

Note that in Figure 14, 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 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. 

8.4.7 C(1, 1) Scheme with Unilateral Key Confirmation Provided by V to U 

Figure 15 depicts a typical flow for a C(1, 1) scheme with unilateral key confirmation from V to U. 

In this situation, party V, the scheme responder, and party U, the scheme initiator, 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 that party V has actively participated in the process. 

MacTagU

Party U Party V

EphemPubKeyU

NonceV, MacTagV

Party U’s Static Public Key

Party V’s Static Public Key

C(1,2) Scheme

Key Confirmation

(Initiator) (Responder)

(KC Recipient) (KC Provider)

(KC Provider) (KC Recipient)
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Figure 15: C(1, 1) Scheme with Unilateral Key Confirmation from Party V to Party U 

To provide (and receive) key confirmation (as described in Section 8.2), both parties set 

EphemDataV = Null, and EphemDataU = EphemPubKeyU: 

Party V provides MacTagV to U (as specified in Section 8.2, 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 {|| Text}. 

The recipient U 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. 

8.4.8 C(0, 2) Scheme with Unilateral Key Confirmation Provided by U to V 

Figure 16 depicts a typical flow for a C(0, 2) scheme with unilateral key confirmation from U to V. 

In this situation, party U, the scheme initiator, and party V, the scheme responder, assume the roles 

of key confirmation provider and recipient, respectively. V shall contribute a nonce (NonceV) to U 

prior to the generation of the MacTag. 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 (see Section 5.4), then party V also obtains assurance that party U has actively 

participated in the process. 

Party U

(Initiator

and KC

Recipient

Party V

(Responder

and KC

Provider

EphemPubKeyU

MacTagV 

Party V’s Static Public Key

C(1,1) Scheme

Key Confirmation
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Figure 16: C(0, 2) Scheme with Unilateral Key Confirmation from Party U to Party V 

To provide (and receive) key confirmation (as described in Section 8.2), U (and V) set 

EphemDataU = NonceU, and EphemDataV = NonceV: 

Party U provides MacTagU to V (as specified in Section 8.2, 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 {|| Text}. 

The recipient V 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 U. 

8.4.9 C(0, 2) Scheme with Unilateral Key Confirmation Provided by V to U 

Figure 17 depicts a typical flow for a C(0, 2) scheme with unilateral key confirmation from V to 

U. In this situation, party V, the scheme responder, and party U, the scheme initiator, assume the 

roles of key confirmation provider and recipient, respectively. The successful completion of the 

key confirmation process provides party U with assurance that party V has derived the same 

secret Z value; if NonceU is a random nonce (see Section 5.4), then party U also obtains 

assurance that party V has actively participated in the process. 

C(0,2) Scheme

MacTagU

Party U

(Initiator

and KC

Provider)

Party V

(Responder

and KC

Recipient) NonceV

Party U’s Static Public Key

Party V’s Static Public Key

Key Confirmation

NonceU
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Figure 17: C(0, 2) Scheme with Unilateral Key Confirmation from Party V to Party U 

To provide (and receive) key confirmation (as described in Section 8.2), Both parties set 

EphemDataV = Null, and EphemDataU = NonceU: 

Party V provides MacTagV to U (as specified in 8.2, 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 {|| Text}. 

The recipient U 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. 

8.4.10 C(0, 2) Scheme with Bilateral Key Confirmation 

Figure 18 depicts a typical flow for a C(0, 2) scheme with bilateral key confirmation. In this method, 

party U, the scheme initiator, and party V, the scheme responder, assume the roles of both the 

provider and the recipient in order to obtain bilateral key confirmation. V shall contribute a nonce 

(NonceV) prior to the generation of MacTagU. The successful completion of the key confirmation 

process provides each party with assurance that the other party has derived the same secret Z value. 

If NonceU is a random nonce (see Section 5.4), then party U obtains assurance that party V has 

actively participated in the process; if NonceV is a random nonce, then party V obtains assurance that 

party U has actively participated in the process. 

To provide bilateral key confirmation (as described in Section 8.3), U and 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 U (as specified in Section 8.2, with P = V and R = U); MacTagV is 

computed by V (and verified by U) using 

 MacDataV = “KC_2_V” || IDV || IDU || NonceV || NonceU {|| Text1}. 

MacTagV

Party U

(Initiator 

and KC 

Recipient)

Party V

(Responder 

and KC 

Provider)

NonceU

Party U’s Static Public Key

Party V’s Static Public Key

Key Confirmation

C(0,2) Scheme
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Party U provides MacTagU to V (as specified in Section 8.2, with P = U and R = V); MacTagU is 

computed by U (and verified by V) using 

 MacDataU = “KC_2_U” || IDU || IDV || NonceU || NonceV {|| Text2}. 

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 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. 

Figure 18: C(0, 2) Scheme with Bilateral Key Confirmation

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 secret 

keying 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 the 

following: 

1. A static key pair may be saved. 

2. An ephemeral public key may be saved. 

3. An ephemeral private key shall be zeroized after use and, therefore, shall not be 

recoverable. 

4. A symmetric key may be saved. 

Note: This implies that keys derived from schemes where both parties generate ephemeral key 

pairs (see Section 6.1) cannot be made recoverable by reconstruction of the secret keying 

MacTagU

Party U Party V

NonceU

NonceV, MacTagV

Party U’s Static Public Key

Party V’s Static Public Key

(KC Recipient) (KC Provider)

(KC Provider)

Key Confirmation

(KC Recipient)

C(0,2) Scheme

(Initiator) (Responder)
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material by parties requiring the ephemeral private key in their calculations. For those schemes 

where only the initiator generates an ephemeral key pair (see Section 6.2), only the responder 

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 [7]. 

10. Implementation Validation 

When the NIST Cryptographic Module Validation Program (CMVP) has established a validation 

program for this Recommendation, a vendor shall have its implementation tested and validated 

by the CMVP in order to claim conformance to this Recommendation. Information on the 

CMVP is available at http://csrc.nist.gov/cryptval/. 

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 5.6.2.4 for FFC or as specified in 

Sections 5.6.2.5 or 5.6.2.6 for ECC. 

• A key agreement scheme from Section 6, together with an Approved key derivation 

function from Section 5.8. 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 140-2 Annex D. If key confirmation is also claimed, the 

appropriate key confirmation technique from Section 8 shall be used. Documentation 

shall include how assurance of private key possession and of 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 the 

parameter set requirements (see Tables 1 and 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 size(s) (see Tables 1 and 2 in Section 5.5.1), 
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• The MAC length(s) (see Tables 1 and 2 in Section 5.5.1), 

• The type of cryptography: FFC or ECC, 

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

• The key derivation function 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), and 

• The key confirmation scheme (see Section 8). 
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Appendix A: Summary of Differences between this 

Recommendation and ANS X9 Standards (Informative) 

This list is informational and not meant to be exhaustive, but is intended to summarize important 

differences between this Recommendation and the indicated ANS X9 standards. In general, this 

Recommendation can be seen as being more restrictive than the ANS X9 standards, but is 

derived from them. The list of differences is as follows: 

1. ANS X9.42 defines MacData for validation testing as “ANSI X9.42 Testing Message”. 

ANS X9.63 does not address implementation validation at this level of detail. Note that 

the implementation test message used for NIST validation is a different text string from 

the implementation test message for ANS X9.42; therefore, conformance to the method 

in this Recommendation does not conform with the ANS X9.42 method. See Section 

5.2.3 of this Recommendation for more information. 

2. Random generation and validation of FFC and ECC domain parameters are being 

extended to (a) specify the use of all Approved hash algorithms for generating and 

validating domain parameters supporting larger key sizes, (b) support the optional use of 

the Shawe-Taylor algorithm to construct and validate FFC primes and (c) support the 

verifiably random generation of the generator of the subgroup. See the FIPS 186-3[3] and 

the ANS X9.62-2[13]. 

3. For FFC domain parameters: (a) The size of the field order p is limited to either 1024 or 

2048 bits, whereas X9.42 allows these sizes along with other multiples of 256 bits that 

are at least 1024 bits; (b) The size of q (subgroup order) is specified, unlike ANS X9.42 

where the size of q has a minimum length of 160 bits; and (c) ANS X9.42 only identifies 

SEED and pgenCounter as being among the domain parameters in its Appendix A, but 

this Recommendation lists them explicitly to be consistent with ANS X9.63. FFC domain 

parameters that conform to this Recommendation in this area also conform to ANS 

X9.42, although the reverse is not necessarily true. See Section 5.5.1.1 of this 

Recommendation for more information. 

4. For ECC domain parameters: The cofactor is 32 bits or less, depending on the selected 

subgroup order, which is more restrictive than ANS X9.63. ECC domain parameters that 

conform to this Recommendation also conform to ANS X9.63, although the reverse is not 

necessarily true. See Section 5.5.1.2 of this Recommendation for more information. 

5. Some schemes in ANS X9.42 and X9.63 allow one set of domain parameters to be used 

with static keys and a different set of domain parameters to be used with ephemeral keys 

in the same scheme. This Recommendation, however, requires the use of only one set of 

domain parameters in one scheme; that is, the same set of domain parameters is used with 

the static and ephemeral keys in any given scheme. Therefore, the ANS X9.42 dhHybrid2 

scheme is not allowed. See Section 5.5 of this Recommendation for more information. 
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Also, the ANS X9.63 Combined Unified Model is not allowed, and the ANS X9.63 

Station to Station (STS) method is not specified in this Recommendation; STS is a 

protocol that is based upon the ECC Ephemeral Unified Model scheme and, as a protocol, 

is not prohibited. 

6. Assurances of the arithmetic validity of a public key are required in this 

Recommendation. Assurance of validity is optional in ANS X9.42, but required in ANS 

X9.63. In both cases, the means of obtaining that assurance is different than in this 

Recommendation. See Section 5.6.2 of this Recommendation for more information. 

7. ANS X9.63 specifies both cofactor and non-cofactor schemes. For this Recommendation, 

only ECC cofactor schemes are used. The use of an ECC scheme that conforms to this 

Recommendation also conforms to ANS X9.63, although the reverse is not necessarily 

true. See Section 5.7 of this Recommendation for details. 

8. Regarding the key derivation function (KDF): 

a. This recommendation specifies two Approved KDFs, the concatenation KDF specified 

in Section 5.8.1 and the ASN.1 KDF specified in Section 5.8.2. 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 140-2 Annex D. 

b. ANS X9.42 provides a concatenation KDF and an ASN.1 KDF, while ANS X9.63 

provides only the concatenation KDF. However, the Approved KDFs of this 

Recommendation require that the counter appears before the shared secret as input to the 

hash function, whereas the ANSI KDFs place the counter after the shared secret 

c. The Approved KDFs in this Recommendation require the input of the identifiers of the 

communicating parties; such information is not required in ANS X9.42 and X9.63. 

d. In this Recommendation, the shared secret is zeroized after a single call to a key 

derivation function, before the key agreement scheme releases any portion of the 

DerivedKeyingMaterial for use by relying applications.. The ANS X9.42 and X9.63 

standards do not indicate when the shared secret needs to be zeroized. An implication of 

this Recommendation’s requirement concerning zeroization is that all of the keying 

material directly derived from the shared secret must be computed during one call to the 

KDF. 

9. FFC and ECC key transport use an Approved key-wrapping algorithm, such as the AES 

key-wrapping algorithm. ANS X9.63 specifies the ECIES method, which is not allowed 

in this Recommendation. Therefore, the use of a key transport method that conforms to 

this Recommendation does not conform to the method in ANS X9.63. See Section 7 of 

this Recommendation for details. ANS X9.42 does not specify a key transport method. 

10. There is a comprehensive specification in this Recommendation of approved ways to do 

key confirmation (KC) when KC is desired as part of the key establishment process. See 

Section 8 of this Recommendation for details. Key confirmation is not discussed in ANS 

X9.42, but a few examples of key confirmation are provided in ANS X9.63. 
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11. This Recommendation specifies that an ephemeral key is used for exactly one 

transaction, with the exception that the sender may use the same ephemeral key pair in 

multiple DLC-based Key Transport transactions if the same secret keying material is 

being transported in each transaction and if all these transactions occur “simultaneously” 

(or within a short period of time). ANS X9.42 and X9.63 do not allow this exception. 
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Appendix B: Rationale for Including Identifiers in the KDF 

Input 

In this Recommendation, it is required that identifiers for both parties to a key agreement 

transaction be included in the OtherInfo that is input to the key derivation function. (See Sections 

5.8.1 and 5.8.2.) 

The inclusion of sufficiently-specific identifiers for the initiator (U) and responder (V) provides 

assurance that the keying material derived by U and V will be different from the keying material 

that is derived by other parties (or by the same parties acting in opposite roles). As a result, key 

agreement schemes gain resilience against unknown key share attacks, which depend on some 

type of confusion over the role played by each party (initiator versus responder). (See, for 

example, references [14], [15], [16], and [17], which all recommend the inclusion of identifiers 

in the KDF as a means of eliminating certain vulnerabilities.) 

In addition to identifiers for the initiator and responder, the inclusion of additional contextspecific information in OtherInfo 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 the included information. Examples of additional context-specific information 

include the intended use of the derived keying material, the choice of key agreement scheme 

used to produce the shared secret Z, ephemeral public keys or nonces exchanged during the 

transaction, and session numbers. 
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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 

When called from an FFC Scheme, n = ⎡t / 8⎤ bytes, where t = ⎡log2 p⎤ where p is 

the large prime field order.

Output: A byte string S of length n bytes. 

1. Let S1, S2,..., Sn be the bytes of S from leftmost to rightmost. 

2. The bytes of S shall satisfy: 

 C = Σ28(n-i)Si for i = 1 to n.

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 α must be a bit string of length m bits. Let s1, s2, ..., sm be the bits of α

from leftmost 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 

the leftmost 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: References (Informative) 

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

[2] FIPS 180-2, Secure Hash Standard, August 2002. 

[3] FIPS 186-3, Digital Signature Standard, anticipated in 2006 

[4] FIPS 197, Advanced Encryption Standard, November 2001. 

[5] FIPS 198, The Keyed-Hash Message Authentication Code (HMAC), March 2002. 

[6] NIST SP 800-38B, Recommendation for Block Cipher Modes of Operation: The CMAC 

Mode for Authentication, May 2005. 

[7] NIST SP 800-57, Recommendation for Key Management, August 2005. 

[8] AES Key Wrap Specification, NIST, November 16, 2001. 

[9] ANS X9.31-1998, Digital Signatures Using Reversible Public Key Cryptography for the 

Financial Services Industry (rDSA). 

[10] ANS X9.42-2001, Public Key Cryptography for the Financial Services Industry: 

Agreement of Symmetric Keys Using Discrete Logarithm Cryptography. 

[11] ANS X9.44 (Draft), Public Key Cryptography for the Financial Services Industry: Key 

Establishment Using Integer Factorization Cryptography, December 2002. 

[12] ANS X9.63-2001, Public Key Cryptography for the Financial Services Industry: Key 

Agreement and Key Transport Using Elliptic Curve Cryptography. 

[13] ANS X9.62-2 Elliptic Curve Digital Signature Algorithm (ECDSA), November 16, 2005. 

[14] On Session Key Construction in Provably-Secure Key Establishment Protocols, KimKwang Raymond Choo, Colin Boyd, and Yvonne Hitchcock, LNCS, Vol. 3715, pp. 116-

131, Springer-Verlag, 2005. Extended version available at: 

http://eprint.iacr.org/2005/206.pdf. 

[15] Another Look at HMQV Alfred Menezes, Nov. 2005.Available at: http://eprint. 

iacr.org/2005/205.pdf. 

[16] Evaluation of Security Level of Cryptography: ECMQVS (from SEC 1), Phillip 

Rogaway, Mihir Bellare, Dan Boneh, Jan. 2001. Available at: 

http://www.ipa.go.jp/security/enc/CRYPTREC/fy15/doc/1069_ks-ecmqv.pdf. 
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[17] Unknown Key-Share Attacks on the Station-to-Station (STS) Protocol, Simon BlakeWilson, Alfred Menezes, Technical Report CORR 98-42, University of Waterloo, 1998. 

Available at: http://cacr.math.uwaterloo.ca.