C O M P U T E R S E C U R I T Y 

Recommendation for Pair-Wise 

Key Establishment Schemes: 

Using Integer Factorization 

Cryptography

Elaine Barker, Lily Chen, Andrew 

Regenscheid, and Miles Smid 

Draft NIST Special Publication 800-56B 

December, 2008 
Draft NIST SP 800-56B: Recommendation for Pair-Wise Key Establishment Schemes: 

Using Integer Factorization Cryptography 

 December, 2008 

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Abstract 

This Recommendation specifies key establishment schemes using integer factorization 

cryptography, based on ANS X9.44, Key Establishment using Integer Factorization 

Cryptography, which was developed by the Accredited Standards Committee (ASC) X9, Inc. 

KEY WORDS: assurances; integer factorization cryptography; key agreement; key confirmation; 

key derivation; key establishment; key management; key recovery; key transport. 

 
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Acknowledgements 

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

appreciates contributions by Rich Davis 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.
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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.” 
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Table of Contents 

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

2 Scope and Purpose ..............................................................................................10

3 Definitions, Symbols and Abbreviations............................................................11

3.1 Definitions...................................................................................................................... 11 

3.2 Symbols and Abbreviations ........................................................................................... 17 

4 Key Establishment Schemes Overview..............................................................23

4.1 Key Establishment Preparations by an Owner............................................................... 24 

4.2 Key Agreement Process................................................................................................. 25 

4.3 IFC-based Key Transport Process ................................................................................. 28 

5 Cryptographic Elements .....................................................................................30

5.1 Cryptographic Hash Functions ...................................................................................... 30 

5.2 Message Authentication Code (MAC) Algorithm......................................................... 30 

5.2.1 MacTag Computation ........................................................................................ 30 

5.2.2 MacTag Checking.............................................................................................. 31 

5.2.3 Implementation Validation Message ................................................................. 31 

5.3 Random Bit Generation ................................................................................................. 31 

5.4 Prime Number Generators ............................................................................................. 31 

5.5 Primality Testing Methods............................................................................................. 32 

5.6 Nonces............................................................................................................................ 32 

5.7 Symmetric Key-Wrapping Algorithms.......................................................................... 32 

5.8 Mask Generation Function (MGF) ................................................................................ 33 

5.9 Key Derivation Functions for Key Establishment Schemes.......................................... 34 

5.9.1 Concatenation Key Derivation Function (Approved Alternative 1).................. 34 

5.9.2 ASN.1 Key Derivation Function (Approved Alternative 2).............................. 37 

6 RSA Key Pairs ....................................................................................................39
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6.1 General Requirements.................................................................................................... 39 

6.2 Criteria for RSA Key Pairs for Key Establishment ....................................................... 40 

6.2.1 Definition of a Key Pair..................................................................................... 40 

6.2.2 Formats ............................................................................................................. 42 

6.2.3 Parameter Length Sets ....................................................................................... 42 

6.3 RSA Key Pair Generators .............................................................................................. 43 

6.3.1 RSAKPG1 Family: RSA Key Pair Generation with a Fixed Public Exponent . 43 

6.3.2 RSAKPG2 Family: RSA key pair generation with a random public exponent. 46 

6.4 Assurances of Validity................................................................................................... 49

6.4.1 Assurance of Key Pair Validity ......................................................................... 49 

6.4.2 Recipient Assurances of Public Key Validity.................................................... 55 

6.5 Assurances of Private Key Possession........................................................................... 56 

6.5.1 Owner Assurance of Private Key Possession .................................................... 57 

6.5.2 Recipient Assurance of Owner’s Possession of a Private Key.......................... 57 

6.6 Key Confirmation .......................................................................................................... 59 

6.6.1 Unilateral Key Confirmation for Key Establishment Schemes ......................... 61 

6.6.2 Bilateral Key Confirmation for Key Establishment Schemes ........................... 62 

6.7 Authentication................................................................................................................ 62 

7 IFC Primitives and Operations ...........................................................................63

7.1 Encryption and Decryption Primitives........................................................................... 63 

7.1.1 RSAEP ............................................................................................................... 63 

7.1.2 RSADP............................................................................................................... 63 

7.2 Encryption and Decryption Operations ......................................................................... 64 

7.2.1 RSA Secret Value Encapsulation (RSASVE).................................................... 64 

7.2.2 RSA with Optimal Asymmetric Encryption Padding (RSA-OAEP)................. 67 

7.2.3 RSA-based Key-Encapsulation Mechanism with a Key-Wrapping Scheme 

(RSA-KEM-KWS)............................................................................................. 74 

8 Key Agreement Schemes....................................................................................79
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8.1 Common Components for Key Agreement ................................................................... 80 

8.2 The KAS1 Family.......................................................................................................... 80

8.2.1 KAS1 Family Prerequisites................................................................................ 81 

8.2.2 KAS1-basic........................................................................................................ 81 

8.2.3 KAS1 Key Confirmation ................................................................................... 83 

8.2.4 KAS1 Security Properties.................................................................................. 85 

8.3 KAS2 Family ................................................................................................................. 86 

8.3.1 KAS2 Family Prerequisites................................................................................ 87 

8.3.2 KAS2-basic........................................................................................................ 88 

8.3.3 KAS2 Key Confirmation ................................................................................... 90 

8.3.4 KAS2 Security Properties..................................................................................... 95 

9 IFC based Key Transport Schemes.....................................................................96

9.1 Additional Input............................................................................................................. 97 

9.2 KTS-OAEP Family: Key Transport Using RSA-OAEP ............................................... 97 

9.2.1 KTS-OAEP Family Prerequisites ...................................................................... 98 

9.2.2 Common components ........................................................................................ 99 

9.2.3 KTS-OAEP-basic............................................................................................... 99 

9.2.4 KTS-OAEP Key Confirmation........................................................................ 100 

9.2.5 KTS-OAEP Security Properties....................................................................... 101 

9.3 KTS-KEM-KWS Family: Key Transport using RSA-KEM-KWS............................. 102 

9.3.1 KTS-KEM-KWS Family Prerequisites............................................................ 103 

9.3.2 Common Components of the KTS-KEM-KWS Schemes............................... 104 

9.3.3 KTS-KEM-KWS-basic.................................................................................... 104 

9.3.4 KTS-KEM-KWS Key Confirmation ............................................................... 105 

9.3.5 KTS-KEM-KWS Security Properties.............................................................. 106 

10 Key Recovery................................................................................................... 107

11.Implementation Validation .............................................................................. 108
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Appendix A: Summary of Differences between this Recommendation and ANS 

X9.44 (Informative) ......................................................................................... 110

Appendix B: Data Conversions (Normative)........................................................ 112

B.1 Integer-to-Byte String (I2BS) Conversion................................................................... 112 

B.2 Byte String to Integer (BS2I) Conversion ................................................................... 112 

Appendix C: Prime Factor Recovery (Normative)............................................... 113

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

Figure 1: Owner Key Establishment Preparations.........................................................................25 

Figure 2: Key Agreement Process .................................................................................................26 

Figure 3: Key Transport Process....................................................................................................28 

Figure 4: RSA-OAEP Encryption Operation.................................................................................70 

Figure 5: RSA-OAEP Decryption Operation ................................................................................73 

Figure 6: RSA-KEM-KWS Encryption Operation........................................................................76 

Figure 7: RSA-KEM-KWS Decryption Operation........................................................................79 

Figure 8: KAS1-basic Scheme.......................................................................................................83 

Figure 9: KAS1-responder-confirmation Scheme (from Party V to Party U) ...............................85 

Figure 10: KAS2-basic Scheme.....................................................................................................90 

Figure 11: KAS2-responder-confirmation Scheme (from Party V to Party U) .............................92 

Figure 12: KAS2-initiator-confirmation Scheme (from Party U to Party V) ................................93 

Figure 13: KAS2-bilateral-confirmation Scheme..........................................................................95 

Figure 14: KTS-OAEP-basic Scheme .........................................................................................100 

Figure 15: KET-OAEP-receiver-confirmation Scheme...............................................................101 

Figure 16: KTS-KEM-KES-basic Scheme ..................................................................................105 

Figure 17: KTS-KEM-KWS-receiver-confirmation Scheme ......................................................106 

Tables 

Table 1: IFC Parameters for Key Establishment ...........................................................................43 
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1 Introduction 

Many U.S. Government Information Technology (IT) systems need to employ strong 

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 198-1 [5] 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, but as the 

number of entities using a system grows, the work involved in the distribution of the secret 

keying material grows 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 a standard developed by the 

Accredited Standards Committee (ASC) X9, Inc.: ANS X9.44, Key Establishment using Integer 

Factorization Cryptography [10]. A key establishment scheme can be characterized as either a 

key agreement scheme or a key transport scheme. This Recommendation provides asymmetricbased key agreement and key transport schemes that are based on the Rivest Shamir Adleman 

(RSA) algorithm. 

When there are differences between this Recommendation and the referenced ANS X9.44[10] 

standard, 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 FIPS 186-3 [3] standard are intended to 

provide information for a vendor to implement secure key establishment using asymmetric 

algorithms in FIPS 140-2/3 [1] validated modules. 

A scheme may be a component of a larger 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. 
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3 Definitions, Symbols and Abbreviations 

3.1 Definitions 

Additional input Information known by two parties that is cryptographically bound to 

keying material using the encryption scheme. 

Algorithm A clearly specified mathematical process for computation; a set of rules 

which, if followed, will give a prescribed result. 

Algorithm identifier A unique identifier for a given cryptographic algorithm, together with 

any required parameters. 

Approved FIPS-approved or NIST-recommended. An algorithm or technique 

that meets at least one of the following: 1) is specified in a FIPS or 

NIST Recommendation, 2) is adopted in a FIPS or NIST 

Recommendation or 3) is specified in a list of NIST-approved security 

functions (e.g., specified as approved in the annexes of FIPS 140-2/3). 

Assurance of 

possession of a 

private key 

Confidence that an entity possesses a private key associated with a 

given 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. 

Bit string An ordered sequence of 0’s and 1’s. 

Byte A bit string of length 8. A byte is represented by a hexadecimal string of 

length 2. The right-most hexadecimal character represents the rightmost four bits of the byte, and the left-most four bits of the byte 

represent the left-most four bits of the byte. For example, 9d represents 

the bit string 10011101. 

Byte string A sequence of bytes. 

Certification 

Authority (CA) 

The entity in a Public Key Infrastructure (PKI) that is responsible for 

issuing public key certificates and exacting compliance to a PKI policy. 

Ciphertext Data in its enciphered form. 
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Cryptographic key 

(Key) 

A parameter used with a cryptographic algorithm that determines its 

operation. Examples include: 

1. the transformation of plaintext data into ciphertext data, 

2. the transformation of ciphertext data into plaintext data, 

3. the computation of a digital signature from data, 

4. the verification of a digital signature, 

5. the computation of an authentication code from data, 

6. the verification of an authentication code from data and a received 

authentication code, and 

7. the computation of a shared secret that is used to derive keying 

material. 

Data integrity A property whereby data has not been altered in an unauthorized 

manner since it was created, transmitted or stored. 

In this Recommendation, the statement that a cryptographic algorithm 

"provides data integrity" means that the algorithm is used to detect 

unauthorized alterations. 

Decryption The process of transforming ciphertext into plaintext using a 

cryptographic algorithm and key. 

Digital signature The result of a cryptographic transformation of data that, when properly 

implemented with supporting infrastructure and policy, provides the 

services of: 

1. origin authentication, 

2. data integrity, and 

3. signer non-repudiation. 

Encryption The process of transforming plaintext into ciphertext using a 

cryptographic algorithm and key. 

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

synonym. 

Entity authentication A process that establishes the origin of information, or determines an 

entity’s identity to the extent permitted by the entity’s identifier. 
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Hash function A function that maps a bit string of arbitrary length to a fixed length bit 

string. Approved hash functions are designed to 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 180-3 [2]. 

Hash of a bit string The hash value produced by applying a hash function to the bit string. 

Hash value The fixed-length bit string produced by a hash function. 

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 Internet Protocol (IP) address and 

timestamp). 

If a party owns a key pair that is used in a key establishment transaction, 

then the identifier assigned to that party is one that is cryptographically 

bound to that key pair. If the party’s key pair is not used in a key 

establishment 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 party. 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 In this Recommendation, the process by which keying material is 

derived from a shared secret and other information. 
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Key establishment The procedure that results in shared secret keying material among 

different parties. 

Key establishment 

transaction 

An instance of establishing secret keying material using a key 

establishment scheme. 

Key management The activities involving the handling of cryptographic keys and other 

related security parameters (e.g., IVs and passwords) during the entire 

life cycle of the keys, including their generation, storage, establishment, 

entry and output, and destruction. 

Key pair A public key and its corresponding private key; a key pair is used with a 

public key algorithm. 

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. 

Length in bits of an 

integer, x

The length, in bits, of the shortest bit string containing the binary 

representation of x. For example, the length in bits of 5 is 3. 

Length in bytes of an 

integer, x 

The length, in bytes, of the shortest byte string containing the binary 

representation of x. For example, the length in bytes of 5 is 1. 

Message 

Authentication Code 

(MAC) algorithm 

A family of one-way cryptographic functions that is parameterized by a 

symmetric key. A given function in the family produces a MacTag on 

input data of arbitrary length. 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. 
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Nonce A time-varying value that has at most a negligible chance of repeating. 

For example, a nonce is a random value that is generated anew for each 

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

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

private key associated with a public key, whether that entity generated 

the key pair itself or a trusted party generated the key pair for the entity. 

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

synonym for party. 

Prime number An integer that is greater than 1 and divisible only by 1 and itself. 

Primitive A low level cryptographic algorithm used as a basic building block for 

higher level cryptographic operations or schemes. 

Private key A cryptographic key, used with a public key cryptographic algorithm 

that is kept secret. A private key is associated with a public key. 

Protocol A special set of rules used by two or more entities that describe the 

message order and data structures for information exchanged between 

the entities. 

Provider A party that provides (1) a public key (e.g., in a certificate); (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 recipient. 

Public key A cryptographic key, used with a public key cryptographic algorithm 

that may be made public. A public key is associated with a private key. 

Public key algorithm A cryptographic algorithm that uses two related keys, a public key and a 

private key. The two keys have the property that determining the private 

key from the public key is computationally infeasible. 

Public key certificate 

(certificate) 

A set of data that uniquely identifies an entity’s identifiers, the entity’s 

public key, and possibly other information, and is digitally signed by a 

trusted party, thereby binding the public key to the included 

identifier(s). 
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Public key 

cryptography 

A form of cryptography that uses two related keys, a public key and a 

private key; the two keys have the property that, given the public key, it is 

computationally infeasible to derive the private key. 

For key establishment, public key cryptography allows different parties 

to communicate securely without having prior access to a shared secret 

key, by using one or more pairs (public key and private key) of 

cryptographic keys. 

Public key validation The procedure whereby the recipient of a public key checks that the key 

conforms to the arithmetic requirements for such a key in order to 

thwart certain types of attacks. 

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

transaction. Contrast with sender. 

Recipient A party that receives (1) a public key (e.g., in a certificate); (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 initiate a key agreement transaction. Contrast 

with initiator. 

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

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, replay 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. 
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Shared secret keying 

material 

As used in this Recommendation, 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 during a key establishment 

scheme and is used as input to a key derivation function to produce 

keying material. 

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. 

Symmetric key A single cryptographic key that is used with a secret (symmetric) key 

algorithm. 

Symmetric key 

algorithm 

A cryptographic algorithm that uses one secret key that is shared 

between authorized parties. 

Target security 

strength 

The desired security strength for a cryptographic application. 

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 that is trusted by its clients to perform certain services, 

such as a CA. (By contrast, the initiator and responder (or sender and 

receiver) in a scheme are considered to be the first and second parties in 

a key establishment transaction.) 

3.2 Symbols and Abbreviations 

A Additional Input that is bound to keying material, a byte string. 

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

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

ASC The American National Standards Institute (ANSI) Accredited 

Standards Committee. 

ANS American National Standard. 
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ASN.1 Abstract Syntax Notation One. 

c Ciphertext, an integer. 

C, C0, C1 Ciphertext, each is a byte string. 

CA Certification Authority. 

context Context string for initiator authentication, a bit string. 

CRT Chinese Remainder Theorem. 

d RSA private exponent, an integer. 

Data A variable-length string of zero or more (eight-bit) bytes 

dataLen The length of Data in bytes. 

DerivedKeyingMaterial Derived Keying Material, a bit string. 

dP RSA private exponent for the prime factor p in the CRT format, 

i.e., d mod (p-1), an integer. 

dQ RSA private exponent for the prime factor q in the CRT format, 

i.e., d mod (q-1), an integer. 

e RSA public exponent, an integer. 

eBits Length in bits of the RSA exponent e. 

EphemDataP, EphemDataR, 

EphemDataU, EphemDataV

Fresh data contributed by the provider or recipient in a key 

confirmation; each is a byte string. 

GCD(a, b) Greatest Common Divisor of two non-negative integers a and b

H An approved hash function 

hBits Length in bits of a hash value 

hLen Length in bytes of a hash value 

HMAC Keyed-hash Message Authentication Code (as specified in FIPS 

198-1 [5]) 
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I2BS Integer to Byte String 

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

IDP, IDR, IDU, IDV Identifier bit strings for parties P, R, U, and V 

IFC Integer Factorization Cryptography 

K Keying material, a byte string. 

KBits Length in bits of the keying material 

KLen Length in bytes of the keying material 

KAS Key Agreement Scheme. 

k Keying material, an integer. 

KAS1-basic The basic form of Key Agreement Scheme 1 

KAS1-responderconfirmation 

Key Agreement Scheme 1 with responder-confirmation 

KAS2-basic The basic form of Key Agreement Scheme 2 

KAS2-responderconfirmation 

Key Agreement Scheme 2 with responder confirmation 

KAS2-initiatorconfirmation 

Key Agreement Scheme 2 with initiator confirmation 

KAS2-bilateralconfirmation 

Key Agreement Scheme 2 with bilateral confirmation 

KC Key Confirmation 

KDF Key Derivation Function 

KEM Key Encapsulation Mechanism 

KeyData Keying material other than that which is used for the MacKey

employed in key confirmation. 

KTS Key Transport Scheme (i.e. KTS-OAEP or KTS-KEM-KWS) 
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KTS-OAEP-basic The basic form of the Key Transport Scheme with Optimal 

Asymmetric Encryption Padding 

KTS-OAEP-receiverconfirmation 

Key Transport Scheme with Optimal Asymmetric Encryption 

Padding and receiver confirmation 

KTS-KEM-KWS-basic The basic form of the Key Transport Scheme with Key 

Encapsulation Mechanism and Key-Wrapping Scheme 

KTS-KEM-KWSreceiver-confirmation 

Key Transport Scheme with Key Encapsulation Mechanism, KeyWrapping Scheme, and receiver confirmation 

KWK Key-Wrapping Key, a byte string 

kwkBits Length in bits of the key-wrapping key 

kwkLen Length in bytes of the key-wrapping key 

KWS (Symmetric) Key-Wrapping Scheme. 

LCM(a, b) Least common multiple of two non-negative integers a and b. 

MAC Message Authentication Code. 

MacData A byte string input to the MacTag computation. 

MacDataU, (or MacDataV) MacData associated with Party U (or Party V, respectively), and 

used to generate MacTagU (or MacTagV, respectively). Each is a 

byte string. 

MacKey Key used to compute the MAC, a byte string. 

MacKeyLen Length in bytes of the MacKey. 

MacTag A byte string that allows an entity to verify the integrity of the 

information. MacTag is the output of the MAC algorithm. The 

literature sometimes refers to MacTag as a Message 

Authentication Code (MAC). 

MacTagV, (MacTagU) The MacTag generated by Party V (or Party U, respectively). Each 

is a byte string. 

MacTagLen The length of MacTag in bytes. 
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Mask Mask, a byte string. 

maskLen Length in bytes of the mask. 

max_hash_inputBits An integer that indicates the maximum length, in bits, of a bit 

string input to the hash function 

MGF Mask Generation Function 

mgfSeed String from which a mask is derived, a byte string. 

 n RSA modulus 

(n, d) RSA private key in the basic format. 

(n, e) RSA public key. 

NV Nonce contributed by party V, a byte string. 

nBits Length in bits of the RSA modulus n. 

nLen Length in bytes of the RSA modulus n.

Null The empty bit string 

OtherInfo Other information for key derivation, a bit string. 

p First prime factor of RSA modulus n. 

PrivKeyU, PrivKeyV Private key of party U or V. 

PubKeyU, PubKeyV Public key of party U or V. 

q Second prime factor of the RSA modulus n. 

qInv Inverse of q modulo p in the CRT format, i.e., q

-1 mod p, an 

integer. 

RBG Random Bit Generator. 

RSASVE RSA Secret Value Encapsulation. 

RSA-KEM-KWS RSA Key Encapsulation Mechanism with a Key-Wrapping 

Scheme 
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RSA-OAEP RSA with Optimal Asymmetric Encryption Padding. 

S String of bytes. 

s Security strength in bits. 

SHA Secure Hash Algorithm. 

TransportedKeyingMaterial Transported Keying Material 

TTP A Trusted Third Party. 

U The initiator or sender of a key establishment process. 

V The responder or receiver in a key establishment process. 

X Byte string to be converted to or from an integer, output of 

conversion from an ASCII string. 

X =? Y Verify that X equals Y. 

x Non-negative integer to be converted to or from a byte string. 

x mod n The modular reduction of the (arbitrary) integer x by the positive 

integer n (the modulus). For the purposes of this Recommendation, 

y = x mod n is the unique integer satisfying the following two 

conditions: 0 ≤ y < n and x − y is divisible by n. 

x

-1 mod n The multiplicative inverse of the integer x modulo the positive 

integer n. This quantity is defined if and only if x is relatively 

prime to n. For the purposes of this Recommendation, y = x

−1

 mod 

n is the unique integer satisfying the following two conditions: 

0 ≤ y < n and 1 = (xy) mod n. 

{X} Indicates that the inclusion of X is optional. 

{x, y} A set containing the integers x and y. 

X || Y Concatenation of two strings X and Y. 

⎡x⎤ The ceiling of x; the smallest integer ≥ x. For example, ⎡5⎤ = 5 and 

⎡5.3⎤ = 6. 
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⎣x⎦ The floor of x, the largest integer less than or equal to x. For 

example, ⎣5⎦ = 5 and ⎣5.3⎦ = 5. 

⎜x⎜ The absolute value of x. 

XOR Exclusive-Or, defined as bit-wise modulo 2 arithmetic with no 

carry. 

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

a key derivation function, a byte string. 

z The integer form of Z

λ(n) Lambda function of the RSA modulus n, i.e., the least positive 

integer i such that a

i ≡ 1 for all a relatively prime to n. If n = pq

where p and q are distinct primes, then λ(n) = LCM(p-1, q-1). 

φ(n) Totient function of the RSA modulus n, i.e., the number of 

integers between 0 and n-1 that are relatively prime to n. If n = pq, 

where p and q are distinct primes, then φ(n) = (p-1)(q-1). 

|| Concatenation operator. 

⊕ XOR Operator. 

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, information is exchanged between both parties that permits each party to 

contribute to and 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), encrypted secret 

keying material is transported from the sender to the receiver. The key transport schemes 

described in this Recommendation use either public key techniques or a combination of public 

key and symmetric key techniques. The party that sends the secret keying material is called the 

sender, and the other party is called the receiver. 
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The security of the Integer Factorization Cryptography (IFC) schemes in this Recommendation is 

based on the intractability of factoring large integers. 

For compliance with this Recommendation, equivalent processes may be used. Two processes 

are equivalent if, whenever the same values are input to each process (either as input parameters 

or as values made available during the process), each process produces the same output as the 

other. 

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, 

4.2, and 4.3. 

4.1 Key Establishment 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 establishment process (i.e., either key agreement or key transport). 

The first step in the process is for the entity to obtain a key pair. Either the entity generates the 

key pair as specified in Section 6.3 or a trusted third party (TTP) generates the key pair as 

specified in Section 6.3, and provides it to the entity. The entity (i.e., the owner) obtains 

assurance of key pair validity, and as part of the process, obtains assurance that it actually 

possesses the (correct) private key. Approved methods for obtaining assurance of key pair 

validity by the owner are addressed in Section 6.4.1. 

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

corresponding to a particular public key (i.e., the identifier names the key pair’s owner). This 

name may uniquely distinguish the entity from all others, in which case it could rightfully be 

considered an identity. However, the name 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 

public key is appropriate for the applications in which the public key will be used. 

This Recommendation requires that there is a trustworthy binding of each entity’s identifier to 

the entity’s public key. The binding of an identifier to a 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 public key and the identifier). The binding 

authority verifies the identifier chosen for the owner. The binding authority is also responsible 

for checking the arithmetic validity of the owner’s public key, and the owner’s possession of the 

private key corresponding to that public key. The methods used by a third party trusted by the 
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recipient to obtain that assurance are beyond the scope of this Recommendation (see Seon 

8.1.5.1.1.2 of SP 800-57 [7]). 

Owner obtains

Assurance of

Key Pair Validity

(6.4.1)

Obtain

Key Pair

(6.3)

Owner Ready for Key Establishment

Owner

generates

TTP

generates

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 key pair owner) is ready to enter 

into a key establishment process. 

4.2 Key Agreement Process 

Figure 2 depicts the steps implemented by an entity when establishing secret keying material 

with another entity using one of the key agreement schemes described in this Recommendation. 

(Some discrepancies in ordering may occur in practice, depending on the communication 

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

scheme, the entity could be either the key agreement initiator or responder. Note that some of the 

actions shown may not be a part of every scheme. For example, key confirmation is not provided 

in the basic key agreement schemes (see Sections 8.2.2 and 8.3.2). The specifications of this 

recommendation indicate when a particular action is required. 
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Retrieve Entity’s Own 

Private Key 

Obtain Other Entity’s 

Public Key 

and 

Obtain Assurance 

of its Validity 

Generate 

(Random) Secret 

or 

Nonce 

Key Agreement Completed 

Receive & Decrypt Ciphertext 

(to obtain Secret value) 

or 

Receive Nonce 

Send 

Public-Key-Encrypted Secret 

or 

(Plaintetext) Nonce 

Generate a Shared Secret (Z) 

and 

Derive Secret Keying Material 

Zeroize Shared Secret (Z) 

Perform Key Confirmation 

(if required by scheme) 

Obtain Assurance that Each 

Key-Pair Owner Possesses the 

(Correct) Private Key 

Figure 2: Key Agreement Process 
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Each participant 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 public key for use in the key agreement scheme 

obtains the public key bound to the other party’s identifier, and obtains assurance of the validity 

of the public key. Approved methods for obtaining assurance of the validity of another entity’s 

public key are provided in Section 6.4.2. 

Each entity generates either a (random) secret value (which becomes a shared secret when 

transmitted to the other entity) or a nonce, as required by the particular key agreement scheme. If 

the scheme requires an entity to generate a secret value, that secret value is generated as 

specified in Section 5.3 and encrypted using the other entity's public key. The resulting 

ciphertext is then provided to the other entity. If the key agreement scheme requires that an entity 

provide a nonce, that nonce is generated as specified in Section 5.6 and provided (in plaintext 

form) to the other party. (See Sections 8.2 and 8.3 for details). 

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

establish a shared secret (Z) as specified Section 8.2.2 or 8.3.2. Each participant then derives 

secret keying material from the shared secret (and other information), as specified in Section 5.9. 

If the key agreement scheme includes key confirmation provided by one or both of the 

participants, then key confirmation is performed as specified in Section 8.2.3 or 8.3.3, thus 

providing assurance that the key pair owner possesses the (correct) private key. 

The owner of any key pair used during the key agreement transaction is required to have 

assurance that the owner is in possession of the correct private key. Likewise, the recipient of 

another entity’s public key is required to have assurance that its owner is in possession of the 

corresponding private key. Assurance of private key possession is obtained prior to using the 

derived keying material for purposes beyond those of the key agreement transaction itself. This 

assurance may be provided/obtained either through key confirmation, or by some other 

approved means (see Sections 6.5.1 and 6.5.2). 
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4.3 IFC-based Key Transport Process 

Figure 3 depicts the steps implemented by two entities when using one of the key-transport 

schemes described in this Recommendation to establish secret keying material. 

Decrypt 

Encrypted Keying Material 

Provide Key Confirmation 

(if required by scheme) 

Receive Encrypted Keying Material

and 

Retrieve Receiver’s Private Key 

Key Transport Receiver

Key Transport Sender

Select the

Keying Material 

Encrypt Keying Material

Transport

Encrypted Keying Material

Obtain Receiver’s Public Key 

and 

Obtain Assurance of its Validity 

Obtain Assurance of 

Receiver’s Possession of

the (Correct) Private Key 

(if Key Confirmation is 

not required by scheme) 

Key Transport Completed 

Figure 3: Key Transport Process 

The entity who will act as the sender obtains the identifier associated with the entity that will act 

as the receiver, and verifies that the receiver’s identifier corresponds to an entity with whom the 

sender wishes to establish secret keying material. 
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Prior to performing key transport, the sender obtains the receiver’s public key and obtains 

assurance of its validity. Approved methods for obtaining assurance of the validity of another 

entity’s public key are provided in Section 6.4.2. The sender is also required to have assurance 

that the receiver is in possession of the private key corresponding to the receiver’s public key 

prior to key transport, unless that assurance is obtained via key confirmation included as part of 

the scheme. (See Sections 9.2 and 9.3 for details). 

The sender selects the secret keying material (and, perhaps, other data) to be transported to the 

other entity. Then, using the intended receiver’s public key, the sender either encrypts that 

material directly (as specified in Section 9.2.3), or, employs a combination of secret value 

encapsulation and key-wrapping (as specified in Section 9.3.3). The resulting ciphertext is 

transported to the receiver. 

Prior to participating in a key establishment transaction, the receiver is required to have 

assurance of the validity of it’s key pair. This assurance may be renewed whenever fresh 

assurance is desired. Upon (or before) receipt of the transported ciphertext, the receiver retrieves 

the private key from its own key pair. Using its private key, the receiver takes the necessary steps 

(as specified in Section 9.2.3 or 9.3.3) to decrypt the ciphertext and obtain the plaintext keying 

material. 

If the key-transport scheme includes key confirmation, then key confirmation is provided by the 

receiver to the sender as specified in Section 9.2.4 or 9.3.4. Through the use of key confirmation, 

the sender can obtain assurance that the receiver has correctly recovered the keying material 

from the ciphertext. The sender can also obtain assurance that the receiver was in possession of 

the correct private key. 
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5 Cryptographic Elements 

This section describes the cryptographic elements that are used in the development of key 

establishment schemes. 

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 198-1 [5], is 

used). FIPS 180-3 [2] specifies approved hash functions. The hash function shall be selected in 

accordance with the parameter lists in Table 1 of Section 6.2.3. 

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. The MAC algorithm is used to provide key 

confirmation as specified in this Recommendation using an appropriate scheme from this 

Recommendation, and is used to validate implementations of the key establishment schemes 

specified in this Recommendation (see Section 5.2.3). 

In the case of key confirmation, an entity is required to compute a MacTag on received or 

derived data using the MAC function determined by a symmetric key derived from a shared 

secret (when a key agreement scheme is used) or from transported keying material (when a key 

transport scheme is used). The MacTag is sent to another entity in order to confirm that the 

keying material is correct. An approved MAC algorithm with appropriate parameter choices 

(see Section 6.2.3) shall be used to compute a MacTag, for example, HMAC [5] or CMAC[6]. 

5.2.1 MacTag Computation 

The computation of the MacTag is represented as follows: 

MacTag = MAC(MacKey, MacTagLen, 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 (when a key agreement scheme is used for key 

confirmation) or from the transported keying material (when a key transport scheme is used for 

key confirmation) (see Section 6.6.1 along with Sections 8.2.3 and 8.3.3 for key agreement and 

9.2.4 and 9.3.4 for key transport); MacTagLen represents the length of MacTag; and MacData 

represents the data on which the MacTag is computed. The minimum for MacTagLen is 

specified in Table 1 of Section 6.2.3. The minimum length for MacKey is also specified in Table 

1. See [5] and [6]. 
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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, MacTagLen, and 

MacData possessed by the recipient/receiver (as specified in Sections 5.2.1 and 5.2.3). The new 

MacTag is compared with the received MacTag. If their values are equal, then it may be inferred 

that the same MacKey, MacTagLen, 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 Algorithm Validation Program), 

the value of MacData shall be the string “Standard Test Message”, followed by a 128-bit 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 purpose of testing when no key confirmation 

capability exists. 

5.3 Random Bit 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) at an appropriate security strength. Approved RBG methods and methods for converting 

the random bits to an integer are provided in SP 800-90. The security strength provided by an 

RBG employed in this Standard shall be greater than or equal to the target security strength for 

the scheme in which it is employed. 

5.4 Prime Number Generators 

A prime number generator employs a random bit generator and a primality test in order to 

produce random prime numbers in a certain range, possibly with certain structure. 

Only approved prime number generation methods shall be employed in this Recommendation. 

Approved prime number generation methods are specified in FIPS 186-3 [3]. 

The prime number generators in FIPS 186-3 accept input values that include the selected public 

exponent e and the desired length (in bits) of the modulus n, and use an approved random bit 

generator to generate the prime factors p and q with (at least) the following properties: 

1. Each is between ⎡ 2(2 nBits/2 – 1)⎤ and 2 nBits/2 – 1, inclusive, 

2. p – 1 and q – 1 are relatively prime to e, and 

3. |p – q| ≤ 2 nBits/2 – 100. 
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5.5 Primality Testing Methods 

A primality testing method determines whether an integer is prime with a negligible probability 

of error. Only approved primality testing methods shall be employed in this Standard. 

approved primality testing methods as of the publication of this Standard are listed in FIPS 186-

3. 

5.6 Nonces 

A nonce is a time-varying value, represented as a byte string 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 RBG used to obtain each random value shall be 

greater than or equal to the security strength associated with the modulus used in the key 

establishment scheme (see SP 800-57-Part 1 [7]). The length of the RBG output shall be 

at least the security strength of the key establishment scheme. 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. 

If a combination of a timestamp and a monotonically increasing sequence number is used 

without a random nonce, the sequence number shall be 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 (see Section 5.2.3), and in 

KAS1 schemes (see Section 8.2). 

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

5.7 Symmetric Key-Wrapping Algorithms 

A symmetric key-wrapping algorithm wraps (i.e., encrypts and integrity-protects) keying 

material using a symmetric key-wrapping key. In this Recommendation, a symmetric keywrapping algorithm is used by the KTS-KEM-KWS schemes specified in Section 9.3. The 

wrapping operation produces a ciphertext C from keying material K, using the key-wrapping key 

KWK and additional input A which is known to both the wrapping and the unwrapping parties, 

but may be null. A is bound to K in that C is a cryptographic function of both values. The 

unwrapping operation recovers K from C using KWK and A; the unwrapping operation then 

verifies the integrity of K and A by means of an integrity test built into the wrapping algorithm. 

Thus, the key-wrapping algorithm shall support both confidentiality and integrity properties. 

There may be restrictions on the length of the keying material and the additional input, but such 

bounds are generally very large. 

In this Recommendation, the wrapping operation is specified as: 
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C = KWA.WRAP(KWK, K, A) , 

and the unwrapping operation is specified as: 

K = KWA.UNWRAP(KWK, C, A) , 

where KWK is the key-wrapping key, K is the plaintext keying material, A is additional input, 

and C is the ciphertext. 

Approved/allowed1

 key-wrapping algorithms shall be used that employ approved block cipher 

algorithms and keys that support a security strength that is equal to or greater than the security 

strength required to protect the data to be cryptographically protected by the wrapped keying 

material (see [7]). For example, if the data requires 112 bits of security (the target security 

strength), the block cipher and keys used for key-wrapping shall support a security strength of at 

least 112 bits. Note, that in this case, the wrapped keying material, together with the algorithm to 

be used to protect the data, shall also support a security strength of at least 112 bits. 

5.8 Mask Generation Function (MGF) 

MGF is a mask generation function based on an approved hash function (see Section 5.1). The 

purpose of the MGF is to generate a string of bits that may be used to “mask” other bit strings. 

The MGF is used by the RSA-OAEP based schemes specified in Section 9.2. The lengths of the 

MGF seed and the mask in MGF are both variable. 

Let H be an approved hash function, and let hLen denote the length of the hash function output 

in bytes. 

For the purposes of this Standard, MGF shall not be run more than once by each party during a 

given transaction, using a given MGF seed (i.e., a mask shall be derived at most once from a 

given MGF seed). 

Function call: MGF(mgfSeed, maskLen) 

Input: 

1. mgfSeed: a string from which the mask is generated, a byte string. 

2. maskLen: the intended length in bytes of the mask. 

Output:

mask: a byte string of length maskLen bytes. 

Errors: 

An indication that the mask is too long. 

 

1

 Allowed key wrap algorithms are specified in FIPS 140-2 Implementation Guidance IG 7.1. 
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Process: 

1. Let T be the empty string. 

2. For counter from 0 to ⎡ maskLen / hLen ⎤ – 1, do the following: 

a. Let D = I2BS(counter, 4) (see Appendix B.1). 

b. Let T = T || H(mgfSeed || D). 

3. Output the first maskLen bytes of T as the byte string mask. 

5.9 Key Derivation Functions for Key Establishment Schemes 

An approved or allowed (i.e., see FIPS 140-2/3 Annexes) key derivation function (KDF) shall

be used to derive secret keying material from a shared secret during the execution of any key 

establishment scheme from the KAS1, KAS2, and KTS-KEM-KWS families of schemes. The 

output from the 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 derive other keys (possibly using a different process). Non-secret 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. 

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

cryptographic keying material (for example, secret initialization vectors). In cases where key 

confirmation is included in a key agreement scheme from the KAS1 family or the KAS2 family, 

MacKey shall be formed from the initial bits of the KDF output. (When key confirmation is 

included in a key transport scheme from the KTS-OAEP family or the KTS-KEM-KWS family, 

MacKey is not obtained from the output of the KDF, but shall be formed from the initial bits of 

the transported keying material.) In all cases, MacKey shall be zeroized after its use (in 

particular, MacKey shall not be used for purposes other than key confirmation). 

Sections 5.9.1 and 5.9.2 specify two approved KDFs for use in key establishment. They differ 

only in the way that they format the OtherInfo bit string. Other allowable methods and the 

protocols that they may be used with are referenced 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 Table 1 in Section 6.2.3). 

5.9.1 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, KBits, OtherInfo). 
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Fixed Values (implementation dependent):

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

to derive the secret keying material. 

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

input to the hash function. 

Auxiliary Function: 

H: an approved hash function. 

Input: 

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

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

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

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

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

 where the subfields are defined as follows: 

a. 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 

after any MacKey is extracted from the derived keying material when key 

confirmation is performed. For example if key confirmation is not performed, 

AlgorithmID might indicate that bits 1-128 are to be used as a 128-bit AES key. If 

key confirmation is performed, then AlgorithmID might indicate that bits 1- 128 

are used as a MacKey, and bits 129-256 are to be used as the 128-bit AES key. 

b. 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, as 

a separate unit of information. 

c. 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, as 

a separate unit of information. When this KDF is used in a KAS1 scheme, the 

nonce, NV, supplied by party V shall be included in PartyVInfo as a separate unit 

of information, immediately following IDV. 

d. (Optional) SuppPubInfo: A bit string containing additional, mutually-known 

public information. 

e. (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). 
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Each of the three subfields AlgorithmID, PartyUInfo, and PartyVInfo shall be the 

concatenation of a fixed, application specific 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 – 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. (In this variable-length 

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 a fixed, application specific 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 substrings that may be used in the allowed subfield(s) and shall specify the 

fixed-length of the dataLen counters. 

Output: 

The bit string DerivedKeyingMaterial of length KBits bits (or an error indicator). 

Any scheme attempting to call this key derivation function with KBits greater than or equal 

to hBits × (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_inputBits bits long shall cause the KDF to output 

an error indicator and stop without outputting DerivedKeyingMaterial. 

Process: 

1. reps = ⎡KBits / hBtis⎤.

2. If reps > (232 −1), then 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_inputBits bits long, 

then output an error indicator and stop. 

5. For i = 1 to reps by 1, do the following: 

a. Compute Hashi = H(counter || Z || OtherInfo). 
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b. Increment counter (modulo 232), treating it as an unsigned 32-bit integer.

6. Let Hhash be set to Hashreps if (KBits / hBits) is an integer; otherwise, let Hhash be set to 

the (KBits mod hBits) leftmost bits of Hashreps. 

7. Set DerivedKeyingMaterial = Hash1 || Hash2 || ... || Hashreps-1 || Hhash. 

Notes: 

1. Party U shall be the initiator or sender, and party V shall be the responder or receiver, as 

assigned by the relying protocol in accordance with the use of those designators by the 

key establishment scheme employing the KDF. 

2. When a party owns a key pair that is used by the key establishment scheme, the identifier 

assigned to that party shall be one that is bound to that key pair. (This will always be the 

case for party V.) If a key establishment scheme does not require a party to contribute a 

public key, then the identifier of that party is a non-null identifier selected in accordance 

with the protocol utilizing the scheme (This may be the case for party U.). The rationale 

for including the identifiers in the KDF input is provided in Appendix B of [SP 800-

56A]. 

5.9.2 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 

5.9.1. 

The ASN.1 KDF is as follows: 

Function call: KDF(Z, KBits, OtherInfo). 

Fixed Values (implementation dependent):

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

to derive the secret keying material. 

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

input to the hash function. 

Auxiliary Function:

H: an approved hash function. 

Input: 

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

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

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

3. OtherInfo: A bit string specified in ASN.1 DER encoding, which consists of the 

following subfields of information in some application-specific order: 
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a. 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 after any MacKey is extracted from the derived keying material when 

key confirmation is performed. For example if key confirmation is not 

performed, AlgorithmID might indicate that bits 1-128 are to be used as a 128-

bit AES key. If key confirmation is performed, then AlgorithmID might 

indicate that bits 1- 128 are used as a MacKey, and bits 129-256 are to be used 

as the 128-bit AES key. 

b. 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, as a separate unit of information. 

c. 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, as a separate unit of information. When this KDF is used in a KAS1 

scheme, the nonce, NV, supplied by party V shall be included in PartyVInfo as 

a separate unit of information, following IDV. 

d. (Optional) SuppPubInfo: A bit string containing additional, mutually-known 

public information. 

e. (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). 

An application using this KDF is responsible for specifying the ASN.1 structure of 

OtherInfo. In particular, applications using this KDF shall specify the ordering, number, 

and ASN.1 type of the separate units of information contained in each of the subfields 

AlgorithmID, PartyUInfo, and PartyVInfo. Applications allowing the use SuppPrivInfo

subfield and/or the SuppPubInfo subfield, shall also specify the ordering, number and 

ASN.1 type of the additional units of information that may be used in the allowed 

subfield(s). 

Output: 

The DerivedKeyingMaterial as a bit string of length KBits bits (or an appropriate error 

indicator). The ASN.1 KDF produces secret keying material that is at most hBits × (232 –1) 

bits in length. Any call to this key derivation function using a KBits value that is greater than 

hBits × (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_inputBits bits long shall cause the KDF to output an 

error indicator and stop without outputting DerivedKeyingMaterial. 

Process: 
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1. reps = ⎡KBits / hBits⎤.

2. If reps > (232 −1), then 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_inputBits bits long, 

then output an error indicator and stop. 

5. For i = 1 to reps by 1, do the following: 

a. Compute Hashi = H(counter || Z || OtherInfo). 

b. Increment counter (modulo 232), treating it as an unsigned 32-bit integer.

6. Let Hhash be set to Hashreps if (KBits / hBits) is an integer; otherwise, let Hhash be set to 

the (KBits mod hBits) leftmost bits of Hashreps. 

7. Set DerivedKeyingMaterial = Hash1 || Hash2 || ... || Hashreps-1 || Hhash. 

Notes: 

1. Party U shall be the initiator or sender, and party V shall be the responder or receiver, as 

assigned by the relying protocol in accordance with the use of those designators by the 

key establishment scheme employing the KDF. 

2. When a party owns a key pair that is used by the key establishment scheme, the identifier 

assigned to that party shall be one that is bound to that key pair. (This will always be the 

case for party V.) If a key establishment scheme does not require a party to contribute a 

public key, then the identifier of that party is a non-null identifier selected in accordance 

with the protocol utilizing the scheme. (This may be the case for party U.). The rationale 

for including the identifiers in the KDF input is provided in Appendix B of [SP 800-

56A]. 

6 RSA Key Pairs 

6.1 General Requirements 

The following are requirements on key pairs (see the Recommendation for Key Management 

[7]): 

1. Each key pair shall be created using an approved key generation method as specified in 

Section 6.3. 

2. The private keys and prime factors shall be protected from unauthorized access, 

disclosure, and modification. 

3. Each Public key-establishment key shall be bound to an identifier corresponding to the 

owner. 
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4. Public keys shall be protected from unauthorized modification. This is often 

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

Authority (CA). 

5. A recipient of a public key shall be assured of the data integrity and correct association of 

(a) the public key and (b) the identifier of the entity that owns the key pair (that is, the 

party with whom the recipient intends to establish secret keying material). 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 public key 

and identifier from the owner, provided that the recipient trusts the owner and distribution 

process to do this. 

6. One key pair shall not be used for different cryptographic purposes (for example, a 

digital signature key pair shall not to be used for key establishment or vice versa) with 

the following possible exception: when requesting the (initial) certificate for a public keyestablishment key, the key establishment private key associated with the public key may 

be used to sign the certificate request. A key pair may be used in more than one key 

establishment scheme. However, a key pair use for schemes specified in this 

recommendation should not be used for any schemes not specified herein. 

7. An owner and a recipient of a public key shall have assurance of the validity of the 

owner’s 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 6.4. 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. 

8. An owner and a recipient of a public key shall have assurance of the owner’s possession 

of the associated private key (see Section 6.5). The owner or a process acting on behalf of 

the owner shall know the method used to obtain assurance of possession of the owner’s 

private key. The recipient or process acting on behalf of 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. 

9. The owner shall have assurance of the validity and possession of the owner’s key pair 

(See Sections 6.4.1 and 6.5.1 ). (Make a higher number) 

6.2 Criteria for RSA Key Pairs for Key Establishment 

6.2.1 Definition of a Key Pair 

An RSA key pair, in its basic form, consists of an RSA public key (n, e) and an RSA private key 

(n, d), where: 
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1. n, the modulus, shall be the product of exactly two odd positive prime factors, p and q

where nBits is the length in bits of n as specified for the desired security strength s (see 

Table 1) and nLen is the corresponding length in bytes. 

2. The public exponent e shall be selected with the following constraints: 

a. The public exponent e shall be selected prior to generating the prime factors p and 

q and the private exponent d. 

b. The exponent e shall be an odd positive integer such that: 

65,537 ≤ e < 2256

Note that the value of e may be the same for different key pairs. 

3. Two secret and randomly generated positive primes p and q shall be selected with the 

following constraints: 

a. The prime factors of the modulus shall be generated independently at random for 

different key pairs. 

b. LCM((p-1), (q-1)) shall be greater than e and relatively prime to e. 

c. The private prime factor p shall be selected randomly from the primes that satisfy 

( 2)(2(nBits/2) – 1) ≤ p ≤ (2nBits/2-1). 

d. The private prime factor q shall be selected randomly from the primes that satisfy 

( 2)(2(nBits/2) – 1) ≤ q ≤ (2nBits/2-1). 

e. The difference between p and q shall be > 2(nBits/2) – 100. 

f. The prime factors p and q shall be generated using an approved method meeting 

the above constraints. Such methods are provided in Appendix B.3 of FIPS 186-3. 

4. The private exponent d shall be selected with the following constraints after the 

generation of p and q: 

a. The exponent d shall be a positive integer value such that 

2nBits/2 < d < LCM((p-1), (q-1)), and 

b. 1 = ed mod LCM((p-1), (q-1)). (That is, d = e

-1 mod (LCM((p-1), (q-1))). 

In the extremely rare event that d ≤ 2nbits/2, then new values for p, q, and d shall be determined 

and a different value of e may be used. 
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To generate key pairs meeting the above requirements see [3]. 

6.2.2 Formats 

Note that the RSA private key may be expressed in several formats. The basic format of the RSA 

private key consists of the modulus n and a private key exponent d that depends on n and the 

public key exponent e; this format is used throughout this Recommendation. The other two 

formats may be used in implementations but may require appropriate modifications for correct 

implementation. To facilitate implementation testing, the private key shall be one of the 

following: 

1. The basic format: (n, d). 

2. The prime factor format: (p, q, d). 

3. The Chinese Remainder Theorem (CRT) format: (n, e, d, p, q, dP, dQ, qInv), where dP = 

d mod (p – 1), dQ = d mod (q – 1), and qInv = q

–1 mod p.

Key pair generators and key pair validation methods are given for each of these formats in 

Section 6.3. 

6.2.3 Parameter Length Sets 

 Federal Government entities shall select a target security strength for the scheme of either 80 

bits or 112 bits and select scheme parameters from the target security strength as shown in Table 

1. The target security strength shall be selected based on the security needs of the information 

that will be protected using the keying material agreed upon using this Recommendation. SP 

800-57 [7] deprecates the use of the 80 bit security strength after 2010. Entities shall select a 

target security strength of 112 bits if the security life of the information extends beyond 2010. 

This Recommendation specifies two choices for the modulus bit length: 1024 and 2048 bits. The 

security strength of the modulus bit length shall meet or exceed that of the target security 

strength of the scheme. See the comparable strengths table in SP 800-57 to assess the 

comparable security strength of a particular modulus bit length. 

Implementations which include a hash function (e.g. for use in a key derivation function or RSAOAEP) shall select any approved hash function. 

Implementations which include a MAC algorithm (e.g. for key confirmation) shall employ an 

approved MAC algorithm. The MacKey length shall meet or exceed the target security 

strength, and should meet or exceed the security strength of the modulus. The MacTag length 

shall meet or exceed 64 bits (8 bytes). 

For example, an entity may select the 112 bit target security strength for an application which 

establishes a 128 bit AES key. An implementation for the scheme may select 2048 bits, SHA256, and HMAC-SHA-256 for the modulus length, hash function and MAC algorithm, along 

with MacTag and MacKey lengths of 64 bits and 128 bits, respectively. 
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Table 1: IFC Parameters for Key Establishment 

Target Security Strength 

IFC Parameter Name 

80 bits 112 bits 

Bit length of n 1024 bits 

or 

2048 bits 

2048 bits 

Minimum MacKey length 80 bits/10 bytes 112 bits/14 bytes 

Minimum MacTag length 64 bits/8 bytes 64 bits/8 bytes 

6.3 RSA Key Pair Generators 

An RSA key pair generator produces a random RSA public-key/private-key pair at a target 

security strength, given an appropriate RSA key length and possibly other inputs. A key pair 

generator requires a random bit generator (RBG) and a prime number generator. Approved

prime number generators may place additional constraints on RSA key pair generation, 

depending on the target security strength (see FIPS 186-3 [3]). Approved RSA key pair 

generators shall be employed. For key pair criteria, see Section 6.2.1. 

6.3.1 RSAKPG1 Family: RSA Key Pair Generation with a Fixed Public Exponent 

The RSAKPG1 family consists of three RSA key pair generators where the public exponent has 

a fixed value (see Section 6.2). 

Three representations are addressed: 

1. rsakpg1-basic generates the private key in the basic format (n, d), 

2. rsakpg1-prime-factor generates the private key in the prime factor format (p, q, d), and 

3. rsakpg1-crt generates the private key in the Chinese Remainder Theorem format (n, e, d, 

p, q, dP, dQ, qInv). 

An implementation may perform a key pair validation before outputting the key pair from the 

generator. The key pair validation methods for this family are specified in Section 6.4.1.2. 

6.3.1.1 rsakpg1-basic 

rsakpg1-basic is the generator in the RSAKPG1 family where the private key is in the basic 

format (n, d). 
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Function call: rsakpg1-basic(s, nBits, e) 

Input: 

1. s: the target security strength for the key-pair generator, an integer in the set {80, 112}, 

2. nBits: the intended length in bits of the RSA modulus, an integer (see Table 1), and 

3. e: a fixed public exponent, an odd integer such that 65,537 ≤ e < 2256. 

Output: 

1. (n, e): the RSA public key, and 

2. (n, d): the RSA private key in the basic format. 

Error: Indications of the following: 

1. The security strength is out of range, 

2. The modulus length is out of range, 

3. The fixed public exponent is out of range, or 

4. Pair-wise consistency failure. 

Process: 

1. Check the ranges: 

a. If s is not an integer in the set {80, 112}, output an indication that the security 

strength is out of range and stop. 

b. If nBits is not an integer in the set {1024, 2048}, or if nBits is less than the 

minimum key length for the target security strength, s, output an indication 

that the modulus length is out of range and stop. 

c. If e is not an odd integer such that 65,537 ≤ e < 2256, output an indication that 

the exponent is out of range and stop. 

2. Generate the prime factors p and q (see Section 5.4). 

3. Determine the private exponent d: 

 d = e

–1 mod LCM(p – 1, q – 1) . 

In the very rare event that d ≤ 2 nBits/2, discard d, and repeat the process, starting at step 2. 

4. Determine the modulus n as n = p · q. 
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5. Perform a pair-wise consistency test by verifying that k = (k

e

)

d

 mod n for some integer k

satisfying 1 < k < n. If an inconsistency is found, output an indication of a pair-wise 

consistency failure and stop. 

6. Output (n, e) as the public key, and (n, d) as the private key. 

Note that key pair validation as specified in Section 6.4.1.2.1 can be performed after step 5 and 

before step 6. If an error is detected, output an indication of key pair validation failure and stop. 

6.3.1.2 rsakpg1-prime-factor 

rsakpg1-prime-factor is the generator in the RSAKPG1 family where the private key is in the 

prime factor format (p, q, d). 

Function call: rsakpg1-prime-factor(s, nBits, e) 

The inputs, outputs and errors are the same as in rsakpg1-basic (see 6.3.1.1), except that the 

private key is in the prime factor format: (p, q, d). 

The steps are the same as in rsakpg1-basic, except that processing Step 6 is replaced by the 

following: 

6. Output (n, e) as the public key, and (p, q, d) as the private key. 

Note that key pair validation as specified in Section 6.4.1.2.2 can be performed after step 5 and 

before step 6. If an error is detected, output an indication of key pair validation failure and stop. 

6.3.1.3 rsakpg1-crt 

rsakpg1-crt is the generator in the RSAKPG1 family where the private key is in the Chinese 

Remainder Theorem format (n, e, d, p, q, dP, dQ, qInv). 

Function call: rsakpg1-crt(s, nBits, e) 

The inputs, outputs and errors are the same as in rsakpg1-basic (see 6.3.1.1), except that the 

private key is in the Chinese Remainder Theorem format: (n, e, d, p, q, dP, dQ, qInv). 

The steps are the same as in rsakpg1-basic, except that processing Steps 5 and 6 are replaced by 

the following: 

5. Determine the components dP, dQ and qInv: 

a. dP = d mod (p – 1). 

b. dQ = d mod (q – 1). 

c. qInv = q

–1 mod p. 

6. Perform a pair-wise consistency test by verifying that k = (k

e

)

d

 mod n for some integer k

satisfying 1 < k < n. If an inconsistency is found, output an indication of a pair-wise 

consistency failure and stop. 
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7. Output (n, e) as the public key, and (n, e, d, p, q, dP, dQ, qInv) as the private key. 

Note that key pair validation as specified in Section 6.4.1.2.3 can be performed after step 6 and 

before step 7. If an error is detected, output an indication of key pair validation failure and stop 

6.3.2 RSAKPG2 Family: RSA key pair generation with a random public exponent 

The RSAKPG2 family consists of three RSA key pair generators where the public exponent e is 

a random value in the range 65,537 ≤ e < 2256.

This family imposes the same constraints on the key pair as in the RSAKPG1 family (see Section 

6.3.1). 

Three representations are addressed: 

1. rsakpg2-basic generates the private key in the basic format (n, d), 

2. rsakpg2-prime-factor generates the private key in the prime factor format (p, q, d), and 

3. rsakpg2-crt generates the private key in the Chinese Remainder Theorem format (n, e, d, 

p, q, dP, dQ, qInv). 

An implementation may perform a key pair validation before outputting the key pair from the 

generation function. The key pair validation methods for this family are specified in Section 

6.4.1.3. 

6.3.2.1 rsakpg2-basic 

rsakpg2-basic is the generator in the RSAKPG2 family where the private key is in the basic 

format (n, d). 

Function call: rsakpg2-basic(s, nBits, eBits) 

Input: 

1. s: the target security strength for the key-pair generator (see Sections 6.2.1 and 6.2.3), an 

integer in the set {80, 112}, 

2. nBits: the intended length in bits of the RSA modulus, an integer (see Table 1), and 

3. eBits: the intended length in bits of the public exponent, an integer such that 17 ≤ eBits ≤

256. 

Output: 

1. (n, e): the RSA public key, and 

2. (n, d): the RSA private key in the basic format. 
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Error: Indications of the following: 

1. The security strength is out of range, 

2. The modulus length is out of range, 

3. The exponent length is out of range, or 

4. Pair-wise consistency failure. 

Process: 

1. Check the ranges: 

a. If s is not an integer in the set {80, 112}, output an indication that the security 

strength is out of range and stop. 

b. If nBits is not an integer in the set {1024, 2048}, or if nBits is less than the 

minimum key length for the target security strength s (see Section 6.2.3), 

output an indication that the modulus length is out of range and stop. 

c. If eBits is not an integer such that 17 ≤ eBits ≤ 256, output an indication that 

the exponent length is out of range and stop. 

2. Generate an odd public exponent e in the range [2 eBits – 1 + 1, 2 eBits – 1] using an 

approved RBG (see Section 5.3). 

3. Generate the prime factors p and q (see Section 5,4). 

4. Determine the private exponent d: 

 d = e

–1 mod LCM(p – 1, q – 1) . 

5. In the very rare event that d ≤ 2 nBits/2, discard d, and repeat the process, starting at either 

step 2 or step 3 (That is, a different value of e may be used when generating a new pair 

of primes, but this is not required). 

6. Determine the modulus n as n = p · q. 

7. Perform a pair-wise consistency test by verifying that k = (k

e

)

d

 mod n for some integer k

satisfying 1 < k < n. If an inconsistency is found, output an indication of a pair-wise 

consistency failure and stop. 

8. Output (n, e) as the public key, and (n, d) as the private key. 

Note that key pair validation as specified in Section 6.4.1.3.1 can be performed after step 7 and 

before step 8. If an error is detected, output an indication of key pair validation failure and stop. 
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6.3.2.2 rsakpg2-prime-factor 

rsakpg2-prime-factor is the generator in the RSAKPG2 family where the private key is in the 

prime factor format (p, q, d). 

Function call: rsakpg2-prime-factor(s, nBits, eBits) 

The inputs, outputs and errors are the same as in rsakpg2-basic (see 6.3.2.1), except that the 

private key is in the prime factor format: 

(p, q, d): RSA private key in prime factor format 

The steps are the same as in rsakpg2-basic except that processing Step 8 is replaced by the 

following: 

8. Output (n, e) as the public key, and (p, q, d) as the private key. 

Note that key pair validation as specified in Section 6.4.1.3.2 can be performed after step 7 and 

before step 8. If an error is detected, output an indication of key pair validation failure and stop. 

6.3.2.3 rsakpg2-crt 

rsakpg2-crt is the generator in the RSAKPG2 family where the private key is in the Chinese 

Remainder Theorem format (n, e, d, p, q, dP, dQ, qInv). 

Function call: rsakpg2-crt(s, nBits, eBits) 

The inputs, outputs and errors are the same as in rsakpg2-basic (see 6.3.2.1), except that the 

private key is in the Chinese Remainder Theorem format: 

(n, e, d, p, q, dP, dQ, qInv): RSA private key in Chinese Remainder Theorem format. 

The steps are the same as in rsakpg2-basic except that processing Steps 7 and 8 are replaced by 

the following: 

7. Determine the components dP, dQ and qInv: 

a. dP = d mod (p – 1). 

b. dQ = d mod (q – 1). 

c. qInv = q–1 mod p . 

 

8. Perform a pair-wise consistency test by verifying that k = (k

e

)

d

 mod n for some integer k

satisfying 1 < k < n. If an inconsistency is found, output an indication of a pair-wise 

consistency failure and stop. 

9. Output (n, e) as the public key, and (n, e, d, p, q, dP, dQ, qInv) as the private key. 

Note that key pair validation as specified in Section 6.4.1.3.3 can be performed after step 8 and 

before step 9. If an error is detected, output an indication of key pair validation failure and stop. 
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6.4 Assurances of Validity 

Secure key establishment depends on the validity of the keys. To explain the assurance 

requirements, some terminology needs to be defined. The owner of a key pair is the entity that is 

authorized to use the private key that corresponds to the owner’s public key, whether or not the 

owner generated the key pair. The recipient of a public key is the entity that is participating in a 

key establishment transaction with the owner and obtains the owner’s public key before or 

during the current transaction. 

6.4.1 Assurance of Key Pair Validity 

Assurance of key pair validity provides assurance that a key pair was generated in accordance 

with the requirements of Section 6.2 and Section 6.3. Key pair validity implies public-key 

validity and assurance of possession of the correct private key. Assurance of key pair validity can 

only be provided by an entity that has the private key (e.g., the owner). The owner shall have 

assurance of key pair validity before using the key pair for other operations. 

6.4.1.1 General Method for Obtaining Assurance of Key Pair Validity 

Assurance of key pair validity shall be obtained by its owner using (all of) the following steps. 

1. Key pair generation: Assurance that the key pair has been correctly formed, in a manner 

consistent with the criteria of Section 6.2, is obtained using one of the following two 

methods: 

a. Owner generation – The owner receives the desired assurance if it generates the 

public/private key pair as specified in Section 6.3. 

b. TTP generation – The owner receives the desired assurance when a trusted third 

party (TTP) that is trusted by the owner generates the public/private key pair as 

specified in Section 6.3 and provides it to the owner. 

2. The owner shall perform a pair-wise consistency test by verifying that k = (k

e

)

d

 mod n for 

some integer k satisfying 1 < k < n. Note that if the owner generated the key pair (see step 

1.a above), an initial pair-wise consistency test was performed during key generation (see 

Section 6.3). Otherwise, the owner shall perform the consistency check separately, prior 

to the first use of the key pair in a key establishment transaction (see Section 4.1). 

Additional pair-wise consistency tests shall be performed by the owner whenever 

assurance of key pair validity needs to be refreshed. 

3. Key pair validation: A key pair shall be validated using one of the following two 

methods: 

a. Owner key pair validation – The owner either performs a successful key pair 

validation during key pair generation (see Section 6.3), or performs a successful 

key pair validation separate from key pair generation (see Sections 6.4.1.2 and 

6.4.1.3). 
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b. TTP key pair validation – A trusted third party (trusted by the owner) either 

performs a successful key pair validation during key pair generation (see Section 

6.3), or performs a successful key pair validation separate from key pair 

generation (see Sections 6.4.1.2 and 6.4.1.3), and indicates the success to the 

owner. Note that if the key pair validation is performed separately from the key 

pair generation, and the TTP does not have the key pair, then the party that 

generated the key pair or owns the key pair must provide it to the TTP. 

A key pair validation shall be performed prior to the first use of the key pair in a key 

establishment transaction (see Section 4.1). The key pair can be revalidated at any time. Note 

that the use of a TTP to generate a key pair or to perform key pair validation 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 or otherwise compromise the key establishment transaction. 

6.4.1.2 RSAKPV1 Family: RSA Key Pair Validation with a Fixed Exponent 

The RSAKPV1 family of key pair validation methods corresponds to the RSAKPG1 family (see 

Section 6.3.1). 

6.4.1.2.1 rsakpv1-basic

rsakpv1-basic is the validation method corresponding to rsakpg1-basic (see Section 6.3.1.1). 

Function call: rsakpv1-basic (s, nBits, efixed, (npub, epub) (npriv, d)) 

Input: 

1. s: the target security strength for the key-pair generator (see Section 6.2.3), an integer in 

the set {80, 112}, 

2. nBits: the expected length in bits of the RSA modulus, an integer (see Table 1), 

3. efixed: the intended fixed public exponent, an odd integer such that 65,537 ≤ efixed < 2256, 

4. (npub, epub): the RSA public key to be validated, and 

5. (npriv, d): the RSA private key to be validated in the basic format. 

Output: 

1. status: An indication that the key pair is valid or an indication of an error: 

a. The security strength is out of range, 

b. The modulus length is out of range, 

c. The fixed exponent is out of range, or 

d. The request is invalid. 

Process: 
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1. Check the ranges: 

a. If s is not an integer in the set {80, 112}, output an indication that the security 

strength is out of range and stop. 

b. If nBits is not an integer in the set {1024, 2048}, or if nBits is less than the 

minimum key length for the target security strength s (see [7]), output an 

indication that the modulus length is out of range and stop. 

c. If efixed is not an odd integer such that 65,537 ≤ efixed < 2256, output an indication 

that the fixed exponent is out of range and stop. 

2. Compare the public exponents: 

If epub ≠ efixed, output an indication that the request is invalid and stop. 

3. Check the modulus: 

a. If npub ≠ npriv, output an indication of an invalid key pair and stop. 

b. If the length in bits of the modulus npub is not nBits, output an indication of an 

invalid key pair and stop. 

4. Prime factor recovery: 

a. Recover the prime factors p and q from the modulus npub, the public exponent 

epub and the private exponent d (see Appendix C): 

(p, q) = RecoverPrimeFactors (npub, epub, d) 

b. If RecoverPrimeFactors outputs an indication that the prime factors were not 

found, output an indication that the request is invalid and stop. 

c. If npub ≠ p · q, then output an indication that the request is invalid and stop. 

5. Check the prime factors: 

a. Apply an approved primality test to test the prime number p (see Section 5.5). 

b. If the primality test indicates that p is not prime, output an invalid key pair and 

stop. 

c. If (p < 2(2 nBits/2 – 1)) or (p > 2 nBits/2 – 1), output an indication of an invalid key 

pair and stop. 

d. If GCD (p – 1, epub) ≠ 1, output an indication of an invalid key pair and stop. 

e. Apply an approved primality test to test the prime number q (see Section 5.5). 

f. If the primality test indicates that q is not prime, output an indication of an invalid 

key pair and stop. 
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g. If (q < 2(2 nBits/2 – 1)) or (q > 2 nBits/2 – 1), output an indication of an invalid key 

pair and stop. 

h. If GCD (q – 1, epub) ≠ 1, output an indication of an invalid key pair and stop. 

i. If |p – q| ≤ 2 nBits/2 – 100, output an indication of an invalid key pair and stop. 

6. Check that the private exponent d satisfies 

a. 2nBits/2 < d < LCM (p – 1, q – 1). 

and 

b. 1 = (d epub) mod LCM (p – 1, q – 1). 

If either check fails, output an indication of an invalid key pair and stop. 

7. Output an indication that the key pair is valid. 

6.4.1.2.2 rsakpv1-prime-factor

rsakpv1-prime-factor is the validation method corresponding to rsakpg1-prime-factor (see 

6.3.1.2). 

Function call: rsakpv1-prime-factor (s, nBits, efixed, (npub, epub), (p, q, d)) 

The inputs, outputs and errors are the same as in rsakpv1-basic (see Section 6.4.1.2.1), except 

that the private key is in the prime factor format: 

(p, q, d) 

The steps are the same as in rsakpv1-basic except that in processing: 

1. Step 3 is replaced by the following: 

3. Check the modulus: 

a. If npub ≠ p · q, output an indication of an invalid key pair and stop. 

b. If the length in bits of the modulus npub is not nBits, output an indication of 

an invalid key pair and stop. 

2. Step 4 (prime factor recovery) is omitted. 

6.4.1.2.3 rsakpv1-crt 

rsakpv1-crt is the validation method corresponding to rsakpg1-crt.

Function call: rsakpv1-crt (s, nBits, efixed, (npub, epub), (npriv, epriv, d, p, q, dP, dQ, qInv)) 

The inputs, outputs and errors are the same as in rsakpv1-basic (see Section 6.4.1.2.1), except 

that the private key is in the Chinese Remainder Theorem format: 

(npriv, epriv, d, p, q, dP, dQ, qInv) 
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The steps are the same as in rsakpv1-basic except that in processing: 

1. Step 2 is replaced by the following: 

2. Compare the public exponents: 

If epub ≠ efixed or epub ≠ epriv, output an indication of an invalid key pair and stop. 

2. Step 3 is replaced by 

3. Check the modulus: 

a. If npub ≠ p · q, or npub ≠ npriv, output an indication of an invalid key pair and 

stop. 

b. If the length in bits of the modulus npub is not nBits, output an indication of 

an invalid key pair and stop. 

3. Step 4 (prime factor recovery) is omitted, 

4. Step 7 is replaced by the following: 

7. Check the CRT components: Check that the components dP, dQ and qInv satisfy 

a. 1 < dP < (p – 1). 

b. 1 < dQ < (q – 1). 

c. 1 < qInv < p . 

d. (dP · efixed) - 1 = 0 mod (p – 1). 

e. (dQ · efixed) - 1 = 0 mod (q – 1). 

f. (qInv · q) - 1 = 0 mod p . 

If any of the criteria are not met, output an indication of an invalid key pair and 

stop. 

8. Output an indication that the key pair is valid. 

6.4.1.3 RSAKPV2 Family: RSA Key Pair Validation with a Random Exponent 

The RSAKPV2 family of key pair validation methods corresponds to RSAKPG2 family (see 

Section 6.3.2). 

6.4.1.3.1 rsakpv2-basic 

rsakpv2-basic is the validation method corresponding to rsakpg2-basic (see Section 6.3.2.1). 

Function call: rsapkv2-basic (s, nBits, eBits, (npub, e), (npriv, d)) 
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The method is the same as the rsapkv1-basic method in Section 6.4.1.2 except that: 

1. The efixed input parameter becomes eBits, which is the expected length in bits of the 

public exponent, an integer such that 17 ≤ eBits ≤ 2256. 

2. Step 1c is replaced by: 

c. If (eBits < 17) or (eBits > 256), output an indication that the exponent is out of 

range and stop. 

3. Step 2 is replaced by: 

2. Check the public exponent. 

If the public exponent epub is not odd, or if the length in bits of the public 

exponent epub is not eBits, output an indication of an invalid key pair and stop. 

6.4.1.3.2 rsakpv2-prime-factor

rsakpv2-prime-factor is the validation method corresponding to rsakpg2-prime-factor (see 

Section 6.3.2.2). 

Function call: rsakpv2-prime-factor (s, nBits, eBits, (npub, epub), (p, q, d)) 

The inputs, outputs and errors are the same as in rsakpv2-basic (see Section 6.4.1.3.1), except 

that the private key is in the prime factor format: 

(p, q, d) 

The steps are the same as in rsakpv2-basic except that in processing: 

1. Step 3 is replaced by the following: 

3. Check the modulus: 

a. If npub ≠ p · q, output an indication of an invalid key pair and stop. 

b. If the length in bits of the modulus npub is not nBits, output an indication of 

an invalid key pair and stop. 

2. Step 4 (prime factor recovery) is omitted. 

6.4.1.3.3 rsakpv2-crt

rsakpv2-crt is the validation method corresponding to rsakpg2-crt (see Section 6.3.1.3). 

Function call: rsakpv2-crt (s, nBits, eBits, (npub, epub), (npriv, epriv, d, p, q, dP, dQ, qInv)) 

The inputs, outputs and errors are the same as in rsakpv2-basic (see Section 6.4.1.3.1), except 

that the private key is in the Chinese Remainder Theorem format: 

(npriv, epriv, d, p, q, dP, dQ, qInv) 

The steps are the same as in rsakpv2-basic except that in processing: 
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1. Step 2 is replaced by the following: 

2. Compare the public exponents: 

If (epub ≠ epriv) or (epub is not odd) or (epub is not eBits), output an indication of an 

invalid key pair and stop. 

2. Step 3 is replaced by 

3. Check the modulus: 

a. If (npub ≠ pq) or (npub ≠ npriv) output an indication of an invalid key pair and 

stop. 

b. If the length in bits of the modulus npub is not nBits, output an indication of 

an invalid key pair and stop. 

3. Step 4 (prime factor recovery) is omitted, 

4. Step 7 is replaced by the following: 

7. Check the CRT components: Check that the components dP, dQ and qInv satisfy 

a. 1 < dP < (p – 1). 

b. 1 < dQ < (q – 1). 

c. 1 < qInv < p . 

d. 1 = (dP · epub) mod (p – 1). 

e. 1 = (dQ · epub) mod (q – 1). 

f. 1 = (qInv · q) mod p. 

If any of the criteria are not met, output an indication of an invalid key pair and 

stop. 

8. Output an indication that the key pair is valid. 

6.4.2 Recipient Assurances of Public Key Validity 

In this Recommendation, the recipient of a public key is an entity that does not (and should not) 

have access to the associated private key. The recipient of a candidate public key shall have 

assurance of the arithmetic validity of that key before using it in a key establishment transaction 

with its owner. 

6.4.2.1 General Method for Obtaining Assurance of Public Key Validity 

The recipient shall obtain assurance of public key validity using one or more of the following 

methods: 
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1. Recipient Partial Public Key Validation - The recipient performs a successful partial 

public key validation (see Section 6.4.2.2). 

2. TTP Partial Public Key Validation – The recipient receives assurance that a trusted third 

party (trusted by the recipient) has performed a successful partial public key validation 

(see Section 6.4.2.2). 

3. TTP Key Pair Validation – The recipient receives assurance that a trusted third party 

(trusted by the recipient and the owner) has performed key pair validation in accordance 

with Section 6.4.1.1 (step 3.b). 

Note that the use of a TTP to perform key pair validation (method 3) implies that both the owner 

and any recipient of the public key trust that the TTP will not use the owner’s private key to 

masquerade as the owner or otherwise compromise the key establishment transaction. 

6.4.2.2 Partial Public Key Validation for RSA 

Partial public key validation for RSA consists of conducting plausibility tests. These tests 

determine whether the public modulus and public exponent are plausible, not necessarily 

whether they are completely valid, i.e., they may not conform to all RSA key generation 

requirements as specified in this Recommendation. Plausibility tests can detect unintentional 

errors with a reasonable probability. Note that full RSA public key validation is not specified in 

this Recommendation, as it is an area of research. Therefore, if an application requires assurance 

of full public key validation, then another approved key establishment method shall be used. 

Plausibility tests shall include the tests specified in SP 800-89, Section 5.3.3 with the caveat that 

the length of the modulus shall be a length that is specified in this Recommendation. 

6.5 Assurances of Private Key Possession 

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

private keys to those who have been authorized to access them (e.g., their respective owners or 

certain trusted third parties). In addition to preventing unauthorized entities from gaining access 

to private keys, it is also important to obtain assurance that authorized owners actually have 

access to their correct private keys. 

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

6.5.1. Parties that interact with the owner (e.g., a public key recipient) also need to obtain 

assurance that the owner possesses the private key; these requirements are specified in Section 

6.5.2. 

When assurance of possession of a private key is initially 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 could lose possession of the 

associated private key, deliberately or due to an error; for this reason, renewing the assurance of 

possession may be appropriate for some applications (i.e., assurance of possession can be 

refreshed). See Section 6.5.2.2 and Section 6.6 for a discussion of the methods that may be used 

by the recipient of a public key to renew the assurance of the owner’s possession of the 
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corresponding private key. A discussion of the effect of time on the assurance of private key 

possession is provided in SP 800-89. 

6.5.1 Owner Assurance of Private Key Possession 

The owner of a 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 Key Generation - The act of generating a key pair, as 

specified in this document, 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. 

2. Owner Receives Assurance via Assurance of Key Pair Validity – The owner obtains 

assurance of key pair validity (See section 6.4.1) thereby also obtaining owner assurance 

of private key possession. 

3. Owner Receives Assurance via Explicit Key Confirmation – The owner employs the key 

pair to successfully engage another party in a key agreement transaction using a scheme 

from the KAS2 family that incorporates 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 6.6 for further 

explanation. 

4. Owner Receives Assurance via an Encrypted Certificate - The owner uses the private key 

while engaging in a key establishment transaction with a Certificate Authority (trusted by 

the owner), after providing the CA with the corresponding public key. As part of this 

transaction, the CA generates a (new) certificate containing the owner’s public key and 

encrypts that certificate using (some portion of) the symmetric keying material that has 

been established. Only the encrypted form of the certificate is provided to the owner. By 

successfully decrypting the certificate and verifying the CA’s signature, the owner 

obtains assurance of possession of the correct private key (at the time of the key 

establishment transaction). 

The owner of a 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 private key is 

sufficient and appropriate to meet the security requirements of the owner’s intended 

application(s) 

6.5.2 Recipient Assurance of Owner’s Possession of a Private Key 

It is assumed that, at the time of binding an identifier to the owner’s public key, the binding 

authority has obtained assurance that the owner is in possession of the correct private key. In 

conjunction with a (successful) key establishment transaction, the recipient of another party’s 

public key shall also obtain this assurance – either indirectly using a trusted third party (see 

Section 6.5.2.1) or directly from the claimed owner (see Section 6.5.2.2) - before using the 

established keying material for purposes beyond those required during the key establishment 

transaction itself. 
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When two parties engage in a key establishment transaction, there is (at least) an implicit claim 

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

is considered to be a claimed owner of the corresponding key pair – as opposed to being a true

owner – until adequate assurance can be provided that the party is actually the one in possession 

of the private key (See Section 6.7). 

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

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

is sufficient and appropriate to meet the security requirements of the owner’s intended 

application(s). 

6.5.2.1 Recipient Indirectly Obtains Assurance of Possession Using a Trusted 

Third Party 

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

the correct private key using a trusted third party, either before or during a key establishment 

transaction that makes use of that 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 6.5.2.2 of this Recommendation and Section 8.1.5.1.1.2 of SP 800-57 Part 1 [7] for 

possible methods). The recipient of a 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). 

6.5.2.2 Recipient Obtains Assurance of Possession Directly from the Claimed 

Owner 

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

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

transaction that explicitly incorporates key confirmation as specified in Sections 8.2.3, 8.3.3, 

9.2.4, or 9.3.4 with the claimed owner serving as the key confirmation provider. Note that the 

recipient of the public key in question will also be the key confirmation recipient. (See Section 

6.6 for further explanation.) Also note that this use of key confirmation is an additional benefit 

beyond its use to confirm that two parties possess the same keying material. 

The recipient of a public key (or agents trusted to act on the recipient’s behalf) shall determine 

whether or not using one of the key establishment schemes in this Recommendation to obtain 

assurance of possession through key confirmation is sufficient and appropriate to meet the 

security requirements of the recipient’s intended application(s). Other approved methods (e.g. 

Section 5.4.4 of SP800-57 Part 1) of directly obtaining this assurance of possession from the 

owner are also allowed. If obtaining assurance of possession directly from the owner is not 

acceptable, then assurance of possession shall be obtained indirectly as discussed in Section 

6.5.2.1. 

Successful key confirmation (performed in the context described in this Recommendation) 

demonstrates that the correct 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 
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The assurance of possession may be useful even when the recipient has previously obtained 

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

be appropriate in situations where the recipient desires renewed assurance that the owner 

possesses the correct private key (and that the owner is still able to use it correctly), including 

situations where there is no access to a trusted party who can provide renewed assurance of the 

owner’s continued possession of the private key. 

Note that the requirement that assurance of possession be obtained before using the established 

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

prohibit the parties to a key establishment transaction from using a portion of the derived or 

transported keying material during the key establishment transaction, for purposes required by 

that key establishment scheme. For example, in a transaction involving a key agreement scheme 

that incorporates key confirmation, the parties establish a (purported) 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. 

6.6 Key Confirmation 

The term key confirmation (KC) refers to actions taken to provide assurance to one party (the key 

confirmation recipient) that another party (the key confirmation provider) is in possession of a 

(supposedly) shared secret and/or the correct version of keying material that was derived or 

transported during a key establishment transaction. (Correct from the perspective of the key 

confirmation recipient.) Such actions are said to provide unilateral key confirmation when they 

provide this assurance to only one of the participants in a key establishment transaction; the 

actions are 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 some means outside of the key 

establishment scheme (for example, by decrypting an encrypted message sent from the other 

party using a symmetric key that was derived, in part, from a “master secret” determined during 

the key establishment transaction), but this is not always the case. Some schemes in the 

Recommendation include the exchange of explicit key confirmation information in order to 

enhance the scheme’s security properties. 

In this Recommendation, key confirmation can be provided only if the provider owns a keyestablishment pair that is used during key establishment. Each family of key agreement schemes 

specified in this Recommendation includes a scheme that incorporates unilateral key 

confirmation provided by the responder to the initiator. Similarly, each family of key transport 

schemes specified in this Recommendation includes a scheme that incorporates unilateral key 

confirmation provided by the receiver to the sender. The KAS2 family of key agreement 

schemes also includes a scheme incorporating unilateral key confirmation provided by the 

initiator to the responder, and a scheme incorporating bilateral key confirmation. 

In each scheme that includes key confirmation, the following steps shall be performed: 

1. The KC recipient sends unpredictable secret data to the KC provider encrypted using the 

provider’s public key to produce (ephemeral) ciphertext. 
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2. The KC provider uses its private key to decrypt the ciphertext before obtaining (or 

deriving) a MacKey that is randomly generated for each transaction, and known only to 

the parties engaged in that transaction. 

3. This MacKey and certain transaction-specific MacData (which includes the parties’ 

identifiers as well as ephemeral data that has been exchanged between the parties) are 

used by the KC provider as input to an approved MAC algorithm to obtain a MacTag

whose value is (for all practical purposes) unique to the transaction. The MacTag value is 

then sent to the KC recipient. (See Sections 5.2, 5.9, 6.2, 8 and 9 for details. 

4. The KC recipient performs an independent computation of the MacTag. If the MacTag

value computed by the KC recipient matches the MacTag value received from the KC 

provider, then key confirmation is successful, and the KC recipient obtains assurance that 

both parties contemporaneously agree on the values of the MacKey and MacData, and 

that they have successfully established a shared secret and/or keying material. The 

MacKey employed during the transaction shall be zeroized after its use. The MacKey

shall not be used for purposes other than key confirmation or implementation validation 

testing. (See Sections 5.2, 5.9, 6.2, 8 and 9 for details). 

A close examination of the KC process shows that any of the six key establishment schemes 

specified in this Recommendation that incorporate key confirmation can be used to provide the 

KC recipient with assurance that the KC provider is currently in possession of the (correct) 

private key – the one corresponding to the KC provider’s public key-establishment key. 

The transaction-specific values of both the MacKey and MacData prevent (for all practical 

purposes) the replay of any previously computed value of MacTag. The receipt of a correctly 

computed MacTag, coupled with the presumed inability of the KC provider (or others) to predict 

the values of either the MacKey or the secret data that was encrypted with the KC provider’s 

public key, provides assurance to the KC recipient that the KC provider has used the correct 

private key during the current transaction – to successfully recover the secret data that is a 

prerequisite to learning the value of the MacKey. 

The three key-agreement schemes in the KAS2 family that incorporate key confirmation can also 

be used to provide assurance to the KC recipient that it is in possession of the correct value of the 

private key corresponding to its own public key-agreement key. In each of these schemes, the 

KC recipient receives an unpredictable secret value from the KC provider that has been 

encrypted using the KC recipient’s public key to form (ephemeral) ciphertext. The KC recipient 

uses its private key to decrypt this ciphertext, and includes the recovered shared secret as part of 

the input to the KDF when deriving the MacKey described above. (See Section 8.3.) If the KC 

recipient used an incorrect value for its private key, it is highly likely that it would not recover 

the correct version of that shared secret, and as a result, would incorrectly compute both the

MacKey and MacTag. Therefore, if the MacTag value computed by the KC recipient in a KAS2 

scheme matches the MacTag value received from the KC provider, then – in addition to the other 

types of assurance described above – the KC recipient obtains assurance that it has used the 

correct private key during the current transaction to successfully recover the secret value sent by 

the KC provider. 
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In order to assert that key confirmation is performed in compliance with this Recommendation, 

key confirmation shall be incorporated into a key establishment scheme as specified in this 

Recommendation. If any other methods are used to provide key confirmation, this 

Recommendation makes no statement as to their adequacy.

6.6.1 Unilateral Key Confirmation for Key Establishment Schemes 

As specified in this Recommendation, unilateral key confirmation occurs when one participant in 

a key establishment scheme (the “provider”) successfully provides assurance to the other 

participant (the “recipient”) that both the provider and the recipient have computed the same 

secret MacKey during a key establishment transaction. This MacKey is either 1) derived from a 

shared secret determined during a key agreement transaction, or 2) included in the secret keying 

material shared by both the provider and the recipient during a key transport transaction. In this 

Recommendation, the inclusion of key confirmation in a scheme is restricted to cases where the 

key confirmation provider owns a key-establishment key pair that is used during the 

establishment transaction. 

As a necessary part of the process of providing/obtaining unilateral key confirmation, the 

following steps shall be incorporated into a key establishment scheme. (Depending upon the 

particular key establishment scheme, additional steps may also be required. Details will be 

provided for each scheme.) Note that the provider may be either the scheme initiator/sender 

(party U) or the scheme responder/receiver (party V), as long as the provider is using a keyestablishment key pair in the key establishment scheme, and the recipient is the other party. 

1. 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 as specified in Section 6.6.2.

IDP is the identifier of the provider. 

IDR is the identifier of the recipient. 

EphemDataP and EphemDataR are ciphertext values or nonces contributed by the 

provider and recipient, respectively. These values are specified in the sections describing 

the schemes that include key confirmation. EphemDataP is null in the key transport cases. 

Text is an optional byte string that may be used during key confirmation and that is known 

by the parties establishing the secret keying material. 

2. In the case of a key agreement scheme: After computing the shared secret and applying 

the key derivation function to obtain the DerivedKeyingMaterial (see Section 5.9), the 

provider parses DerivedKeyingMaterial into two parts, the MacKey and the KeyData: 

MacKey || KeyData = DerivedKeyingMaterial.
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In the case of a key transport scheme, the provider parses the TransportedKeyingMaterial

into the same two parts: 

MacKey || KeyData = TransportedKeyingMaterial.

3. The provider computes MacTagP (see Section 5.2.1) and sends it to the recipient: 

MacTagP = MAC(MacKey, MacTagLen, MacDataP). 

4. The recipient determines MacDataP, MacKey, and MacTagP in the same manner as the 

provider, and then compares its computed MacTagP to the value received from the 

provider. If they are equal, then the recipient is assured that the provider has used the 

same value for MacKey in its computations and that the provider shares the recipient’s 

value of MacDataP. The assurance of a shared value for MacKey provides additional 

assurance to the recipient that: 

a. For key agreement schemes: the provider shares the secret value (Z) from which 

MacKey and KeyData are derived (see Section 5.9). Thus, the recipient also has 

assurance that the provider could compute KeyData correctly. 

b. For key transport schemes: The provider has recovered all of the transported 

keying material including KeyData.

5. Zeroize the MacKey once it is no longer needed for MacTag computations. 

6.6.2 Bilateral Key Confirmation for Key Establishment Schemes 

Bilateral key confirmation is accomplished by performing unilateral key confirmation in both 

directions (with U providing MacTagU to recipient V, and V providing MacTagV to recipient U) 

during the same scheme. In addition to replacing P and R by U and V (or by V and U), there are 

also a few clarifications to the process described in Section 6.6.1: 

1. When computing MacTagU, the value of the six-byte message_stringU that forms the 

initial segment of MacDataU is “KC_2_U”. 

2. When computing MacTagV, the value of the six-byte message_stringV that forms the 

initial segment of MacDataV is “KC_2_V”. 

3. If used at all, the value of the (optional) byte string Text used to form the final segment of 

MacDataU can be different than the value of Text used to form the final segment of 

MacDataV. 

6.7 Authentication 

Successful key confirmation, when performed as specified in this Recommendation, can supply 

entity authentication with respect to the key confirmation provider; i.e., the key confirmation 

recipient can obtain assurance concerning the identity of the provider. 

The correct computation of MacTagP by the provider requires knowledge of the private key 

corresponding to a particular public key-establishment key that has been bound to the key-pair 
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owner’s identifier. The recipient of a correctly computed MacTagP (correct from the recipient’s 

perspective) obtains assurance that the provider is in possession of the correct private key, and 

may infer that the provider is the owner of that key pair. 

In addition to the security level associated with the cryptographic elements and parameters 

employed during the key-establishment/key-confirmation process, the level of assurance 

associated with this entity-authentication technique is dependent upon the specificity of the keypair owner’s identifier. 

7 IFC Primitives and Operations 

7.1 Encryption and Decryption Primitives 

RSAEP and RSADP are the basic encryption and decryption primitives from the RSA 

cryptosystem [16]. RSAEP produces ciphertext from keying material using a public key; RSADP 

recovers the keying material from the ciphertext using the corresponding private key. 

7.1.1 RSAEP 

RSAEP produces ciphertext from keying material using an RSA public key. 

Function call: RSAEP((n, e), k) 

Input: 

1. (n, e): the RSA public key. 

2. k: the keying material, an integer such that 1 < k < n – 1. 

Output: 

c: the ciphertext, an integer such that 1 < c < n – 1. 

Errors: An indication that the keying material is out of range. 

Assumption: The RSA public key is valid (see Section 6.4). 

Process: 

1. If the keying material k is not such that 1 < k < n – 1, output an indication that the keying 

material is out of range and stop. 

2. Let c = (k) e

 mod n. 

3. Output c. 

7.1.2 RSADP 

RSADP is the basic decryption operation. It recovers keying material from ciphertext using an 

RSA private key. 

Function call: RSADP((n, d), c) 
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Input: 

1. (n, d): the RSA private key. 

2. c: the ciphertext, such that 1 < c < n – 1. 

Output: 

k: the keying material, an integer such that 1 < k < n – 1. 

Errors: An indication that the ciphertext is out of range. 

Assumption: The RSA private key is part of a valid key pair (see Section 6.4). 

Process: 

1. If the ciphertext c is not such that 1 < c < n – 1, output an indication that the ciphertext is 

out of range and stop. 

2. Let k = c

d

 mod n. 

3. Output k. 

Note: 

Care needs to be taken to ensure that an implementation of RSADP does not reveal even 

partial information about the value of k. An opponent who can reliably obtain particular bits 

of k for sufficiently many chosen ciphertexts may be able to obtain the full decryption of an 

arbitrary ciphertext by applying the bit-security results of Håstad and Näslund [17]. 

7.2 Encryption and Decryption Operations 

7.2.1 RSA Secret Value Encapsulation (RSASVE) 

Secret value encapsulation generates and encrypts a secret value to produce ciphertext using a 

public key-establishment key. The recovery operation recovers the secret value (now the shared 

secret) from the ciphertext using the corresponding private key-establishment key. Secret value 

encapsulation employs a Random Bit Generator (RGB) to generate the secret value. 

The RSASVE generate and recovery operations specified in Sections 7.2.1.2 and 7.2.1.3, 

respectively, are based on the RSAEP and RSADP primitives (see Section 7.1). These operations 

are used by the KAS1 and KAS2 key agreement families (see Sections 8.2 and 8.3), and by the 

RSA-KEM KWS key transport family (see Sections 9.3 and 7.2.3). 

7.2.1.1 RSASVE Components 

RSASVE requires an approved RBG (see Section 5.3). The security strength for the RBG shall

be equal to or greater than the target security strength for the schemes in which RSASVE is 

employed. 
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1. RSAEP: RSA Encryption Primitive (see Section 7.1.1). 

2. RSADP: RSA Decryption Primitive (see Section 7.1.2). 

7.2.1.2 RSASVE Generate Operation 

RSASVE.GENERATE generates a shared secret and corresponding ciphertext using an RSA 

public key. 

Function call: RSASVE.GENERATE((n, e)) 

Input: 

(n, e): an RSA public key. 

Output: 

1. Z: the shared secret; a byte string of length nLen bytes.

2. C: the ciphertext; a byte string of length nLen bytes.

Assumptions: The RSA public key is part of a valid key pair. 

Process: 

1. Compute the value of nLen as the length in bytes of the modulus n. 

2. Generation: 

a. Using the RBG (see Section 5.3), generate an nLen byte string, Z. 

b. Convert Z to an integer z (See Appendix B.2): 

 z = BS2I(Z, nLen). 

c. If z does not satisfy 1 < z < n – 1, then go to a. 

3. RSA encryption: 

a. Apply the RSAEP encryption primitive (see Section 7.1.1) to the integer z

using the public key (n, e) to produce an integer ciphertext c: 

c = RSAEP((n, e), z). 

b. Convert the ciphertext c to a ciphertext byte string C of length nLen bytes 

(see Appendix B.1): 

C = I2BS(c, nLen). 
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4. Output the string Z as the shared secret, and the ciphertext C. 

7.2.1.3 RSASVE Recovery Operation 

RSASVE.RECOVER recovers a shared secret from ciphertext using an RSA private key. 

Function call: RSASVE.RECOVER((n, d), C) 

Input: 

1. (n, d): an RSA private key.

2. C: the ciphertext; a byte string of length nLen bytes.

Output: 

Z: the shared secret; a byte string of length nLen bytes.

Errors: An indication of a decryption error. 

Assumptions: The RSA private key is part of a valid key pair. 

Process: 

1. Compute the value of nLen as the length in bytes of the modulus n. 

2. Length checking: 

If the length of the ciphertext C is not nLen bytes, output an indication of a decryption 

error and stop. 

3. RSA decryption: 

a. Convert the ciphertext C to an integer ciphertext c (see Appendix B.2): 

c = BS2I(C). 

b. Apply the RSADP decryption primitive (see Section 7.1.2) to the ciphertext 

c using the private key (n, d) to produce an integer z: 

z = RSADP((n, d), c) . 

c. If RSADP indicates that the ciphertext is out of range, output an indication 

of a decryption error and stop. 

d. Convert the integer z to a byte string Z of length nLen bytes (see Appendix 

B.1): 

Z = I2BS(z, nLen). 
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4. Output the string Z as the shared secret. 

Note: 

Care needs to be taken to ensure that an implementation does not reveal information about 

the encapsulated secret value Z. For instance, the observable behavior of the I2BS routine 

must not reveal even partial information about the byte string Z. An opponent who can 

reliably obtain particular bits of Z for sufficiently many chosen ciphertexts may be able to 

obtain the full decryption of an arbitrary RSA-encrypted value by applying the bit-security 

results of Håstad and Näslund [17]. 

7.2.2 RSA with Optimal Asymmetric Encryption Padding (RSA-OAEP) 

RSA-OAEP consists of asymmetric encryption and decryption operations that are based on an 

approved hash function, an approved random bit generator, a mask generation function, and the 

RSAEP and RSADP primitives. These operations are used by the KTS-OAEP key transport 

schemes (see Section 9.2). 

In the RSA-OAEP encryption operation, a data block is constructed from the keying material to 

be transported and the hash of additional input (see Section 9.1) that is shared by the sender and 

the intended receiving party. A random byte string is generated, after which both the random 

byte string and the data block are masked in a way that binds their values. The masked values are 

used to form the plaintext that is input to the RSAEP primitive, along with the public keyestablishment key of the intended receiving party. The resulting RSAEP output further binds the 

random byte string, the keying material and the hash of the additional data in the ciphertext that 

is sent to the receiving party. 

In the RSA-OAEP decryption operation, the ciphertext and the receiving party’s private keyestablishment key are input to the RSADP primitive, recovering the masked values as output. 

The mask generating function is then used to reconstruct and remove the masks that obscure the 

random byte string and the data block. After removing the masks, the receiving party can 

examine the format of the recovered data, and can compare its own computation of the hash of 

the additional data to the hash value contained in the unmasked data block, thus obtaining some 

measure of assurance of the integrity of the recovered data – including the transported keying 

material. 

RSA-OAEP can process up to nLen – 2(hLen) – 2 bytes of keying material, where nLen is the 

length of the recipient’s RSA modulus, and hLen is the length (in bytes) of the values output by 

the underlying hash function. 

7.2.2.1 RSA-OAEP Components 

RSA-OAEP uses the following components: 

1. H: An approved hash function (see Section 5.1). hLen is used to denote the 

length (in bytes) of the hash function output. 

2. MGF: The mask generation function (see Section 5.8). 
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3. RBG: An approved random bit generator (see Section 5.3). 

4. RSAEP: RSA Encryption Primitive (see Section 7.1.1). 

5. RSADP: RSA Decryption Primitive (see Section 7.1.2). 

The security strength for the RBG shall be at least the target security strength for the scheme. 

7.2.2.2 RSA-OAEP Encryption Operation 

The RSA-OAEP.Encrypt operation produces a ciphertext from keying material and additional 

input using an RSA public key as shown in Figure 4. See Section 9.1 for more information on the 

additional input. Let hLen be the length of the hash function output in bytes. 

Function call: RSA-OAEP.ENCRYPT((n, e), K, A) 

Input: 

1. (n, e): the receiver’s RSA public key. 

2. K: the keying material; a byte string of length at most nLen – 2hLen – 2. 

3. A: additional input; a byte string (may be the empty string) to be cryptographically bound 

to the keying material (see Section 9.1). 

Output: 

C: the ciphertext; a byte string of length nLen bytes. 

Errors: An indication that the keying material is too long. 

Assumptions: The RSA public key is valid. 

Process:

1. nLen = the length of n in bytes. 

2. Length checking: 

a. KLen = the length of K in bytes. 

b. If KLen > nLen – 2hLen – 2, then output an indication that the keying material 

is too long and stop. 

3. OAEP encoding: 

a. Apply the selected hash function to compute: 

HA = H(A). 
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HA is a byte string of length hLen. If A is an empty string, then HA is the 

hash value for the empty string. 

b. Construct a byte string PS consisting of nLen – KLen – 2hLen – 2 zero 

bytes. The length of PS may be zero. 

c. Concatenate HA, PS, a single byte with hexadecimal value of 01, and the 

keying material K to form data DB of length nLen – hLen – 1 bytes as 

follows: 

DB = HA || PS || 01 || K, 

where 01 represents a byte. 

d. Using the RBG (see Section 5.3), generate a random byte string mgfSeed of 

length hLen bytes. 

e. Apply the mask generation function in Section 5.8 to compute: 

dbMask = MGF(mgfSeed, nLen – hLen – 1). 

f. Let maskedDB = DB ⊕ dbMask. 

g. Apply the mask generation function in Section 5.8 to compute: 

mgfSeedMask = MGF(maskedDB, hLen). 

h. Let maskedMGFSeed = mgfSeed ⊕ mgfSeedMask. 

i. Concatenate a single byte with hexadecimal value 00, maskedMGFSeed, and 

maskedDB to form an encoded message EM of length nLen bytes as follows: 

EM = 00 || maskedMGFSeed || maskedDB 

 where 00 represents a byte.

4. RSA encryption: 

a. Convert the encoded message EM to an integer em (see Appendix B.2): 

em = BS2I(EM). 

b. Apply the RSAEP encryption primitive (see Section 7.1.1) to the integer em

using the public key (n, e) to produce a ciphertext integer c: 

c = RSAEP((n, e), em). 
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c. Convert the ciphertext integer c to a ciphertext byte string C of length nLen

bytes (see Appendix B.1): 

C = I2BS(c, nLen). 

5. Output the ciphertext C. 

Figure 4: RSA-OAEP Encryption Operation 

7.2.2.3 RSA-OAEP Decryption Operation 

RSA-OAEP.DECRYPT recovers keying material from a ciphertext and additional input using an 

RSA private key as shown in Figure 5. Let hlen be the length of the hash function output in 

bytes. 

Function call: RSA-OAEP.DECRYPT((n, d), C, A) 

Input: 

1. (n, d): the receiver’s RSA private key. 

2. C: the ciphertext; a byte string. 

RSAEP

MGF

mgfSeed

MGF

DB = lHash PS 01 K

EM = 00 maskedMGFSeed maskedDB

C

BS2I

I2BS
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3. A: additional input; a byte string (may be the empty string) whose cryptographic binding 

to the keying material is to be verified (see Section 9.1). 

Output: 

K: the recovered keying material, a byte string of length at most = nLen – 2hLen-2 bytes; or, 

Errors: Indications of the following: 

1. Erroneous input. 

2. Decryption error. 

Assumptions: The RSA private key is valid. 

Process: 

1. Initializations: 

a. nLen = the length of n in bytes. 

b. DecryptErrorFlag = False. 

2. Check for erroneous input: 

a. If nLen < 2hLen + 2, or if the length of the ciphertext C is not nLen bytes, 

output an indication of erroneous input and stop. 

b. Convert the ciphertext byte string C to a ciphertext integer c 

(see Section B.2): 

c = BS2I(C) 

c. If the ciphertext integer c is not such that 1 < c < n – 1, output an indication of 

erroneous input and stop. 

3. RSA decryption: 

a. Apply the RSADP decryption primitive (see Section 7.1.2) to the ciphertext 

integer c using the private key (n, d) to produce an integer em: 

 em = RSADP((n, d), c). 

b. Convert the integer em to an encoded message EM, a byte string of length 

nLen bytes (see Appendix B.1): 

EM = I2BS(em, nLen). 

4. OAEP decoding: 
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a. Apply the selected hash function (see Section 5.1) to compute: 

HA = Hash(A). 

HA is a byte string of length hLen bytes. 

b. Separate the encoded message EM into a single byte Y, a byte string 

maskedMGFSeed′ of length hLen bytes, and a byte string maskedDB′ of 

length nLen – hLen – 1 bytes as follows: 

EM = Y || maskedMGFSeed′ || maskedDB′. 

c. Apply the mask generation function specified in Section 5.8 to compute: 

mgfSeedMask′ = MGF(maskedDB′, hLen). 

d. Let mgfSeed′ = maskedMGFSeed′ ⊕ mgfSeedMask′. 

e. Apply the mask generation function specified in Section 5.8 to compute: 

dbMask′= MGF(mgfSeed′, nLen – hLen – 1). 

f. Let DB′ = maskedDB′ ⊕ dbMask′. 

g. Separate DB′ into a byte string HA′ of length hLen bytes and a byte string X

of length nLen – 2hLen – 1 bytes as follows: 

DB′ = HA′ || X. 

5. Check for RSA-OAEP decryption errors: 

a. If Y is not a 00 byte, then DecryptErrorFlag = True. 

b. If HA′ does not equal HA, then DecryptErrorFlag = True. 

c. If X does not have the form PS || 01 || K, where PS consists of zero or more 

consecutive 00 bytes, then DecryptErrorFlag = True. 

The type(s) of any error(s) found shall not be reported. 

(See the notes below for more information.) 

6. Output of the decryption process: 

a. If DecryptErrorFlag = True then output an indication of an (unspecified) 

decryption error and stop. (See the notes below for more information.) 

b. Otherwise, output K, the portion of the byte string X that follows the leading 

01 byte. 
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Notes: 

1. Care must be taken to ensure that the different error conditions that may be detected in 

Step 5 above cannot be distinguished from one another by an opponent, whether by error 

message or by process timing. Otherwise, an opponent may be able to obtain useful 

information about the decryption of a chosen ciphertext C, leading to the attack observed 

by Manger [15]. A single error message must be employed and output the same way for 

each type of decryption error. There should be no difference in the observable behavior 

for the different RSA-OAEP decryption errors. 

2. In addition, care needs to be taken to ensure that even if there are no errors, an 

implementation does not reveal partial information about the encoded message EM. For 

instance, the observable behavior of the mask generation function must not reveal even 

partial information about the MGF seed employed in the process (since that could 

compromise portions of the maskedDB′ segment of EM). An opponent who can reliably 

obtain particular bits of EM for sufficiently many chosen ciphertexts may be able to 

obtain the full decryption of an arbitrary ciphertext by applying the bit-security results of 

Håstad and Näslund [17]. 

RSADP

C

MGF

MGF

DB = lHash PS 01 K

EM = 00 maskedMGFSeed maskedDB

BS2I

I2BS

Figure 5: RSA-OAEP Decryption Operation 
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7.2.3 RSA-based Key-Encapsulation Mechanism with a Key-Wrapping Scheme 

(RSA-KEM-KWS) 

RSA-KEM-KWS is used by the KTS-KEM-KWS key-transport schemes (see Section 9.3). RSAKEM-KWS operations include a key-encapsulation method based on the RSASVE secret-value 

encapsulation operations and an approved key-derivation function (which depend, in turn, upon 

an approved random bit generator, the RSAEP and RSADP primitives, and an approved hash 

function). These operations are used to communicate a symmetric key-wrapping key to the 

intended receiving party. RSA-KEM-KWS operations also include an approved symmetric keywrapping algorithm, which is used to convey the actual keying material to the intended receiving 

party. 

RSA-KEM-KWS can process keying material of any length supported by the key-wrapping 

algorithm. 

7.2.3.1 RSA-KEM-KWS Components 

RSA-KEM-KWS uses the following components: 

1. KDF: A key derivation function (see Section 5.9). 

2. KWA: A symmetric key-wrapping algorithm, consisting of a wrapping operation 

KWA.WRAP and an unwrapping operation KWA.UNWRAP (see Section 5.7). 

3. RSASVE: A Secret value encapsulation operation that generates and encrypts a shared 

secret value to produce ciphertext (using the RSASVE.GENERATE operation 

in Section 7.2.1.2) or recovers the shared secret value from the ciphertext 

(using the RSASVE.RECOVER operation in Section 7.2.1.3). 

4. RBG: A random bit generator (see Section 5.3). The security strength for the RBG 

shall be at least the target security strength for the key establishment scheme. 

7.2.3.2 RSA-KEM-KWS Encryption Operation 

RSA-KEM-KWS.ENCRYPT is illustrated in Figure 6. The public key-establishment key of the 

intended receiving party is input to RSASVE.GENERATE, obtaining a secret value Z and 

corresponding ciphertext byte string C0. This secret value, along with any required OtherInfo

shared by the sender and the intended receiving party (see Section 5.9), is used as input to the 

KDF to obtain a key-wrapping key. This key-wrapping key, along with (optional) additional 

input A (see Section 9.1) that is known to both the sender and the intended receiving party, is 

used by the key-wrapping algorithm to encrypt the keying material, producing a ciphertext byte 

string C1. The output of the RSA-KEM-KWS encryption operation is the concatenation of C0

and C1. 

Function call: RSA-KEM-KWS.ENCRYPT((n, e), kwkBits, K A) 

Input: 

1. (n, e): the receiver’s RSA public key. 
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2. kwkBits: the length of the key-wrapping key in bits. 

3. K: the keying material to be wrapped, a byte string. 

4. A: the additional input (see Section 9.1), a byte string (may be the empty string). 

Output: 

C: the ciphertext; a byte string.

Errors: An indication that the keying material length is not supported. 

Assumptions: The RSA public key is valid and the value of KLen is known. 

Process: 

1. nLen = the length of n in bytes. 

2. Length checking: 

a. KLen = the length of K in bytes. 

b. If KLen is not among the lengths supported by the key-wrapping algorithm, 

output an indication that the keying material length is not supported and stop. 

3. Secret value generation and encapsulation: 

Use the RSASVE GENERATE operation specified in Section 7.2.1.2 to generate a secret 

value byte string Z and a corresponding ciphertext byte string C0 using the responder’s 

public key, where both Z and C0 are nLen bytes in length. 

(Z, C0) = RSASVE GENERATE((n, e)). 

4. Key derivation: 

Derive a key-wrapping key KWK of length kwkBits bits from the byte string Z 

KWK = KDF(Z, kwkBits, OtherInfo), 

where the OtherInfo is known by both parties (see Section 5.9). 

5. Key-wrapping: 

Wrap the keying material K (see Section 5.7) using the key-wrapping key KWK, 

associating it with the additional input, A to produce a KWA-ciphertext byte string C1: 

C1 = KWA.WRAP(KWK, K, A) . 
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6. Concatenation: 

Concatenate the RSA-ciphertext byte string C0 and the KWA-ciphertext byte string C1 to 

form a ciphertext byte string C: 

C = C0 || C1. 

7. Output the ciphertext byte string C. 

K A

Z

Z keydataBits OtherInfo

RSASVE

GENERATE

KDF

KWA

Wrap

C = C0 C1

KWK

C0

Figure 6: RSA-KEM-KWS Encryption Operation 

7.2.3.3 RSA-KEM-KWS Decryption Operation 

RSA-KEM-KWS.DECRYPT is illustrated in Figure 7. The private key-establishment key of the 

intended receiving party and C0 are input to RSASVE.RECOVER, which returns the secret value 

Z. This secret value (along with any required OtherInfo) is used as input to the KDF to recover 

the key-wrapping key. The key-wrapping key (together with the additional data A – if that option 

was exercised) is then used to decrypt C1 and recover the transported keying material. 

Function call: RSA-KEM-KWS.DECRYPT((n, d), C, kwkBits A) 

Input: 

1. (n, d): the recipient’s RSA private key..
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2. C: the ciphertext; a byte string. 

3. kwkBits: the length of the key-wrapping key in bits. 

4. A: additional input; a byte string (may be the empty string). 

Output: 

K: the recovered keying material that was wrapped; a byte string. 

Errors: An indication of a decryption error. 

Assumptions: The RSA private key is valid, and the value of KBits is known. 

Process:

1. nLen = the length of n in bytes. 

2. Length checking: 

a. cLen = the length of the ciphertext string C in bytes. 

b. If cLen ≤ nLen, or if cLen − nLen is not among the lengths supported by the 

symmetric key-wrapping algorithm, output an indication of a decryption error 

and stop. 

3. Separation: 

Separate the ciphertext byte string C into an RSA-ciphertext byte string C0 of length nLen

bytes and a KWA-ciphertext byte string C1 of length cLen − nLen bytes: 

C = C0 || C1. 

4. Recover Shared Secret: 

 Recover the shared secret byte string Z from the ciphertext byte string C0 using the 

RSASVE.RECOVER operation specified in Section 7.2.1.3. 

Z = RSASVE.RECOVER((n, d), C0) 

If an indication of a decryption error is returned, output an indication of a decryption 

error and stop.

5. Key derivation: 

Derive a key-wrapping key KWK of length kwkBits bits from the byte string Z 

KWK = KDF(Z, KBits,OtherInfo), 
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where the OtherInfo is known by both parties (see Section 5.9). 

6. Key unwrapping: 

Unwrap the KWA-ciphertext byte string C1 using the key-wrapping key KWK to recover 

the keying material K (see Section 5.7), verify the correctness of A: 

K = KWA.UNWRAP(KWK, C1 A). 

If the unwrapping operation outputs an error indicator, output an indication of a 

decryption error and stop. 

7. Output the keying material K. 

Notes: 

1. Care needs be taken to ensure that the different error conditions in Steps 2.2, 4, and 6 

cannot be distinguished from one another by an opponent, whether by error message 

or timing. Otherwise, an opponent may be able to obtain useful information about the 

decryption of a chosen ciphertext C, leading to the attack observed by Manger [15]. A 

single error message shall be employed and output the same way for each error type. 

There should be no difference in timing or other behavior for the different errors. In 

addition, care needs to be taken to ensure that even if there are no errors, an 

implementation does not reveal partial information about the shared secret Z. An 

opponent who can reliably obtain particular bits of Z for sufficiently many chosen 

ciphertexts may be able to obtain the full decryption of an arbitrary ciphertext by 

applying the bit-security results mentioned in Annex B5.2.2 (last paragraph) of ANS 

X9.44. 

2. In addition, care must be taken to ensure that an implementation does not reveal 

information about the encapsulated secret value Z. For instance, the observable 

behavior of the KDF must not reveal even partial information about the Z value 

employed in the key derivation process. An opponent who can reliably obtain 

particular bits of Z for sufficiently many chosen ciphertexts may be able to obtain the 

full decryption of an arbitrary ciphertext by applying the bit-security results of Håstad 

and Näslund [17]. 
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K

A

Z

RSASVE

RECOVER

KWA

Unwrap

Z keydataBits OtherInfo

KDF

C = C0 C1

KWK

C0

Figure 7: RSA-KEM-KWS Decryption Operation 

8 Key Agreement Schemes 

In a key agreement scheme, two parties, the initiator and the responder, establish keying 

material over which neither has direct control of the result, but both have influence. This 

Recommendation provides two families of key agreement schemes, KAS1 and KAS2. These 

schemes are based on secret value encapsulation (see Section 7.2.1). 

Key confirmation is included in some of these schemes to provide assurance that the participants 

share the same keying material; see Section 6.6 for the details of key confirmation. When 

possible, 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. 

The scheme initiator, party U, shall have an identifier IDU, and the scheme responder, party V, 

shall have an identifier IDV. The identifiers shall be non-null and selected in accordance with the 

protocol utilizing the scheme. When a party’s public key is employed in a scheme, that party’s 

identifier shall be bound to its public key. 

A general flow diagram is provided for each key agreement scheme. The dotted-line arrows 

represent the distribution of 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 nonces or cryptographically protected values that occur during the key agreement 

scheme. Note that the flow diagrams in this Recommendation omit explicit mention of various 
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validation checks that are required. The flow diagrams and descriptions in this Recommendation 

assume a successful completion of the key agreement process. 

8.1 Common Components for Key Agreement 

The key agreement schemes in this Recommendation have the following common components: 

1. RSASVE: RSA secret value encapsulation, consisting of a generation operation 

RSASVE.GENERATE and a recovery operation RSASVE.RECOVER (see 

Section 7.2.1). 

2. KDF: A key derivation function (see Section 5.9). 

8.2 The KAS1 Family 

In this family of key agreement schemes, only the responder’s key-establishment key pair is 

used. Note that both parties may actually have key-establishment key pairs, but only the 

responder’s key pair is used in the scheme. 

The schemes in this family have the following general form: 

1. Party U (the initiator) generates a secret value (which will become a shared secret) and a 

corresponding ciphertext using the RSASVE.GENERATE operation and party V’s (the 

responder’s) public key-establishment key, and sends the ciphertext to party V. 

2. Party V recovers the shared secret from the ciphertext using the RSASVE.RECOVER

operation and its private key-establishment key. Party V generates a nonce and sends it to 

party U. 

3. Both parties then derive keying material from the shared secret and “other information”, 

including party V’s nonce, using a key derivation function. The length of the keying 

material that can be agreed on is limited only by the length that can be output by the key 

derivation function. 

4. If key confirmation is incorporated, then the derived keying material is parsed into two 

parts, MacKey and KeyData. The MacKey and MacData are used to compute a MacTag 

of length MacTagLen (see Section 6.6.1). The MacTag is sent from the provider to the 

recipient. If the MacTag computed by the provider matches the MacTag computed by the 

recipient, then the successful establishment of keying material is confirmed to the 

recipient. 

The following schemes are defined: 

1. KAS1-basic, the basic scheme without key confirmation (see Section 8.2.2). 

2. KAS1-responder-confirmation, a variant of KAS1-basic with unilateral key 

confirmation from party V to party U (see Section 8.2.3). 

For the security properties of this family, see Section 8.2.4. 
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8.2.1 KAS1 Family Prerequisites 

1. The responder shall have been designated as the owner of a key-establishment key pair 

that was generated as specified in Section 6.3. The responder shall have assurance of its 

possession of the correct value for its private key as specified in Section 6.5.1. 

2. The initiator and responder shall have agreed upon an approved key derivation function 

(see Section 5.9), an approved hash function appropriate for use with the key derivation 

function and associated parameters (see Sections 5.1 and 6.2.3), the value of KBits, and 

the contents of the OtherInfo field used during key derivation. 

3. When key confirmation is used, the initiator and responder shall have agreed upon an 

approved MAC algorithm and associated parameters (see Sections 5.2 and 6.2.3). 

4. Prior to or during the key agreement process, each party shall obtain the identifier that is 

to be associated with the other party during the key agreement transaction. The initiator 

shall obtain the public key-establishment key that is bound to the responder’s identifier in 

a trusted manner (e.g., from a certificate signed by a trusted CA). The initiator shall also 

obtain the assurance of validity of this public key as specified in Section 6.4.2. 

5. The following is a prerequisite for using any keying material derived during a KAS1 key 

agreement scheme for purposes beyond those of the scheme itself. 

The initiator of a particular KAS1 key agreement transaction shall obtain assurance 

that the intended responder is (or was) in possession of the private key-establishment 

key corresponding to the public key-establishment key used by the initiator during 

that transaction, as specified in Section 6.5.2. 

This requirement recognizes the possibility that assurance of private-key possession may 

be provided/obtained by means of key confirmation performed as part of a particular 

KAS1 transaction. 

8.2.2 KAS1-basic 

KAS1-basic is the basic key agreement scheme in the KAS1 family. In this scheme, the 

responder does not contribute to the formation of the shared secret; instead, a nonce is used as a 

responder-selected contribution to the KDF, ensuring that both parties influence the derived 

keying material. 

Let (PubKeyV, PrivKeyV) be party V’s key-establishment key pair. Let KBits be the intended 

length in bits of the keying material to be established. The parties shall perform the following or 

an equivalent sequence of steps, as illustrated in Figure 8. 

Party U shall execute the following key agreement steps in order to a) establish a shared secret Z 

with party V, and b) derive shared secret keying material from Z. 

Actions: Party U shall derive secret keying material as follows: 

1. Use the RSASVE.GENERATE operation in Section 7.2.1.2 to generated a secret value Z

and a corresponding ciphertext C using party V’s public key-establishment key PubKeyV. 

Note that the secret value Z will become a shared secret when recovered by Party V. 
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2. Send the ciphertext C to party V. 

3. Obtain party V’s nonce NV from party V. If NV is not available, output an error indicator 

and stop. 

4. Construct the other information OtherInfo for key derivation (see Section 5.9) using (at a 

minimum) the identifiers IDU and IDV, and the nonce NV. 

5. Use the agreed-upon key derivation function (see Section 5.9) to derive secret keying 

material DerivedKeyingMaterial of length KBits from the shared secret Z and OtherInfo. 

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 the DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length KBits or an error indicator. 

Party V shall execute the following key agreement steps in order to a) establish a shared secret Z 

with party U, and b) derive shared secret keying material from Z. 

Actions: Party V shall derive secret keying material as follows: 

1. Receive a ciphertext C from party U. 

2. Use the RSASVE.RECOVER operation in Section 7.2.1.3 to recover the shared secret Z

from the ciphertext C using the private key-establishment key PrivKeyV. If the call to 

RSASVE.RECOVER outputs an error indicator, zeroize the result of all intermediate 

calculations used in the attempted recovery of Z, output an error indicator and stop (see 

Note). 

3. Obtain a nonce NV (see Section 5.6) and send NV to party U. 

4. Construct the other information OtherInfo for key derivation (see Section 5.9) using (at a 

minimum) the identifiers IDV and IDU, and the nonce NV. 

5. Use the agreed-upon key derivation function (see Section 5.9) to derive secret keying 

material DerivedKeyingMaterial of length KBits from the shared secret Z and OtherInfo. 

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 the DerivedKeyingMaterial. 

Output: The bit string DerivedKeyingMaterial of length KBits or error indicator. 
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Initiator (Party U) Responder (Party V) 

 (PubKeyV, PrivKeyV) 

Obtain responder’s public keyestablishment key 

PubKeyV

(Z, C) = 

RSASVE.GENERATE(PubKeyV) 

C 

Z = RSASVE.RECOVER(PrivKeyV, C)

Derived keying material = KDF(Z, 

KBits, OtherInfo) 

NV

Derived keying material = KDF(Z, 

KBits, OtherInfo)

Figure 8: KAS1-basic Scheme 

The messages may be sent in a different order. Even though party U remains the designated 

initiator, NVmay be sent before C. 

It is extremely important that an implementation not reveal any sensitive information. It is also 

important to conceal partial information about the shared secret Z. 

8.2.3 KAS1 Key Confirmation 

The KAS1-responder-confirmation scheme is based on the KAS1-basic scheme. 

8.2.3.1 KAS1 Key Confirmation Components 

The components for KAS1 key confirmation are the common components listed in Section 8.1, 

plus the following: 

3. MAC: A message authentication code algorithm with the following parameters 

(see Section 5.2), 

a. MacKeyLen: the length in bytes of MacKey (see Table 1 in Section 

6.2.3), and 

b. MacTagLen: the length in bytes of the MacTag (see Table 1 in 

Section 6.2.3). 
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For KAS1 key confirmation, the length of the keying material shall be at least MacKeyLen bytes,

where MacKeyLen is the length of the MacKey (see Section 6.2.3), and usually longer so that 

other keying material is available for subsequent operations. The MacKey shall be the first 

MacKeyLen bytes of the keying material and shall be used only for the key confirmation 

operation. 

8.2.3.2 KAS1-responder-confirmation 

Figure 9 depicts a typical flow for a KAS1 scheme with unilateral key confirmation from party V 

to party U. In this scheme, party V, the scheme responder, and party U, the scheme initiator, 

assume the roles of key confirmation provider and recipient, respectively. 

To provide (and receive) key confirmation (as described in Section 6.6.1), both parties set 

EphemDataV = NV, and EphemDataU = C: 

Party V provides MacTagV to party U (as specified in Section 6.6.1, with P = V and R = U), 

where MacTagV is computed (as specified in Section 5.2.1) using 

MacDataV = “KC_1_V” || IDV || IDU || NV || C{ || Text}. 

The recipient (party U) uses the identical format and values to compute MacTagV and then 

verifies that the newly computed MacTagV matches the MacTagV value provided by party V. 

The MacKey used during Key Confirmation shall be zeroized by Party V immediately after the 

computation of MagTagV, and by Party U immediately after the verification of the received 

MacTagV or a (final) determination that the received MacTagV is in error. 
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Initiator (Party U) Responder (Party V) 

(PubKeyV, PrivKeyV) 

Obtain responder’s public keyestablishment key 

PubKey 

(Z, C) = 

RSASVE.GENERATE(PubKeyV) 

C 

Z = RSASVE.RECOVER(PrivKeyV, C)

MacKey || KeyData = KDF(Z, 

KBits, OtherInfo) 

NV MacKey || KeyData = KDF(Z, KBits, 

OtherInfo) 

MacTagV =? MAC(MacKey, 

MacTagLen, MacDataV) 

MacTagV

MacTagV = MAC(MacKey, 

MacTagLen, MacDataV) 

Figure 9: KAS1-responder-confirmation Scheme (from Party V to Party U) 

Certain messages may be combined or sent in a different order. E.g., NV and MacTagV may be 

sent together. Alternatively, NV may be sent before C even though party U remains the 

designated initiator. 

8.2.4 KAS1 Security Properties 

In each scheme included in this family, only the identifier of V (the responder) is required to be 

bound to a public key-establishment key. U (the initiator) has assurance that no unintended party 

can recover Z from C (without the compromise of private information). 

The responder, however, has no such assurance. In particular, V has no assurance as to the 

accuracy of the identifier claimed by the initiator and, therefore, has no assurance as to the true 

source of the ciphertext C. 

Due to the initiator’s unilateral selection of the random Z value, U has assurance that fresh 

keying material will be derived in each instantiation of these schemes. V has similar assurance 

owing to its contribution of the nonce NV to the KDF input. 

A compromise of the responder’s private key will allow an adversary to masquerade as the 

responder in future key establishment transactions, and compromise all shared secrets (and 
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derived keying material) resulting from past and future KAS1 transactions having that party as 

the responder – assuming that a malicious party has access to the publicly exchanged data 

(including C and NV). Other applications that rely on the private key will be affected. 

It is important to understand that a scheme may be just one of several components of a protocol. 

The combination of these components may endow the protocol with additional security 

properties beyond those provided by any particular component. Note that protocols, per se, are 

not specified in this Recommendation. 

Through the inclusion of MacTagV in the KAS1-responder-confirmation scheme, and by 

successfully comparing the received value of MacTagV with its own computation, the initiator 

(U) obtains assurance that 

1. the responder (V) has correctly recovered Z from C; 

2. that both parties agree on the values of IDV, IDU, NV, and C; 

3. that (at least the MacKey portion of) the derived keying material has been correctly 

computed by V; 

4. V is in possession of the correct private key that corresponds to the public key used in the 

transaction, and 

5. V has actively participated in the transaction. 

Consequently, responder authentication is implicitly provided by the binding of party V’s 

identifier to the public key-establishment key (see Section 6.7). 

8.3 KAS2 Family 

In this family of key agreement schemes, both the initiator’s and responder’s key-establishment 

key pairs are used. 

The schemes in this family have the following general form: 

1. Party U (the initiator) generates a secret value (which will become a part of the shared 

secret) and a corresponding ciphertext using the RSASVE.GENERATE operation and 

party V’s (the responder’s) public key-establishment key, and sends the ciphertext to 

party V. 

2. Party V recovers the shared secret value from the ciphertext received from party U using 

the RSASVE.RECOVER operation and its private key-establishment key. 

3. Party V generates a secret value (which will become a second part of the shared secret) 

and the corresponding ciphertext using RSASVE GENERATE operation and party U’s (the 

initiator’s) public key-establishment key, and sends the ciphertext to party U. 

4. Party U recovers the shared secret value from the ciphertext received from party V using 

the RSASVE.RECOVER operation and its private key-establishment key. 
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5. Both parties concatenate the shared secret values to form the shared secret, and then 

derive keying material from the shared secret and “other information” using a key 

derivation function. The length of the keying material that can be agreed on is limited 

only by the length that can be output by the key derivation function. 

6. Party U and/or party V may additionally provide key confirmation. If key confirmation is 

incorporated, then the derived keying material is parsed into two parts, MacKey and 

KeyData. The MacKey is then used to compute a MacTag of MacTagLen bytes on 

MacData (see Section 6.6.1). The MacTag is sent from the provider to the recipient. If 

the MacTag computed by the provider matches the MacTag computed by the recipient, 

then the successful establishment of keying material is confirmed to the recipient. 

The following schemes are defined: 

1. KAS2-basic, the basic scheme without key confirmation (see Section 8.3.2).

2. KAS2-responder-confirmation, a variant of KAS2-basic with unilateral key 

confirmation from party V to party U (see Section 8.3.3.2).

3. KAS2-initiator-confirmation, a variant of KAS2-basic with unilateral key confirmation 

from party U to party V (see Section 8.3.3.3).

4. KAS2-bilateral-confirmation, a variant of KAS2-basic with bilateral key confirmation 

between party U and party V (see Section 8.3.3.4). 

8.3.1 KAS2 Family Prerequisites 

1. Each party (initiator and responder) shall have been designated as the owner of a keyestablishment key pair that was generated as specified in Section 6.3. Prior to or during 

the key agreement process, each party shall obtain assurance of its possession of the 

correct value for its own private key as specified in Section 6.5.1. 

2. The initiator and responder shall have agreed upon an approved key derivation function 

(see Section 5.9), an approved hash function appropriate for use with the key derivation 

function and associated parameters (see Sections 5.1 and 6.2.3), the value of KBits, and 

the contents of the OtherInfo field used during key derivation. 

3. Prior to or during the key agreement process, each party shall obtain the identifier that is 

to be associated with the other party during the key agreement transaction and shall

obtain the public key-establishment key that is bound to that identifier. These 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 validity of the public key bound to the other party’s 

identifier, as specified in Section 6.4.2. 

4. The following is a prerequisite for using any keying material derived during a KAS2 key 

agreement scheme for purposes beyond those of the scheme itself. 

The recipient of a public key-establishment key that is used by the recipient during a 

particular KAS2 key agreement transaction shall obtain assurance that its (claimed) 

owner is (or was) in possession of the corresponding private key-establishment key, as 

specified in Section 6.5.2. That is, 
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a. The initiator of a particular KAS2 key agreement transaction shall obtain 

assurance that the intended responder is (or was) in possession of the private keyestablishment key corresponding to the public key-establishment key used by the 

initiator during that transaction (as specified in Section 6.5.2); 

b. The responder in a particular KAS2 key agreement transaction shall obtain 

assurance that the apparent initiator is (or was) in possession of the private keyestablishment key corresponding to the public key-establishment key used by the 

responder during that transaction (as specified in Section 6.5.2). 

This requirement recognizes the possibility that assurance of private-key possession may 

be provided/obtained by means of key confirmation performed as part of a particular 

KAS2 transaction. 

8.3.2 KAS2-basic 

Figure 10 depicts the typical flow for the KAS2-basic scheme. The parties exchange secret 

values that are concatenated together to form the mutually determined shared secret to be input 

to the key derivation function. 

Party U shall execute the following key agreement steps in order to a) establish a mutually 

determined shared secret Z with party V, and b) derive secret keying material from Z. 

Actions: Party U shall derive secret keying material as follows: 

1. Use the RSASVE.GENERATE operation in Section 7.2.1.2 to generate a secret value ZU

and a corresponding ciphertext CU using party V’s public key-establishment key 

PubKeyV. 

2. Send the ciphertext CU to party V. 

3. Receive a ciphertext CV from party V. If CV is not available, output an error indicator and 

stop. 

4. Use the RSASVE.RECOVER operation in Section 7.2.1.3 to recover the ZV from the 

ciphertext CV using the private key-establishment key PrivKeyU. If the call to 

RSASVE.RECOVER outputs an error indicator, zeroize the results of all intermediate 

calculations used in the attempted recovery of ZV, zeroize Zu, and output an error 

indicator and stop. 

5. Construct the mutually determined shared secret Z from ZU and ZV: 

Z = ZU || ZV. 

6. Construct the other information OtherInfo for key derivation (see Section 5.9) using (at a 

minimum) the identifiers IDU and IDV. 

7. Use the agreed-upon key derivation function (see Section 5.9) to derive secret keying 

material DerivedKeyingMaterial of length KBits from the shared secret Z and OtherInfo. 
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If the key derivation function outputs an error indicator, zeroize all copies of Z, ZU, and

ZV, and output an error indicator and stop. 

8. Zeroize all copies of Z, ZU, and ZV, and output the DerivedKeyingMaterial or an error 

indicator. 

Output: The byte string DerivedKeyingMaterial of length KBits or error indicator. 

Party V shall execute the following key agreement steps in order to a) establish a mutually 

determined shared secret Z with party U, and b) derive secret keying material from Z. 

Actions: Party V shall derive secret keying material as follows: 

1. Receive a ciphertext CU from party U. 

2. Use the RSASVE.RECOVER operation in Section 7.2.1.3 to recover ZU from the 

ciphertext CU using the private key-establishment key PrivKeyV. If the call to 

RSASVE.RECOVER outputs an error indicator, zeroize the result of all intermediate 

calculations used in the attempted recovery of ZU, output an error indicator and stop. 

3. Use the RSASVE.GENERATE operation in Section 7.2.1.2 to generate a secret value ZV

and a corresponding ciphertext CV using party U’s public key-establishment key 

PubKeyU. 

4. Send the ciphertext CV to party U. 

5. Determine the mutually determined shared secret Z from ZU and ZV: 

Z = ZU || ZV. 

6. Construct the other information OtherInfo for key derivation (see Section 5.9) using (at a 

minimum) the identifiers IDV and IDU. 

7. Use the agreed-upon key derivation function (see Section 5.9) to derive secret keying 

material DerivedKeyingMaterial of length KBits from the shared secret Z and OtherInfo. 

If the key derivation function outputs an error indicator, zeroize all copies of Z, ZU, and

ZV, and output an error indicator and stop. 

8. Zeroize all copies of Z, ZU, and ZV, and output the DerivedKeyingMaterial. 

Output: The byte string DerivedKeyingMaterial of length KBits or error indicator. 
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Initiator (Party U) Responder (Party V) 

(PubKeyU, PrivKeyU) (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key 

PubKeyV

← ⎯ ⎯ ⎯ 

PubKeyU 

 ⎯ ⎯ ⎯ →

Obtain party U’s public keyestablishment key 

(ZU, CU) = 

RSASVE.GENERATE(PubKeyV) 

CU

⎯⎯⎯⎯→

ZU = 

RSASVE.RECOVER(PrivKeyV, 

CU) 

ZV = 

RSASVE.RECOVER(PrivKeyU, 

CV)

CV

←⎯⎯⎯⎯

(ZV, CV) = 

RSASVE.GENERATE(PubKeyU)

Z = ZU || ZV Z = ZU || ZV

DerivedKeyingMaterial = 

KDF(Z, KBits, OtherInfo) 

DerivedKeyingMaterial = 

KDF(Z, KBits, OtherInfo) 

Figure 10: KAS2-basic Scheme 

The messages may be sent in a different order. Even though party U remains the designated 

initiator, CV may be sent before CU. 

It is extremely important that an implementation not reveal any sensitive information. It is also 

important to conceal partial information about ZU, ZV and Z to prevent chosen-ciphertext attacks 

on the secret value encapsulation scheme. In particular, the observable behavior of the keyagreement process should not reveal partial information about the shared secret Z. 

8.3.3 KAS2 Key Confirmation 

The KAS2 key confirmation schemes are based on the KAS2-basic scheme. 

8.3.3.1 KAS2 Key Confirmation Components 

The scheme components for KAS2 key confirmation are the common components in Section 

8.1, plus the following: 

3. MAC: A message authentication code algorithm (see Section 5.2) 

a. MacKeyLen: the length in bytes of the MacKey (see Table 1 in 

Section 6.2.3). 
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b. MacTagLen: the length in bytes of the MacTag (see Table 1 in 

Section 6.2.3). 

For this scheme, the length of the keying material shall be at least MacKeyLen, where 

MacKeyLen is the length in bytes of the MacKey (see Section 6.2.3), and usually longer so that 

other keying material is available for subsequent operations. The MacKey shall be the first 

MacKeyLen bytes of the keying material and shall be used only for the key confirmation 

operation. 

8.3.3.2 KAS2-responder-confirmation 

Figure 11 depicts a typical flow for a KAS2 scheme with unilateral key confirmation from party 

V to party U. In this scheme, party V, the scheme responder, and party U, the scheme initiator, 

assume the roles of the key confirmation provider and recipient, respectively. 

To perform key confirmation (as described in Section 6.6.1), both parties set EphemDataV = 

CV, and EphemDataU = CU. 

Party V provides MacTagV to party U (as specified in Section 6.6.1, with P = V and R = U), 

where MacTagV is computed (as specified in Section 5.2.1) on 

MacDataV = “KC_1_V” || IDV || IDU || CV || CU{ || Text}. 

The recipient (party U) uses the identical format and values to compute MacTagV and then 

verifies that the newly computed MacTagV equals the MacTagV value provided by party V. 

The MacKey used during key confirmation shall be zeroized by Party V immediately after the 

computation of MagTagV, and by Party U immediately after the verification of the received 

MacTagV or a (final) determination that the received MacTagV is in error. 

Initiator (Party U) Responder (Party V) 

(PubKeyU, PrivKeyU) (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key 

PubKeyV

← ⎯ ⎯ ⎯ 

PubKeyU 

⎯ ⎯ ⎯ →

Obtain party U’s public key 

establishment-key 

(ZU, CU) = 

RSASVE.Generate(PubKeyV) 

CU

⎯⎯⎯⎯⎯→

ZU = 

RSASVE.Recover(PrivKeyV, CU)

ZV = 

RSASVE.Recover(PrivKeyU, 

CV) 

CV

←⎯⎯⎯⎯⎯

(ZV, CV) = 

RSASVE.Generate(PubKeyU) 
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Z = ZU || ZV Z = ZU || ZV

K= KDF(Z, KBits, OtherInfo) 

= MacKey || KeyData

K = KDF(Z, KBits, OtherInfo) 

= MacKey || KeyData

MacTagV =? MAC(MacKey, 

MacTagLen, MacDataV) 

MacTagV

MacTagV = MAC(MacKey, 

MacTagLen, MacDataV) 

Figure 11: KAS2-responder-confirmation Scheme (from Party V to Party U) 

Certain messages may be combined or sent in a different order. E.g., CV and MacTagV may be 

sent together. Alternatively, CV may be sent before CU, even though party U remains the 

designated initiator. 

8.3.3.3 KAS2-initiator-confirmation 

Figure 12 depicts a typical flow for a KAS2 scheme with unilateral key confirmation from party 

U to party V. In this scheme, party U, the scheme initiator, and party V, the scheme responder, 

assume the roles of key confirmation provider and recipient, respectively. 

To provide (and receive) key confirmation (as described in Section 6.6.1), both parties set 

EphemDataV = CV, and EphemDataU = CU. 

Party U provides MacTagU to party V (as specified in Section 6.6.1, with P = U and R = V), 

where MacTagU is computed (as specified in Section 5.2.1) on 

MacDataU = “KC_1_U” || IDU || IDV || CU || CV{ || Text}. 

The recipient (party V) uses the identical format and values to compute MacTagU and then 

verifies that the newly computed MacTagU matches the MacTagU value provided by party U. 

The MacKey used during key confirmation shall be zeroized by Party U immediately after the 

computation of MagTagU, and by Party V immediately after the verification of the received 

MacTagU or a (final) determination that the received MacTagU is in error. 
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Initiator (Party U) Responder (Party V) 

(PubKeyU, PrivKeyU) (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key

PubKeyV 

PubKeyU Obtain party U’s public keyestablishment key 

(ZU, CU) = 

RSASVE.GENERATE(PubKeyV)

CU 

ZU = 

RSASVE.RECOVER(PrivKeyV, CU)

ZV = RSASVE.RECOVER(PrivKeyU, 

CV)

CV (ZV, CV) = 

RSASVE.GENERATE(PubKeyU)

Z = ZU ⎜⎜ZV Z = ZU ⎜⎜ZV

MacKey || KeyData = KDF(Z, KBits, 

OtherInfo)

MacKey || KeyData = KDF(Z, KBits, 

OtherInfo)

MacTagU = MAC(MacKey, 

MacTagLen, MacDataU)

MacTagU

⎯⎯⎯⎯→

MacTagU =? MAC(MacKey, 

MacTagLen, MacDataU)

Figure 12: KAS2-initiator-confirmation Scheme (from Party U to Party V) 

Certain messages may be sent in a different order (and combined with others). Even though party 

U remains the designated initiator, CV may be sent before CU; in which case CU and MacTagU

may be sent together. 

8.3.3.4 KAS2-bilateral-confirmation 

Figure 13 depicts a typical flow for a KAS2 scheme with bilateral key confirmation. In this 

scheme, 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. 

To provide bilateral key confirmation (as described in Section 6.6.2), party U and party V 

exchange and verify MacTags that have been computed (as specified in Section 6.6.1) using 

EphemDataU = CU, and EphemDataV = CV. 
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Party V provides MacTagV to party U (as specified in Section 6.6.1, with P = V and R = U); 

MacTagV is computed by party V (and verified by party U) on 

MacDataV = “KC_2_V” || IDV || IDU || CV || CU{ || Text1}. 

Party U provides MacTagU to party V (as specified in Section 6.6.1, with P = U and R = V); 

MacTagU is computed by party U (and verified by party V) on 

MacDataU = “KC_2_U” || IDU || IDV || CU || CV{ || Text2}. 

The MacKey used during key confirmation shall be zeroized by each party immediately 

following its use to compute and verify the MacTag for key confirmation. Once Party U has 

computed MacTagU and has either verified the received MacTagV or made a (final) 

determination that the received MacTagU is in error, Party U shall immediately zeroize its copy 

of MacKey. Similarly, after Party V has computed MacTagV and has either verified the received 

MacTagU or made a (final) determination that the received MacTagU is in error; Party V shall

immediately zeroize its copy of MacKey. 

Initiator (Party U) Responder (Party V) 

(PubKeyU, PrivKeyU) (PubKeyV, PrivKeyV)

Obtain party V’s public keyestablishment key 

PubKeyV (Z, C) = 

RSASVE.GENERATE(PubKeyV)

PubKeyU Obtain party U’s public keyestablishment key 

(ZU, CU) = 

RSASVE.GENERATE(PubKeyV)

CU ZU = 

RSASVE.RECOVER(PrivKeyV, 

CU) 

ZV = 

RSASVE.RECOVER(PrivKeyU, 

CV) 

CV (ZV, CV) = 

RSASVE.GENERATE(PubKeyV)

Z = ZU ⎜⎜ZV Z = ZU ⎜⎜ZV

MacKey || KeyData = KDF(Z,

KBits, OtherInfo)

MacKey || KeyData= KDF(Z, 

KBits, OtherInfo)

MacTagV =? MAC(MacKey, 

MacTagLen, MacDataV)

MacTagV MacTagV = MAC(MacKey, 

MacTagLen, MacDataV)
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MacTagU = MAC(MacKey, 

MacDataU)

MacTagU MacTagU =? MAC(MacKey, 

MacDataU)

Figure 13: KAS2-bilateral-confirmation Scheme 

Certain messages may be sent in a different order (and/or combined with others). Even though 

party U remains the designated initiator, CV may be sent before CU and/or MacTagV may be sent 

before MacTagU. If CU is sent immediately before MacTagU, then CU and MacTagU may be sent 

together. If CV is sent immediately before MacTagV, then CV and MacTagV may be sent together. 

8.3.4 KAS2 Security Properties 

In the schemes included in this family, each party has an identifier that is bound to a public keyestablishment key. Therefore, U (the initiator) has assurance that no unintended party can 

recover ZU from CU, and V (the responder) has assurance that no unintended party can recover ZV

from CV (without the compromise of private information). Consequently, U and V both have 

assurance that they are the only two parties capable of deriving the keying material 

corresponding to ZU || ZV. 

By virtue of their random contributions (ZU by U and ZV by V) to the KDF input, each party also 

has assurance that fresh keying material will be derived in each instantiation of these schemes. 

The compromise of one party’s private key will allow an adversary to masquerade as that party 

in future key establishment transactions. However, the compromise of the private key of a single 

participant will not, by itself, permit the compromise of keying material derived in KAS2 

transactions (between honest parties). 

Through the inclusion of MacTagV in the KAS2-responder-confirmation and KAS2-bilateralconfirmation schemes, and by successfully comparing the received value of MacTagV with its 

own computation, the initiator (U) obtains assurance that 

1. the responder (V) has correctly recovered ZU from CU; 

2. that both parties agree on the values of IDV, IDU, CV, and CU; 

3. that (at least the MacKey portion of) the derived keying material has been correctly 

computed by V; 

4. V is in possession of the correct private key that corresponds to the public key used by 

Party U in the transaction; 

5. V has actively participated in the process; and 

6. U has correctly recovered ZV from CV and therefore possesses the correct value for its 

private key. 

Through the inclusion of MacTagU in the KAS2-initiator-confirmation and KAS2-bilateralconfirmation schemes, and by successfully comparing the received value of MacTagV with its 

own computation, the responder (V) obtains assurance that 

1. the initiator (U) has correctly recovered ZV from CV; 
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2. that both parties agree on the values of IDV, IDU, CV, and CU; 

3. that (at least the MacKey portion of) the derived keying material has been correctly 

computed by U; 

4. U is in possession of the correct private key that corresponds to the public key used by 

Party V in the transaction; 

5. U has actively participated in the process; and 

6. V has correctly recovered ZU from CU and, therefore, possesses the correct value for its 

private key. 

9 IFC based Key Transport Schemes 

In a key transport scheme, two parties, the sender and the receiver, establish keying material 

selected initially by the sender. The keying material may be cryptographically bound to 

additional input (see Section 9.1). 

Two families of key transport schemes are specified: KTS-OAEP and KTS-KEM-KWS. 

Key confirmation is included in some of these schemes to provide assurance to the sender that 

the participants share the same keying material (see Section 6.6 for further details on key 

confirmation.). 

The keying material to be transported is determined by the sender in a key transport scheme and 

has the general form: 

TransportedKeyingMaterial = MacKey || KeyData. 

In key transport schemes that provide key confirmation (see Sections 9.2.4.2 and 9.3.4.2), the 

transported keying material shall contain a MacKey as the first bits of the keying material; the 

MacKey will be used for the computation and verification of the MacTag. KeyData is the keying 

material that follows the MacKey. The MacKey shall be generated anew for each instance of a 

key establishment transaction using an approved random bit generator at the security strength 

required for the key establishment transaction. The MacKey length shall be equal to or greater 

than the security strength associated with the modulus used in the key establishment scheme (see 

SP 800-57-Part 1 [7]). The KeyData may be null, or may contain keying material to be used 

subsequent to the key transport transaction. The MacKey shall be used during Key Confirmation 

and then immediately zeroized. 

In key transport schemes that do not provide key confirmation (see Sections 9.2.4.1 and 9.3.4.1), 

the TransportedKeyingMaterial = KeyData. The KeyData contains keying material to be used 

subsequent to the key transport transaction. 

A general flow diagram is provided for each key transport scheme. The dotted-line arrows 

represent the distribution of 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 cryptographically protected values that occur during the key transport or key 

confirmation process. Note that the flow diagrams in this Recommendation omit explicit mention 
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of various validation checks that are required. The flow diagrams and descriptions in this 

Recommendation assume a successful completion of the key transport process. 

9.1 Additional Input 

Additional input, A, is supported by the key transport schemes specified in Section 9.2 and 9.3. 

1. It may include a representation of shared information either exchanged by the parties or 

obtained from higher-level protocols, such as: 

a. the names or other identifying information (e.g., e-mail address, etc.) of the 

sender and receiver; 

b. nonces or other fresh data contributed by the parties; 

c. the type, length, or intended use of the keying material (and/or of individual keys 

within the keying material, if the keying material consists of more than one key); 

d. a counter value; 

e. secret data shared by the parties; 

f. a hash value, and/or 

g. other public data shared by the parties. 

2. A may consist of an empty string. 

3. Each party to the key establishment shall know the contents of A before it is required by 

the scheme. 

One purpose of the additional input could be to enable the sender to indicate that it intends to 

employ the keying material in a specified context and to bind the keying material to this context. 

The method for formatting and distributing the additional input is application-defined. 

9.2 KTS-OAEP Family: Key Transport Using RSA-OAEP 

The KTS-OAEP family of key transport schemes is based on RSA-OAEP encrypt and decrypt 

operations (see Section 7.2.2), which are, in turn, based on the asymmetric encryption and 

decryption primitives, RSAEP and RSADP (see Section 7.1). In this family, only the receiver’s 

key pair is used. 

The key transport schemes of this family have the following general form: 

1. Party U (the sender) encrypts the keying material to be transported using the RSA-OAEP 

ENCRYPT operation and party V’s (the receiver’s) public key-establishment key to 

produce a ciphertext, and sends the ciphertext to party V. 
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2. Party V decrypts the ciphertext using its private key-establishment key and the RSAOAEP DECRYPT operation to recover the transported keying material. 

3. If key confirmation is incorporated, then the transported keying material is parsed into 

two parts, a transaction-specific (random) MacKey followed by KeyData. The MacKey

portion of the keying material and an approved MAC algorithm are used by each party 

to compute a MacTag (of an appropriate, agreed-upon length) on what should be the 

same MacData (see Section 6.6.1). The MacTag computed by the key-confirmation 

provider (V) is sent to the key-confirmation recipient (U). If the value of the MacTag sent 

by V matches the MacTag value computed by U, then U obtains a confirmation of the 

success of the key-transport transaction. 

The common components of the schemes in the KTS-OAEP family are listed in Section 9.2.2. 

The following schemes are then defined: 

1. KTS-OAEP-basic, the basic scheme without key confirmation (see Section 9.2.3). 

2. KTS-OAEP -receiver-confirmation, a variant of KTS-OAEP-basic with unilateral key 

confirmation from party V to party U (see Section 9.2.4). 

For the security attributes of the KTS-OAEP family, see Section 9.2.5. 

9.2.1 KTS-OAEP Family Prerequisites 

1. The receiver shall have been designated as the owner of a key-establishment key pair that 

was generated as specified in Section 6.3. The receiver shall have assurance of its 

possession of the correct value for its private key as specified in Section 6.5.1. 

2. The sender and receiver shall have agreed upon an approved hash function appropriate 

for use with the mask generation function used by RSA-OAEP (see Sections 5.1, 5.8, 

6.2.3, and 7.2.2). 

3. Prior to or during the transport process, the sender and receiver shall have either agreed 

upon the form and content of the additional input A (a byte string to be cryptographically 

bound to the transported keying material in that the cipher is a cryptographic function of 

both values), or agreed that A will be an empty string (see Section 9.1 above). 

4. When key confirmation is used, the sender and receiver shall have agreed upon an 

approved MAC algorithm and associated parameters (see Sections 5.2 and 6.2.3). 

5. Prior to or during the key transport process, each party shall obtain the identifier that is to 

be associated with the other party during the key transport transaction. The sender shall

obtain the public key-establishment key that is bound to the receiver’s identifier. The 

sender shall obtain this public key in a trusted manner (e.g., from a certificate signed by a 

trusted CA). The sender shall obtain assurance of validity of this public key as specified 

in Section 6.4.2. 

6. Prior to or during the key transport process, the sender shall obtain assurance that the 

intended receiver is (or was) in possession of the (correct) private key corresponding to 
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the public key-establishment key used during the transaction, as specified in Section 

6.5.2. 

7. Prior to or during the key transport process, the keying material to be transported shall be 

determined as specified at the beginning of Section 9. 

9.2.2 Common components 

The schemes in the KTS-OAEP family have the following common component: 

1. RSA-OAEP: asymmetric operations, consisting of an encryption operation RSAOAEP.ENCRYPT and a decryption operation RSA-OAEP.DECRYPT (see Section 7.2.2). 

9.2.3 KTS-OAEP-basic 

KTS-OAEP-basic is the basic key transport scheme in the KTS-OAEP family without key 

confirmation. 

Let (PubKeyV, PrivKeyV) be party V’s (the receiver’s) key-establishment key pair. Let K be the 

keying material to be transported from party U (the sender) to party V. The parties shall perform 

the following or an equivalent sequence of steps, which are also illustrated in Figure14. 

Party U shall execute the following steps in order to transport keying material to party V. 

Party U Actions: 

1. Encrypt the keying material K using party V’s public key-establishment key PubKeyV

and the additional input A, to produce a ciphertext C (see Section 7.2.2.2): 

C = RSA-OAEP.ENCRYPT(PubKeyV, K, A). 

2. Send the ciphertext C to party V. 

Party V shall execute the following steps when receiving keys transported from party V. 

Party V Actions: 

1. Receive the ciphertext C. 

2. Decrypt the ciphertext C using the private key-establishment key PrivKeyV and the 

additional input A, to recover the transported keying material K (see Section 7.2.2.3): 

K = RSA-OAEP.DECRYPT(PrivKeyV, C, A). 

If the decryption operation outputs an error indicator, output an error indication and stop. 

Output: The byte string K or an error indicator. 
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Sender (Party U) Receiver (Party V) 

K to be transported (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key 

PubKeyV

← ⎯ ⎯ ⎯ 

C = RSA-OAEP. 

ENCRYPT(PubKeyV, K, A) 

C

⎯⎯⎯⎯⎯→

K = RSA-OAEP. 

DECRYPT(PrivKeyV, C, A) 

Figure 14: KTS-OAEP-basic Scheme 

9.2.4 KTS-OAEP Key Confirmation 

The KES-OAEP-receiver-confirmation scheme is based on the KTS-OAEP-basic scheme. 

9.2.4.1 KTS-OAEP Common Components for Key Confirmation 

The components for KTS-OAEP key confirmation are the same as for KTS-OAEP-basic (see 

Section 9.2.2), plus the following: 

2. MAC: A message authentication code algorithm (see Section 5.2). 

a. MacKeyLen: the length in bytes of the MacKey (see Table 1 in Section 

6.2.3). 

b. MacTagLen: the length in bytes of the MacTag (see Table 1 in Section 

6.2.3). 

For this scheme, the length of the keying material shall be at least MacKeyLen, where 

MacKeyLen is the length in bytes of the MacKey (see Section 6.2.3) and usually longer so that 

other keying material is available for subsequent operations. The MacKey shall be the first 

MacKeyLen bytes of the keying material and shall be used only for the key confirmation 

operation. 

9.2.4.2 KTS-OAEP-receiver-confirmation 

KTS-OAEP-receiver-confirmation is a variant of KTS-OAEP-basic with unilateral key 

confirmation from party V to party U. 

Figure 15 depicts a typical flow for the KTS-OAEP-receiver-confirmation scheme. In this scheme, 

party V, the receiver, and party U, the sender, assume the roles of key confirmation provider and 

recipient, respectively. 

To provide (and receive) key confirmation (as described in Section 6.6.1), both parties form 

MacData with EphemDataV = Null, and EphemDataU = C: 
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Party V provides MacTagV to party U (as specified in Section 6.6.1, with P = V and R = U), 

where MacTagV is computed (as specified in Section 5.2.1) using 

MacDataV = “KC_1_V” || IDV || IDU || Null || C{ || Text}. 

Party U uses the identical format and values to compute MacTagV and then verifies that the 

newly computed MacTagv matches the MacTagV value provided by party V. 

The MacKey used during key confirmation shall be zeroized by Party V immediately after the 

computation of MagTagV, and by Party U immediately after the verification of the received 

MacTagV or a (final) determination that the received MacTagV is in error. 

Sender (Party U) Receiver (Party V) 

K = MacKey ll KeyData (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key

PubKeyV

← ⎯ ⎯ ⎯ 

C = RSA-OAEP. 

ENCRYPT(PubKeyV, K, A) 

C

⎯⎯⎯⎯⎯→

K = RSA-OAEP. 

DECRYPT(PrivKeyV, C, A)

 MacKey || KeyData = K 

MacTagV =? MAC(MacKey, 

MacTagLen, MacDataV) 

MacTagV

←⎯⎯⎯⎯⎯

MacTagV = MAC(MacKey, 

MacTagLen, MacDataV) 

Figure 15: KET-OAEP-receiver-confirmation Scheme 

9.2.5 KTS-OAEP Security Properties 

In each scheme included in this family, only the identifier of V (the receiver) is bound to a public 

key-establishment key. U (the sender) has assurance that no unintended party can recover K

from the ciphertext C (without the compromise of private information). 

The receiver, however, has no such assurance. In particular, V has no assurance as to the 

accuracy of the identifier claimed by the sender and, therefore, has no assurance as to the true 

source of the ciphertext C (or the transported K). 

Due to the sender’s unilateral selection of K, U has assurance that fresh keying material has been 

transported. V has no such assurance. 

A compromise of the receiver’s private key will allow an adversary to masquerade as the 

receiver in future key establishment transactions, and compromises all keying material 

transported to the receiver in both past and future transactions. 
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In the key confirmation case, if one assumes that the transported keying material includes a 

MacKey that is (as required by this Recommendation) unique to the particular KTS-OAEPreceiver-confirmation transaction between U and V, then by successfully comparing the received 

value of MacTagV with its own computation, the sender (U) can obtain these assurances: 

1. the receiver (V) has correctly recovered (at least the MacKey portion of) K from C; 

2. both parties agree on the values of IDV, IDU, and C; 

3. V possesses the correct value of the private key corresponding to the public key used in 

the transaction; and 

4. V has actively participated in the transaction. 

If U has transported the same K to multiple parties, and/or (in violation of this Recommendation) 

U has re-used a MacKey, then the return of a correct MacTagV value to U does not provide 

assurance that V has correctly obtained the keying material (or anything else). Anyone in 

possession of (at least the MacKey portion of) K could have computed MacTagV. 

9.3 KTS-KEM-KWS Family: Key Transport using RSA-KEM-KWS 

The KTS-KEM-KWS family of key transport schemes is based on the RSA-KEM-KWS encrypt 

and decrypt operations. These operations employ the asymmetric RSASVE secret-value 

encapsulation operations and an approved KDF to establish a key-wrapping key that is 

transaction-specific. The key-wrapping key is used with an approved symmetric key-wrapping 

algorithm to wrap (and unwrap) the keying material to be transported. In this family, only party 

V’s key pair is used. 

The key transport schemes of this family have the following general form: 

1. Using the RSA-KEM-KWS.ENCRYPT operation, party U (the sender) first generates a 

secret byte string Z and a corresponding ciphertext component by employing the 

RSASVE.GENERATE operation and the public key-establishment key of party V (the 

receiver). The byte string Z (along with OtherInfo available to U and V) is then used as 

input to the KDF, to derive a transaction-specific key-wrapping key (KWK) of an 

appropriate, agreed-upon bit length kwkBits. The keying material to be transported is 

wrapped using the KWK and the symmetric key-wrapping algorithm to produce a second 

ciphertext component. The two ciphertext components are sent to party V. 

2. Using the RSA-KEM-KWS Decrypt operation, Party V begins by employing the 

RSASVE.RECOVER operation and its private key-establishment key to obtain Z from the 

first ciphertext component. Party V then employs the KDF (with inputs Z, kwkBits, and 

OtherInfo) to derive the same KWK that was used by U. The KWK and the symmetric 

key-unwrapping algorithm are used to obtain the transported keying material from the 

second ciphertext component. 

3. If key confirmation is incorporated, the transported keying material consists of a 

transaction-specific (random) MacKey followed by KeyData. The MacKey portion of the 

keying material and an approved MAC algorithm are used by each party to compute a 
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MacTag (of an appropriate, agreed-upon length) on what should be the same MacData

(see Section 6.6.1). The MacTag computed by the key-confirmation provider (V) is sent 

to the key-confirmation recipient (U). If the value of the MacTag sent by V matches the 

MacTag value computed by U, then U obtains a confirmation of the success of the keytransport transaction. 

Common components of the schemes in the KTS-KEM-KWS family are listed in Section 9.3.2. 

Two schemes are then defined: 

1. KTS-KEM-KWS-basic, the basic scheme without key confirmation (see Section 9.3.3). 

2. KTS-KEM-KWS-receiver-confirmation, a variant with unilateral key confirmation 

from the receiver (Party V) to the sender (Party U) (see Section 9.3.4). 

For the security attributes of the KTS-KEM-KWS family, see Section 9.3.5. 

9.3.1 KTS-KEM-KWS Family Prerequisites 

1. The receiver shall have been designated as the owner of a key-establishment key pair that 

was generated as specified in Section 6.3. The receiver shall obtain assurance of the 

validity of its key pair as specified in Section 6.4.1, and shall obtain assurance of its 

possession of the correct value for its private key as specified in Section 6.5.1. 

2. The sender and receiver shall have agreed upon an approved key derivation function, an 

approved hash function appropriate for use with the key derivation function, and 

associated parameters (see Sections 5.1, 5.9, and 6.2.3). 

3. The sender and receiver shall have agreed upon an approved symmetric key-wrapping 

algorithm and key length (kwkBits) employing an approved block cipher algorithm 

whose security strength is equal to or greater than the target security strength of the 

applicable key transport scheme (see Sections 5.7 and 7.2.3). 

4. Prior to or during the transport process, the sender and receiver shall have either agreed 

upon the form and content of the additional input A (a byte string to be cryptographically 

bound to the transported keying material in that the cipher is a cryptographic function of 

both values), or agreed that A will be an empty string (see Section 9.1 above). 

5. When key confirmation is used, the sender and receiver shall have agreed upon an 

approved MAC algorithm and associated parameters (see Sections 5.2 and 6.2.3). 

6. Prior to or during the key transport process, each party shall obtain the identifier that is to 

be associated with the other party during the key transport transaction. The sender shall

obtain the public key-establishment key that is bound to the receiver’s identifier in a 

trusted manner (e.g., from a certificate signed by a trusted CA). The sender shall obtain 

assurance of the validity of this public key as specified in Section 6.4.2. 

7. Prior to or during the key transport process, the sender shall obtain assurance that the 

intended receiver is (or was) in possession of the private key corresponding to the public 

key-establishment key used during the transaction, as specified in Section 6.5.2. 

8. Prior to or during the key transport process, the keying material to be transported shall be 

determined as specified in Section 9. 
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9.3.2 Common Components of the KTS-KEM-KWS Schemes 

The schemes in the KTS-KEM-KWS family have the following common component: 

1. RSA-KEM-KWS: Consisting of an encryption operation RSA-KEM-KWS.ENCRYPT

and a decryption operation RSA-KEM-KWS.DECRYPT (see Section

7.2.3). 

9.3.3 KTS-KEM-KWS-basic 

KTS-KEM-KWS-basic is the basic key transport scheme in the KTS-KEM-KWS family 

without key confirmation. 

Let (PubKeyV, PrivKeyV) be party V’s key-establishment key pair. Let K be the keying material 

to be transported from party U to party V. The parties shall perform the following or an 

equivalent sequence of steps, which are also illustrated in Figure 16. 

Party U shall execute the following steps in order to transport keying material to party V. 

Party U Actions: 

1. Using party V’s public key-establishment key PubKeyV, the length kwkBits of key to be 

used for key-wrapping, keying material K, and the additional input A, generate a 

ciphertext C (see Section 7.2.3.2), which includes an encrypted KWK as C0 and the 

wrapped keying material as C1: 

C = RSA-KEM-KWS.ENCRYPT(PubKeyV, kwkBits, K, A). 

2. Send the ciphertext C to party V. 

Party V shall execute the following steps when receiving keys transported from party V. 

Party V Actions: 

1. Receive the transported keying material. 

2. Using the ciphertext C, the private key-establishment key PrivKeyV, the length kwkBits of 

the key-wrapping key, and the additional input A, recover the keying material K (see 

Section 7.2.3.3): 

K = RSA-KEM-KWS.DECRYPT(PrivKeyV, C, kwkBits, A). 

If the decryption operation outputs an error indicator, output an error indication and stop. 

Output: The byte string K or an error indicator. 
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Sender (Party U) Receiver (Party V) 

K to be transported (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key 

PubKeyV

← ⎯ ⎯ ⎯ 

C = RSA-KEM-KWS. 

ENCRYPT(PubKeyV, K, A) 

C

⎯⎯⎯⎯⎯→

K = RSA-KEM-KWS. 

DECRYPT(PrivKeyV, C, A) 

Figure 16: KTS-KEM-KES-basic Scheme 

9.3.4 KTS-KEM-KWS Key Confirmation 

The KTS-KEM-KWS-receiver-confirmation scheme offers receiver confirmation and is based 

on the KTS-KEM-KWS-basic scheme. 

9.3.4.1 KTS-KEM-KWS Common Components for Key Confirmation 

The components for KTS-KEM-KWS-receiver-confirmation are the same as for KTS-KEMKWS-basic (see Section 9.3.2), plus the following: 

2. MAC: A message authentication code algorithm (see Section 5.2). 

a. the MacKeyLen: length in bytes of the MacKey (see Table 1 in Section 

6.2.3). 

b. the MacTagLen: length in bytes of the MacTag (see Table 1 in Section 

6.2.3). 

For this scheme, the length of the keying material shall be at least MacKeyLen, where 

MacKeyLen is the length of the MacKey (see Section 6.2.3) and usually longer so that other 

keying material is available for subsequent operations. The MacKey shall be the first 

MacKeyLen bytes of the keying material and shall be used only for key confirmation. 

9.3.4.2 KTS-KEM-KWS-receiver-confirmation 

KTS-KEM-KWS-receiver-confirmation is a variant of KTS-KEM-KWS-basic with unilateral 

key confirmation from party V to party U. 

Figure 17 depicts a typical flow for the KTS-KEM-KWS-receiver-confirmation scheme. In this 

scheme, party V, the receiver, and party U, the sender, assume the roles of key confirmation provider 

and recipient, respectively. 

To provide (and receive) key confirmation (as described in Section 6.6.1), both parties set 

EphemDataV = Null, and EphemDataU = C: 
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Party V provides MacTagV to party U (as specified in Section 6.6.1, with P = V and R = U), 

where MacTagV is computed (as specified in Section 5.2.1) using 

MacDataV = “KC_1_V” || IDV || IDU || Null || C{ || Text}. 

Party U uses the identical format and values to compute MacTagV and then verifies that the 

newly computed MacTagV matches the MacTagV value provided by party V. 

The MacKey used during Key Confirmation shall be zeroized by Party V immediately after the 

computation of MagTagV, and by Party U immediately after the verification of the received 

MacTagV or a (final) determination that the received MacTagV is in error. 

Sender (Party U) Receiver (Party V) 

K = MacKey ll KeyData (PubKeyV, PrivKeyV) 

Obtain party V’s public keyestablishment key

PubKeyV

← ⎯ ⎯ ⎯

C = RSA-KEM-KWS. 

ENCRYPT(PubKeyV, K, A)

C

⎯⎯⎯⎯⎯→

K = RSA-KEM.KWS. 

DECRYPT(PrivKeyV, C, A)

 MacKey || KeyData = K 

MacTagV =? MAC(MacKey, 

MacTagLen, MacDataV) 

MacTagV

←⎯⎯⎯⎯

MacTagV = MAC(MacKey, 

MacTagLen, MacDataV) 

Figure 17: KTS-KEM-KWS-receiver-confirmation Scheme 

9.3.5 KTS-KEM-KWS Security Properties 

In each scheme included in this family, only the identifier of V (the receiver) is bound to a public 

key-establishment key. U (the sender) has assurance that no unintended party can recover the 

shared secret Z, and hence the KWK, from the ciphertext component C0, and then use KWK to 

obtain the keying material K from the ciphertext component C1 (without the compromise of some 

private information). 

The receiver, however, has no such assurance. In particular, V has no assurance as to the 

accuracy of the identifier claimed by the sender and, therefore, has no assurance as to the true 

source of the ciphertext C = C0 || C1 (or the transported K). 

Due to the sender’s unilateral selection of K, U can obtain assurance that fresh keying material 

has been transported. V has no such assurance. 

A compromise of the receiver’s private key will allow an adversary to masquerade as the 

receiver in future key establishment transactions, and compromises all keying material 

transported to the receiver in both past and future transactions. 
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In the key confirmation case, if one assumes that the transported keying material includes a 

MacKey that is (as required by this Recommendation) randomly generated for the particular 

KTS-KEM-KWS-receiver-confirmation transaction between U and V, then by successfully 

comparing the received value of MacTagV with its own computation, the sender (U) can obtain 

these assurances: 

1. V has correctly recovered Z, and hence the KWK; 

2. both parties agree on the values of IDV, IDU, and C; 

3. the receiver (V) has correctly recovered (at least the MacKey portion of) K from C; 

4. V possesses the correct value of the private key corresponding to the public key used in 

the transaction; and 

5. V has actively participated in the transaction. 

The use of a transaction-specific (random) Z (and hence the transaction-specific KWK) provides 

assurance to U that both C0 and C1 are also random for a given transaction, even if K is not. 

However, if U has transported the same K to multiple parties, and/or (in violation of the 

Recommendation) U has re-used a MacKey, then the return of a correct MacTagV value to U 

does not provide assurance that V has correctly obtained the keying material (or anything else). 

Anyone in possession of (at least the MacKey portion of) K could have computed MacTagV. 

10 Key Recovery 

For some applications, the secret keying material used to protect data or to process protected 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 

and other inputs to the scheme used to perform the key establishment process). 

For example, the following information that is used during key establishment may need to be 

saved: 

1. One or both keys of a key pair, as needed. 

2. The nonce(s), 

3. The ciphertext. 

4. Additional input. 

5. OtherInfo, 

6. A symmetric key. 

General guidance on key recovery and the protections required for each type of key is provided 

in the Recommendation for Key Management [7]. 
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11. Implementation Validation 

When the NIST Cryptographic Algorithm Validation System (CAVS) 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: 

1. Key pair generation as specified in Section 6.3. 

2. Explicit public key validation as specified in Section 6.4.3. 

3. A key agreement scheme from Section 8, together with an approved key derivation 

function from Section 5.9. Other key derivation methods with specific protocols may be 

temporarily allowed for backward compatibility if agreed upon by the participating 

entities (i.e., party U and party V). These other allowable methods and the protocols that 

they may be used with are referenced in FIPS 140-2 Annex D. Documentation shall

include how assurance of private key possession and assurance of public key validity are 

expected to be achieved by both the owner and the recipient. 

4. A key transport scheme as specified in Section 9, together with an approved random bit 

generator, an approved hash function, an approved symmetric key-wrapping algorithm, 

and an approved key derivation function from Section 5.9 for RSA-KEM-KWS based 

schemes. Other key derivation methods with specific protocols may be temporarily 

allowed for backward compatibility if agreed upon by the participating entities (i.e., party 

U and party V). These other allowable methods and protocols are referenced in FIPS 140-

2 Annex D. 

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

1. The security strength(s) of supported cryptographic algorithms; this will determine the 

parameter set requirements (see Table 1 in Section 6.2.3), 

2. The hash function to be used (see Section 5.1), 

3. The MacKey length(s) (see Table 1 in Section 6.2.3), 

4. The MacTag length (see Table 1 in Section 6.2.3), 

5. The key establishment schemes available (see Sections 8 and 9), 

6. The key derivation function to be used if a key agreement scheme is implemented, 

including the format of OtherInfo (see Section 5.9), 

7. The type of nonces to be generated (see Section 5.6). 

8. How assurance of private key possession and assurance of public key validity are 

expected to be achieved by both the owner and the recipient. 
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9. If a key transport scheme is implemented, indicate whether a capability is available to 

handle additional input. 
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Appendix A: Summary of Differences between this 

Recommendation and ANS X9.44 (Informative) 

This list is informational and not meant to be exhaustive, but is intended to summarize important 

differences between this Recommendation and ANS X9.44. In general, this Recommendation 

can be seen as being more restrictive than ANS X9.44. The list of differences is as follows: 

1. For purposes of validating an implementation of the schemes in this Recommendation 

durring an implementation validation test (under the NIST Cryptographic Algorithm 

Validation System), the value of MacData is set to the string “Standard Test Message”, 

followed by a 128-bit 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 purpose of 

testing when no key confirmation capability exists. ANS X9.44 does not address 

implementation validation at this level of detail. 

2. ANS X9.44 allows the public key exponent e to be as small as 3, whereas this 

Recommendation requires that e be at least 65537. 

3. ANS X9.44 requires that separate keys be used for key transport and key agreement. This 

Recommendation allows the same key to be used for both purposes. 

4. Regarding the key derivation function (KDF): 

a. This Recommendation specifies two approved KDFs, the concatenation KDF 

specified in Section 5.9.1 and the ASN.1 KDF specified in Section 5.9.2. 

Additional KDFs may be allowed for a transition period. 

b. ANS X9.44 provides two forms of a concatenation KDFs, KDF2 and KDF3. 

KDF2 is compatible with the concatenation KDFs specified in IEEE 1363[13], 

IEEE 1363a [14], ANS X9.42 [9], and ANS X9.63[11]. KDF3 can be used in a 

mode that is compatible with this Recommendation. The significant difference 

between KDF2 and KDF3 is that in KDF2, the counter is hashed after the shared 

secret, whereas in KDF3, the counter is hashed before the shared secret. 

c. The approved KDFs in this Recommendation require the input of the identifiers 

of the communicating parties; such information is allowed, but not required, in 

ANS X9.44. 

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 intent in ANS X9.44 

is to prohibit the re-use of the shared secret, but the zeroization requirement is not 

specifically stated. 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. 
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5. ANS X9.44 requires the use of X9-approved key-wrapping algorithms, whereas this 

Recommendation requires the use of NIST-approved/allowed key-wrapping algorithms. 

ASC X9 allows both AES and TDES key wrap algorithms [12], whereas NIST currently 

specifies only AES for key wrapping [8]. 

6. This Recommendation uses a more stringent definition of key confirmation than does 

ANS X9.44. ANS X9.44 does not require that the confirmation provider be authenticated. 

Therefore, schemes that qualify as offering key confirmation under ANS X9.44 may not 

qualify as offering key confirmation under this Recommendation. For example, the kas1-

bilateral-confirmation scheme of ANS X9.44 does not exist in this Recommendation, 

since the identity of the initiator is not authenticated. 

7. The KAS2 schemes are not provided in ANS X9.44. However, they are included in this 

Recommendation to provide schemes that are similar to the C(1,1) schemes in SP 800-

56A. 
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Appendix B: Data Conversions (Normative) 

B.1 Integer-to-Byte String (I2BS) Conversion 

Input: A non-negative integer C and the intended length n of the byte string satisfying 

 28n > C 

Output: A byte string S of length n bytes. 

1. Let S1, S2,..., Sn be the bytes of S from leftmost to rightmost. 

2. The bytes of S shall satisfy: 

 C = Σ28(n-i)Si for i = 1 to n.

B.2 Byte String to Integer (BS2I) Conversion 

Input: A byte string S (SLen is used to denote the length of the byte string).

Output: A non-negative integer C. 

Steps:

1. Let S1 S2 ... SsLen be the bytes of S from first to last, and let xSLen–i be the integer value of 

the octet xi for 1 ≤ i ≤ SLen, where the integer value is represented as a byte (i.e., an 

eight-bit string), with the most significant bit first (i.e., on the left). 

2. Let x = xSLen–1 · 256 SLen–1 + xSLen–2 · 256 SLen–2 + ... + x1 · 256 + x0. 

3. Output x.
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Appendix C: Prime Factor Recovery (Normative) 

The following algorithm recovers the prime factors of a modulus, given the public and private 

exponents. The algorithm is based on Fact 1 in [18].

Function call: RecoverPrimeFactors(n, e, d) 

Input: 

1. n: modulus 

2. e: public exponent 

3. d: private exponent 

Output: 

1. (p, q): prime factors of modulus 

Errors: “prime factors not found” 

Assumptions: The modulus n is the product of two prime factors p and q; the public and private 

exponents satisfy d × e ≡ 1 (mod λ(n)) where λ(n) = LCM(p – 1, q – 1) 

Process: 

1. Let k = d × e – 1. If k is odd, then go to Step 4. 

2. Write k as k = r 2t , where r is the largest odd integer dividing k, and t ≥ 1. 

3. For i = 1 to 100 do: 

a. Generate a random integer g in the range [0, n−1]. 

b. Let y = g r

 mod n. 

c. If y = 1 or y = n – 1, then go to Step g. 

d. For j = 1 to t – 1 do: 

I. Let x = y 2 mod n. 

II. If x = 1, go to Step 5 

III. If x = n – 1, go to Step3.7. 

IV. Let y = x. 
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e. Let x = y

2

 mod n. 

f. If x = 1, go to Step 5. 

g. Continue. 

4. Output “prime factors not found” and stop. 

5. Let p = GCD(y – 1, n) and let q = n/p. 

6. Output (p, q) as the prime factors. 

Notes: 

1. According to Fact 1 in [18], the probability that one of the values of y in an iteration of 

Step 3 reveals the factors of the modulus is at least 1/2, so on average, at most two 

iterations of that step will be required. If the prime factors are not revealed after 100 

iterations, then the probability is overwhelming that the modulus is not the product of two 

prime factors, or that the public and private exponents are not consistent with each other. 

2. The algorithm bears some resemblance to the Miller-Rabin primality testing algorithm 

(see, e.g., ANS X9.80). 

3. The order of the recovered prime factors p and q may be the reverse of the order in which 

the factors were generated originally. 
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Appendix D: References (Informative) 

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

FIPS 140-3 is currently under development. 

[2] FIPS 180-3, Secure Hash Standard, June 2007 (Draft). 

[3] FIPS 186-3, Digital Signature Standard, March 2006 (Draft) 

[4] FIPS 197, Advanced Encryption Standard, November 2001. 

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

[6] NIST SP 800-38B, Recommendation for Block Cipher Modes of Operation: The 

CMAC Mode for Authentication, May 2005. 

[7] NIST SP 800-57-Part 1, Recommendation for Key Management, August 2005. 

[8] AES Key Wrap Specification, NIST, November 16, 2001. 

[9] ANS X9.42-2001, Public Key Cryptography for the Financial Services Industry: 

Agreement of Symmetric Keys Using Discrete Logarithm Cryptography. 

[10] [ANS X9.44 Public Key Cryptography for the Financial Services Industry: Key 

Establishment Using Integer Factorization Cryptography, August 24, 2007. 

[11] ANS X9.63-2001, Public Key Cryptography for the Financial Services Industry: 

Key Agreement and Key Transport Using Elliptic Curve Cryptography. 

[12] ANSI X9.102-2008, Symmetric Key Cryptography for the Financial Services 

Industry – Wrapping of Keys and Associated Data. 

[13] IEEE 1363-2000, IEEE Standard Specifications for Public Key Cryptography. 

[14] IEEE 1363a-2004, IEEE Standard Specifications for Public Key Cryptography -

Amendment 1: Additional Techniques. 

[15] A Chosen Ciphertext Attack on RSA Optimal Asymmetric Encryption Padding 

(OAEP) as Standardized in PKCS #1 v2.0, J. Manger, In J. Kilian, editor, 

Advances in Cryptology – Crypto 2001, pp. 230 – 238, Springer Verlag, 2001. 

[16] A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. R. 

Rivest, A. Shamir and L. Adleman, Communications of the ACM, 21(2), pp. 120 – 

126, February 1978. 
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[17] The Security of all RSA and Discrete Log Bits, J. Håstad and M. Näslund. Proc. of 

the 39th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 510

− 521, 1998. 

[18] Twenty Years of Attacks on the RSA Cryptosystem, D. Boneh, Notices of the 

American Mathematical Society (AMS), 46(2), 203 – 213. 1999.