FIPS PUB 186-4

FEDERAL INFORMATION PROCESSING STANDARDS 

PUBLICATION

Digital Signature Standard (DSS)

CATEGORY: COMPUTER SECURITY SUBCATEGORY: CRYPTOGRAPHY

Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8900

Issued July 2013

U.S. Department of Commerce

Cameron F. Kerry, Acting Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director
FOREWORD

The Federal Information Processing Standards Publication Series of the National Institute 

of Standards and Technology (NIST) is the official series of publications relating to 

standards and guidelines adopted and promulgated under the provisions of the Federal 

Information Security Management Act (FISMA) of 2002. 

Comments concerning FIPS publications are welcomed and should be addressed to the 

Director, Information Technology Laboratory, National Institute of Standards and 

Technology, 100 Bureau Drive, Stop 8900, Gaithersburg, MD 20899-8900. 

Charles Romine, Director 

Information Technology Laboratory

Abstract

This Standard specifies a suite of algorithms that can be used to generate a digital signature. 

Digital signatures are used to detect unauthorized modifications to data and to authenticate the 

identity of the signatory. In addition, the recipient of signed data can use a digital signature as 

evidence in demonstrating to a third party that the signature was, in fact, generated by the 

claimed signatory. This is known as non-repudiation, since the signatory cannot easily repudiate 

the signature at a later time.

Key words: computer security, cryptography, digital signatures, Federal Information Processing 

Standards, public key cryptography.
i

Federal Information Processing Standards Publication 186-4

July 2013

Announcing the

DIGITAL SIGNATURE STANDARD (DSS)

Federal Information Processing Standards Publications (FIPS PUBS) are issued by the National 

Institute of Standards and Technology (NIST) after approval by the Secretary of Commerce 

pursuant to Section 5131 of the Information Technology Management Reform Act of 1996 

(Public Law 104-106), and the Computer Security Act of 1987 (Public Law 100-235).

1. Name of Standard: Digital Signature Standard (DSS) (FIPS 186-4).

2. Category of Standard: Computer Security. Subcategory. Cryptography.

3. Explanation: This Standard specifies algorithms for applications requiring a digital 

signature, rather than a written signature. A digital signature is represented in a computer as a 

string of bits. A digital signature is computed using a set of rules and a set of parameters that 

allow the identity of the signatory and the integrity of the data to be verified. Digital signatures 

may be generated on both stored and transmitted data.

Signature generation uses a private key to generate a digital signature; signature verification uses 

a public key that corresponds to, but is not the same as, the private key. Each signatory 

possesses a private and public key pair. Public keys may be known by the public; private keys 

are kept secret. Anyone can verify the signature by employing the signatory’s public key. Only 

the user that possesses the private key can perform signature generation.

A hash function is used in the signature generation process to obtain a condensed version of the 

data to be signed; the condensed version of the data is often called a message digest. The 

message digest is input to the digital signature algorithm to generate the digital signature. The 

hash functions to be used are specified in the Secure Hash Standard (SHS), FIPS 180. FIPS 

approved digital signature algorithms shall be used with an appropriate hash function that is 

specified in the SHS.

The digital signature is provided to the intended verifier along with the signed data. The 

verifying entity verifies the signature by using the claimed signatory’s public key and the same 

hash function that was used to generate the signature. Similar procedures may be used to 

generate and verify signatures for both stored and transmitted data.

4. Approving Authority: Secretary of Commerce.
ii

5. Maintenance Agency: Department of Commerce, National Institute of Standards and 

Technology, Information Technology Laboratory, Computer Security Division.

6. Applicability: This Standard is applicable to all Federal departments and agencies for the 

protection of sensitive unclassified information that is not subject to section 2315 of Title 10, 

United States Code, or section 3502 (2) of Title 44, United States Code. This Standard shall be 

used in designing and implementing public key-based signature systems that Federal 

departments and agencies operate or that are operated for them under contract. The adoption and 

use of this Standard is available to private and commercial organizations.

7. Applications: A digital signature algorithm allows an entity to authenticate the integrity of 

signed data and the identity of the signatory. The recipient of a signed message can use a digital 

signature as evidence in demonstrating to a third party that the signature was, in fact, generated 

by the claimed signatory. This is known as non-repudiation, since the signatory cannot easily 

repudiate the signature at a later time. A digital signature algorithm is intended for use in 

electronic mail, electronic funds transfer, electronic data interchange, software distribution, data 

storage, and other applications that require data integrity assurance and data origin 

authentication. 

8. Implementations: A digital signature algorithm may be implemented in software, firmware, 

hardware or any combination thereof. NIST has developed a validation program to test 

implementations for conformance to the algorithms in this Standard. Information about the 

validation program is available at http://csrc.nist.gov/cryptval. Examples for each digital 

signature algorithm are available at http://csrc.nist.gov/groups/ST/toolkit/examples.html.

Agencies are advised that digital signature key pairs shall not be used for other purposes.

9. Other Approved Security Functions: Digital signature implementations that comply with 

this Standard shall employ cryptographic algorithms, cryptographic key generation algorithms, 

and key establishment techniques that have been approved for protecting Federal government 

sensitive information. Approved cryptographic algorithms and techniques include those that are 

either:

a. specified in a Federal Information Processing Standard (FIPS),

b. adopted in a FIPS or a NIST Recommendation, or

c. specified in the list of approved security functions for FIPS 140.

10. Export Control: Certain cryptographic devices and technical data regarding them are 

subject to Federal export controls. Exports of cryptographic modules implementing this Standard 

and technical data regarding them must comply with these Federal regulations and be licensed by 

the Bureau of Industry and Security of the U.S. Department of Commerce. Information about 

export regulations is available at: http://www.bis.doc.gov.

11. Patents: The algorithms in this Standard may be covered by U.S. or foreign patents.
iii

12. Implementation Schedule: This Standard becomes effective immediately upon 

approval by the Secretary of Commerce. A transition strategy for validating algorithms 

and cryptographic modules will be posted on NIST’s Web page at 

http://csrc.nist.gov/groups/STM/cmvp/index.html under Notices. The transition plan 

addresses the transition by Federal agencies from modules tested and validated for 

compliance to previous versions of this Standard to modules tested and validated for 

compliance to FIPS 186-4 under the Cryptographic Module Validation Program. The 

transition plan allows Federal agencies and vendors to make a smooth transition to FIPS 

186-4.

13. Specifications: Federal Information Processing Standard (FIPS) 186-4 Digital Signature 

Standard (affixed).

14. Cross Index: The following documents are referenced in this Standard. Unless a specific 

version or date is indicated with the document number, the latest version of the given document 

is intended as the reference.

a. FIPS PUB 140, Security Requirements for Cryptographic Modules.

b. FIPS PUB 180 Secure Hash Standard.

c. ANS X9.31-1998, Digital Signatures Using Reversible Public Key Cryptography for the 

Financial Services Industry (rDSA).

d. ANS X9.62-2005, Public Key Cryptography for the Financial Services Industry: The 

Elliptic Curve Digital Signature Algorithm (ECDSA).

e. ANS X9.80, Prime Number Generation, Primality Testing and Primality Certificates.

f. Public Key Cryptography Standard (PKCS) #1, RSA Encryption Standard.

g. Special Publication (SP) 800-57, Recommendation for Key Management.

h. Special Publication (SP) 800-89, Recommendation for Obtaining Assurances for Digital 

Signature Applications.

i. Special Publication (SP) 800-90A, Recommendation for Random Number Generation 

Using Deterministic Random Bit Generators.

j. Special Publication (SP) 800-102, Recommendation for Digital Signature Timeliness.

k. Special Publication (SP) 800-131A, Transitions: Recommendation for Transitioning the 

Use of Cryptographic Algorithms and Key Lengths.

l. IEEE Std. 1363-2000, Standard Specifications for Public Key Cryptography.

15. Qualifications: The security of a digital signature system is dependent on maintaining the 

secrecy of the signatory’s private keys. Signatories shall, therefore, guard against the disclosure 

of their private keys. While it is the intent of this Standard to specify general security 

requirements for generating digital signatures, conformance to this Standard does not assure that 
iv

a particular implementation is secure. It is the responsibility of an implementer to ensure that 

any module that implements a digital signature capability is designed and built in a secure 

manner.

Similarly, the use of a product containing an implementation that conforms to this Standard does 

not guarantee the security of the overall system in which the product is used. The responsible 

authority in each agency or department shall assure that an overall implementation provides an 

acceptable level of security. 

Since a standard of this nature must be flexible enough to adapt to advancements and 

innovations in science and technology, this Standard will be reviewed every five years in order 

to assess its adequacy.

16. Waiver Procedure: The Federal Information Security Management Act (FISMA) does not 

allow for waivers to Federal Information Processing Standards (FIPS) that are made mandatory 

by the Secretary of Commerce. 

17. Where to Obtain Copies of the Standard: This publication is available by accessing 

http://csrc.nist.gov/publications/. Other computer security publications are available at the same 

web site. 
v

Table of Contents

1. INTRODUCTION .................................................................................................................................... 1

2. GLOSSARY OF TERMS, ACRONYMS AND MATHEMATICAL SYMBOLS ....................................... 2

2.1 TERMS AND DEFINITIONS ................................................................................................................ 2

2.2 ACRONYMS..................................................................................................................................... 5

2.3 MATHEMATICAL SYMBOLS................................................................................................................ 6

3. GENERAL DISCUSSION....................................................................................................................... 9

3.1 INITIAL SETUP ............................................................................................................................... 11

3.2 DIGITAL SIGNATURE GENERATION.................................................................................................. 12

3.3 DIGITAL SIGNATURE VERIFICATION AND VALIDATION ....................................................................... 13

4 THE DIGITAL SIGNATURE ALGORITHM (DSA) ............................................................................... 15

4.1 DSA PARAMETERS ....................................................................................................................... 15

4.2 SELECTION OF PARAMETER SIZES AND HASH FUNCTIONS FOR DSA................................................ 15

4.3 DSA DOMAIN PARAMETERS........................................................................................................... 16

4.3.1 Domain Parameter Generation ...................................................................................... 17

4.3.2 Domain Parameter Management................................................................................... 17

4.4 KEY PAIRS.................................................................................................................................... 17

4.4.1 DSA Key Pair Generation .............................................................................................. 17

4.4.2 Key Pair Management ................................................................................................... 18

4.5 DSA PER-MESSAGE SECRET NUMBER .......................................................................................... 18

4.6 DSA SIGNATURE GENERATION ...................................................................................................... 19

4.7 DSA SIGNATURE VERIFICATION AND VALIDATION............................................................................ 19

5. THE RSA DIGITAL SIGNATURE ALGORITHM.................................................................................. 22

5.1 RSA KEY PAIR GENERATION ......................................................................................................... 22

5.2 KEY PAIR MANAGEMENT................................................................................................................ 23

5.3 ASSURANCES................................................................................................................................ 23

5.4 ANS X9.31 .................................................................................................................................. 24

5.5 PKCS #1 ..................................................................................................................................... 24

6. THE ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM (ECDSA)............................................. 26

6.1 ECDSA DOMAIN PARAMETERS...................................................................................................... 26

6.1.1 Domain Parameter Generation ...................................................................................... 26

6.1.2 Domain Parameter Management................................................................................... 28

6.2 PRIVATE/PUBLIC KEYS .................................................................................................................. 28

6.2.1 Key Pair Generation....................................................................................................... 29

6.2.2 Key Pair Management ................................................................................................... 29

6.3 SECRET NUMBER GENERATION...................................................................................................... 29

6.4 ECDSA DIGITAL SIGNATURE GENERATION AND VERIFICATION ........................................................ 29

6.5 ASSURANCES................................................................................................................................ 30

APPENDIX A: GENERATION AND VALIDATION OF FFC DOMAIN PARAMETERS ........................... 31
vi

A.1 GENERATION OF THE FFC PRIMES P AND Q ................................................................................... 31

A.1.1 Generation and Validation of Probable Primes.............................................................31

A.1.2 Construction and Validation of the Provable Primes p and q.........................................36

A.2 GENERATION OF THE GENERATOR G .............................................................................................. 41

A.2.1 Unverifiable Generation of the Generator g ................................................................... 41

A.2.2 Assurance of the Validity of the Generator g .................................................................42

A.2.3 Verifiable Canonical Generation of the Generator g ......................................................42

A.2.4 Validation Routine when the Canonical Generation of the Generator g Routine Was 

Used............................................................................................................................... 44

APPENDIX B: KEY PAIR GENERATION.................................................................................................. 46

B.1 FFC KEY PAIR GENERATION.......................................................................................................... 46

B.1.1 Key Pair Generation Using Extra Random Bits .............................................................46

B.1.2 Key Pair Generation by Testing Candidates.................................................................. 47

B.2 FFC PER-MESSAGE SECRET NUMBER GENERATION ...................................................................... 48

B.2.1 Per-Message Secret Number Generation Using Extra Random Bits ............................49

B.2.2 Per-Message Secret Number Generation by Testing Candidates.................................49

B.3 IFC KEY PAIR GENERATION........................................................................................................... 50

B.3.1 Criteria for IFC Key Pairs ............................................................................................... 50

B.3.2 Generation of Random Primes that are Provably Prime................................................53

B.3.3 Generation of Random Primes that are Probably Prime................................................55

B.3.4 Generation of Provable Primes with Conditions Based on Auxiliary Provable Primes ..56

B.3.5 Generation of Probable Primes with Conditions Based on Auxiliary Provable Primes ..58

B.3.6 Generation of Probable Primes with Conditions Based on Auxiliary Probable Primes..60

B.4 ECC KEY PAIR GENERATION......................................................................................................... 61

B.4.1 Key Pair Generation Using Extra Random Bits .............................................................62

B.4.2 Key Pair Generation by Testing Candidates.................................................................. 63

B.5 ECC PER-MESSAGE SECRET NUMBER GENERATION...................................................................... 64

B.5.1 Per-Message Secret Number Generation Using Extra Random Bits ............................64

B.5.2 Per-Message Secret Number Generation by Testing Candidates.................................65

APPENDIX C: GENERATION OF OTHER QUANTITIES.......................................................................... 67

C.1 COMPUTATION OF THE INVERSE VALUE .......................................................................................... 67

C.2 CONVERSION BETWEEN BIT STRINGS AND INTEGERS...................................................................... 68

C.2.1 Conversion of a Bit String to an Integer......................................................................... 68

C.2.2 Conversion of an Integer to a Bit String ......................................................................... 68

C.3 PROBABILISTIC PRIMALITY TESTS................................................................................................... 69

C.3.1 Miller-Rabin Probabilistic Primality Test......................................................................... 71

C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test .......................................................72
vii

C.3.3 (GENERAL) LUCAS PROBABILISTIC PRIMALITY TEST ........................................................................ 74

C.4 CHECKING FOR A PERFECT SQUARE .............................................................................................. 75

C.5 JACOBI SYMBOL ALGORITHM.......................................................................................................... 76

C.6 SHAWE-TAYLOR RANDOM_PRIME ROUTINE.................................................................................... 77

C.7 TRIAL DIVISION ............................................................................................................................. 80

C.8 SIEVE PROCEDURE ....................................................................................................................... 80

C.9 COMPUTE A PROBABLE PRIME FACTOR BASED ON AUXILIARY PRIMES............................................. 81

C.10 CONSTRUCT A PROVABLE PRIME (POSSIBLY WITH CONDITIONS), BASED ON CONTEMPORANEOUSLY 

CONSTRUCTED AUXILIARY PROVABLE PRIMES................................................................................ 83

APPENDIX D: RECOMMENDED ELLIPTIC CURVES FOR FEDERAL GOVERNMENT USE................ 87

D.1 NIST RECOMMENDED ELLIPTIC CURVES ........................................................................................ 87

D.1.1 Choices .......................................................................................................................... 87

D.1.2 Curves over Prime Fields............................................................................................... 89

D.1.3 Curves over Binary Fields.............................................................................................. 92

D.2 IMPLEMENTATION OF MODULAR ARITHMETIC................................................................................. 101

D.2.1 Curve P-192................................................................................................................. 101

D.2.2 Curve P-224................................................................................................................. 102

D.2.3 Curve P-256................................................................................................................. 102

D.2.4 Curve P-384................................................................................................................. 103

D.2.5 Curve P-521................................................................................................................. 104

D.3 NORMAL BASES .......................................................................................................................... 104

D.4 SCALAR MULTIPLICATION ON KOBLITZ CURVES............................................................................. 106

D.5 GENERATION OF PSEUDO-RANDOM CURVES (PRIME CASE) ......................................................... 109

D.6 VERIFICATION OF CURVE PSEUDO-RANDOMNESS (PRIME CASE) ................................................... 110

D.7 GENERATION OF PSEUDO-RANDOM CURVES (BINARY CASE)......................................................... 111

D.8 VERIFICATION OF CURVE PSEUDO-RANDOMNESS (BINARY CASE).................................................. 111

D.9 POLYNOMIAL BASIS TO NORMAL BASIS CONVERSION .................................................................... 112

D.10 NORMAL BASIS TO POLYNOMIAL BASIS CONVERSION .................................................................... 113

APPENDIX E: A PROOF THAT V = R IN THE DSA................................................................................ 115

APPENDIX F: CALCULATING THE REQUIRED NUMBER OF ROUNDS OF TESTING USING THE 

MILLER-RABIN PROBABILISTIC PRIMALITY TEST ...................................................................... 117

F.1 THE REQUIRED NUMBER OF ROUNDS OF THE MILLER-RABIN PRIMALITY TESTS.............................. 117

F.2 GENERATING DSA PRIMES.......................................................................................................... 118

F.3 GENERATING PRIMES FOR RSA SIGNATURES .............................................................................. 119

APPENDIX G: REFERENCES ................................................................................................................. 121
1

Federal Information Processing Standards Publication 186-4

July 2013

Specifications for the

DIGITAL SIGNATURE STANDARD (DSS)

1. Introduction

This Standard defines methods for digital signature generation that can be used for the protection 

of binary data (commonly called a message), and for the verification and validation of those 

digital signatures. Three techniques are approved.

(1) The Digital Signature Algorithm (DSA) is specified in this Standard. The specification 

includes criteria for the generation of domain parameters, for the generation of public and 

private key pairs, and for the generation and verification of digital signatures.

(2) The RSA digital signature algorithm is specified in American National Standard (ANS) 

X9.31 and Public Key Cryptography Standard (PKCS) #1. FIPS 186-4 approves the use 

of implementations of either or both of these standards and specifies additional 

requirements.

(3) The Elliptic Curve Digital Signature Algorithm (ECDSA) is specified in ANS X9.62. 

FIPS 186-4 approves the use of ECDSA and specifies additional requirements. 

Recommended elliptic curves for Federal Government use are provided herein.

This Standard includes requirements for obtaining the assurances necessary for valid digital 

signatures. Methods for obtaining these assurances are provided in NIST Special Publication 

(SP) 800-89, Recommendation for Obtaining Assurances for Digital Signature Applications.
2

2. Glossary of Terms, Acronyms and Mathematical Symbols

2.1 Terms and Definitions

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

that is either 1) specified in a FIPS or NIST Recommendation, or 2) 

adopted in a FIPS or NIST Recommendation or 3) specified in a list of 

NIST approved security functions.

Assurance of domain 

parameter validity

Confidence that the domain parameters are arithmetically correct.

Assurance of 

possession

Confidence that an entity possesses a private key and any associated 

keying material.

Assurance of public 

key validity

Confidence that the public key is arithmetically correct.

Bit string An ordered sequence of 0’s and 1’s. The leftmost bit is the most 

significant bit of the string. The rightmost bit is the least significant bit 

of the string.

Certificate A set of data that uniquely identifies a key pair and an owner that is 

authorized to use the key pair. The certificate contains the owner’s 

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

Certification Authority (i.e., a trusted party), thereby binding the 

public key to the owner. 

Certification Authority 

(CA)

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

issuing certificates and exacting compliance with a PKI policy.

Claimed signatory From the verifier’s perspective, the claimed signatory is the entity that 

purportedly generated a digital signature. 

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

properly implemented, provides a mechanism for verifying origin 

authentication, data integrity and signatory non-repudiation.

Domain parameter seed A string of bits that is used as input for a domain parameter generation 

or validation process. 

Domain parameters Parameters used with cryptographic algorithms that are usually 

common to a domain of users. A DSA or ECDSA cryptographic key 

pair is associated with a specifc set of domain parameters.
3

Entity An individual (person), organization, device or process. Used 

interchangeably with “party”.

Equivalent process Two processes are equivalent if, when the same values are input to 

each process (either as input parameters or as values made available 

during the process or both), the same output is produced.

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

bit string. Approved hash functions are specified in FIPS 180 and are 

designed to satisfy the following properties:

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

maps to any new pre-specified output, and

2. (Collision resistant) It is computationally infeasible to find any 

two distinct inputs that map to the same output.

Hash value See “message digest”.

Intended signatory An entity that intends to generate digital signatures in the future. 

Key A parameter used in conjunction with a cryptographic algorithm that 

determines its operation. Examples applicable to this Standard 

include:

1. The computation of a digital signature from data, and

2. The verification of a digital signature.

Key pair A public key and its corresponding private key.

Message The data that is signed. Also known as “signed data” during the 

signature verification and validation process.

Message digest The result of applying a hash function to a message. Also known as 

“hash value”.

Non-repudiation A service that is used to provide assurance of the integrity and origin 

of data in such a way that the integrity and origin can be verified and 

validated by a third party as having originated from a specific entity in 

possession of the private key (i.e., the signatory).

Owner A key pair owner is the entity that is authorized to use the private key 

of a key pair. 

Party An individual (person), organization, device or process. Used 

interchangeably with “entity”.

Per-message secret 

number

A secret random number that is generated prior to the generation of 

each digital signature.
4

Public Key 

Infrastructure (PKI)

A framework that is established to issue, maintain and revoke public 

key certificates.

Prime number 

generation seed

A string of random bits that is used to determine a prime number with 

the required characteristics. 

Private key A cryptographic key that is used with an asymmetric (public key) 

cryptographic algorithm. For digital signatures, the private key is 

uniquely associated with the owner and is not made public. The 

private key is used to compute a digital signature that may be verified 

using the corresponding public key. 

Probable prime An integer that is believed to be prime, based on a probabilistic 

primality test. There should be no more than a negligible probability 

that the so-called probable prime is actually composite. 

Provable prime An integer that is either constructed to be prime or is calculated to be 

prime using a primality-proving algorithm.

Pseudorandom A process or data produced by a process is said to be pseudorandom 

when the outcome is deterministic, yet also effectively random as long 

as the internal action of the process is hidden from observation. For 

cryptographic purposes, “effectively” means “within the limits of the 

intended security strength.”

Public key A cryptographic key that is used with an asymmetric (public key) 

cryptographic algorithm and is associated with a private key. The 

public key is associated with an owner and may be made public. In the 

case of digital signatures, the public key is used to verify a digital 

signature that was signed using the corresponding private key.

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

operations) that is required to break a cryptographic algorithm or 

system. Sometimes referred to as a security level.

Shall Used to indicate a requirement of this Standard.

Should Used to indicate a strong recommendation, but not a requirement of 

this Standard. 

Signatory The entity that generates a digital signature on data using a private 

key.

Signature generation The process of using a digital signature algorithm and a private key to 

generate a digital signature on data.
5

Signature validation The (mathematical) verification of the digital signature and obtaining 

the appropriate assurances (e.g., public key validity, private key 

possession, etc.).

Signature verification The process of using a digital signature algorithm and a public key to 

verify a digital signature on data.

Signed data The data or message upon which a digital signature has been 

computed. Also, see “message”.

Subscriber An entity that has applied for and received a certificate from a 

Certificate Authority.

Trusted third party 

(TTP)

An entity other than the owner and verifier that is trusted by the owner 

or the verifier or both. Sometimes shortened to “trusted party”.

Verifier The entity that verifies the authenticity of a digital signature using the 

public key.

2.2 Acronyms

ANS American National Standard.

CA Certification Authority.

DSA Digital Signature Algorithm; specified in this Standard. 

ECDSA Elliptic Curve Digital Signature Algorithm; specified in ANS X9.62.

FIPS Federal Information Processing Standard.

NIST National Institute of Standards and Technology.

PKCS Public Key Cryptography Standard.

PKI Public Key Infrastructure.

RBG Random Bit Generator.

RSA Algorithm developed by Rivest, Shamir and Adleman; specified in 

ANS X9.31 and PKCS #1.

SHA Secure Hash Algorithm; specified in FIPS 180.

SP NIST Special Publication

TTP Trusted Third Party.
6

2.3 Mathematical Symbols

a mod n The unique remainder r, 0 ≤ r ≤ (n – 1), when integer a is divided by

the positive integer n. For example, 23 mod 7 = 2.

b ≡ a mod n There exists an integer k such that b – a = kn; equivalently, a mod n

= b mod n.

counter The counter value that results from the domain parameter generation 

process when the domain parameter seed is used to generate DSA 

domain parameters.

d 1. For RSA, the private signature exponent of a private key.

2. For ECDSA, the private key.

domain_parameter_seed A seed used for the generation of domain parameters.

e The public verification exponent of an RSA public key.

g One of the DSA domain parameters; g is a generator of the q-order 

cyclic group of GF(p)*; that is, an element of order q in the 

multiplicative group of GF(p).

GCD (a, b) Greatest common divisor of the integers a and b.

Hash (M) The result of a hash computation (message digest or hash value) on 

message M using an approved hash function.

index A value used in the generation of g to indicate its intended use (e.g., 

for digital signatures).

k For DSA and ECDSA, a per-message secret number.

L For DSA, the length of the parameter p in bits.

(L, N) The associated pair of length parameters for a DSA key pair, where L

is the length of p, and N is the length of q.

LCM (a, b) The least common multiple of the integers a and b.

len (a) The length of a in bits; the integer L, where 2L-1 ≤ a < 2L

.

M The message that is signed using the digital signature algorithm. 

m For ECDSA, the degree of the finite field GF m2 .

N For DSA, the length of the parameter q in bits.
7

n 1. For RSA, the modulus; the bit length of n is considered to be the 

key size.

2. For ECDSA, the order of the base point of the elliptic curve; the 

bit length of n is considered to be the key size.

(n, d) An RSA private key, where n is the modulus, and d is the private 

signature exponent.

(n, e) An RSA public key, where n is the modulus, and e is the public 

verification exponent.

nlen The length of the RSA modulus n in bits.

p 1. For DSA, one of the DSA domain parameters; a prime number 

that defines the Galois Field GF(p) and is used as a modulus in the 

operations of GF(p).

2. For RSA, a prime factor of the modulus n.

q 1. For DSA, one of the DSA domain parameters; a prime factor of 

p – 1.

2. For RSA, a prime factor of the modulus n.

Q An ECDSA public key.

r One component of a DSA or ECDSA digital signature. See the 

definition of (r, s).

(r, s) A DSA or ECDSA digital signature, where r and s are the digital 

signature components.

s One component of a DSA or ECDSA digital signature. See the 

definition of (r, s).

seedlen The length of the domain_parameter_seed in bits.

SHAx(M) The result when M is the input to the SHA-x hash function, where 

SHA-x is specified in FIPS 180. 

x The DSA private key.

y The DSA public key.

⊕ Bitwise logical “exclusive-or” on bit strings of the same length; for 

corresponding bits of each bit string, the result is determined as 

follows: 0 ⊕ 0 = 0, 0 ⊕ 1 = 1, 1 ⊕ 0 = 1, or 1 ⊕ 1 = 0. 

Example: 01101 ⊕ 11010 = 10111

+ Addition.
8

∗ Multiplication.

/ Division.

a || b The concatenation of two strings a and b. Either a and b are both bit 

strings, or both are byte strings. 

a The ceiling of a: the smallest integer that is greater than or equal to 

a. For example, 5 = 5, 5.3 = 6, and –2.1 = –2. 

a The floor of a; the largest integer that is less than or equal to a. For 

example, 5 = 5, 5.3 = 5, and -2.1. = -3.

|a| The absolute value of a; |a| is – a if a < 0; otherwise, it is simply a. 

For example, |2| = 2, and |–2| = 2.

[a, b] The interval of integers between and including a and b. For example, 

[1, 4] consists of the integers 1, 2, 3 and 4.

{, a, b, ...} Used to indicate optional information.

0x The prefix to a bit string that is represented in hexadecimal characters.
9

3. General Discussion

A digital signature is an electronic analogue of a written signature; the digital signature can be 

used to provide assurance that the claimed signatory signed the information. In addition, a 

digital signature may be used to detect whether or not the information was modified after it was 

signed (i.e., to detect the integrity of the signed data). These assurances may be obtained whether 

the data was received in a transmission or retrieved from storage. A properly implemented 

digital signature algorithm that meets the requirements of this Standard can provide these 

services.

A digital signature algorithm includes a signature generation process and a signature verification 

process. A signatory uses the generation process to generate a digital signature on data; a verifier 

uses the verification process to verify the authenticity of the signature. Each signatory has a 

public and private key and is the owner of that key pair. As shown in Figure 1, the private key is 

used in the signature generation process. The key pair owner is the only entity that is authorized 

to use the private key to generate digital signatures. In order to prevent other entities from 

claiming to be the key pair owner and using the private key to generate fraudulent signatures, the 

Private

Key

Public

Signature Key

Generation

Message/Data

Signature 

Verification

Message/Data

Hash Function Hash Function

Message Digest Message Digest

Signature Generation Signature Verification

Signature

Valid or

Invalid

Figure 1: Digital Signature Processes
10

private key must remain secret. The approved digital signature algorithms are designed to 

prevent an adversary who does not know the signatory’s private key from generating the same 

signature as the signatory on a different message. In other words, signatures are designed so that 

they cannot be forged. A number of alternative terms are used in this Standard to refer to the 

signatory or key pair owner. An entity that intends to generate digital signatures in the future 

may be referred to as the intended signatory. Prior to the verification of a signed message, the 

signatory is referred to as the claimed signatory until such time as adequate assurance can be 

obtained of the actual identity of the signatory. 

The public key is used in the signature verification process (see Figure 1). The public key need 

not be kept secret, but its integrity must be maintained. Anyone can verify a correctly signed 

message using the public key.

For both the signature generation and verification processes, the message (i.e., the signed data) is 

converted to a fixed-length representation of the message by means of an approved hash 

function. Both the original message and the digital signature are made available to a verifier. 

A verifier requires assurance that the public key to be used to verify a signature belongs to the 

entity that claims to have generated a digital signature (i.e., the claimed signatory). That is, a 

verifier requires assurance that the signatory is the actual owner of the public/private key pair 

used to generate and verify a digital signature. A binding of an owner’s identity and the owner’s 

public key shall be effected in order to provide this assurance. 

A verifier also requires assurance that the key pair owner actually possesses the private key

associated with the public key, and that the public key is a mathematically correct key.

By obtaining these assurances, the verifier has assurance that if the digital signature can be 

correctly verified using the public key, the digital signature is valid (i.e., the key pair owner 

really signed the message). Digital signature validation includes both the (mathematical) 

verification of the digital signature and obtaining the appropriate assurances. The following are 

reasons why such assurances are required.

1. If a verifier does not obtain assurance that a signatory is the actual owner of the key pair 

whose public component is used to verify a signature, the problem of forging a signature 

is reduced to the problem of falsely claiming an identity. For example, anyone in 

possession of a mathematically consistent key pair can sign a message and claim that the 

signatory was the President of the United States. If a verifier does not require assurance 

that the President is actually the owner of the public key that is used to mathematically 

verify the message’s signature, then successful signature verification provides assurance 

that the message has not been altered since it was signed, but does not provide assurance 

that the message came from the President (i.e., the verifier has assurance of the data’s 

integrity, but source authentication is lacking).

2. If the public key used to verify a signature is not mathematically valid, the arguments 

used to establish the cryptographic strength of the signature algorithm may not apply. 
11

The owner may not be the only party who can generate signatures that can be verified 

with that public key. 

3. If a public key infrastructure cannot provide assurance to a verifier that the owner of a 

key pair has demonstrated knowledge of a private key that corresponds to the owner’s 

public key, then it may be possible for an unscrupulous entity to have their identity (or an 

assumed identity) bound to a public key that is (or has been) used by another party. The 

unscrupulous entity may then claim 

to be the source of certain messages 

signed by that other party. Or, it 

may be possible that an 

unscrupulous entity has managed to 

obtain ownership of a public key 

that was chosen with the sole 

purpose of allowing for the 

verification of a signature on a 

specific message.

Technically, a key pair used by a digital 

signature algorithm could also be used for 

purposes other than digital signatures (e.g., 

for key establishment). However, a key pair 

used for digital signature generation and 

verification as specified in this Standard 

shall not be used for any other purpose. See

SP 800-57 on Key Usage for further 

information.

A number of steps are required to enable a 

digital signature generation or verification 

capability in accordance with this Standard. 

All parties that generate digital signatures 

shall perform the initial setup process as 

discussed in Section 3.1. Digital signature 

generation shall be performed as discussed 

in Section 3.2. Digital signature verification

and validation shall be performed as 

discussed in Section 3.3.

3.1 Initial Setup

Figure 2 depicts the steps that are

performed prior to generating a digital 

Obtain

Assurance of Possession

of the 

DS Private Key

Obtain

Assurance of

Public Key Validity

Obtain

DS Key Pair

Obtain Assurance of

Domain Parameter 

Validity

Obtain

Domain Parameters

Intended Signatory Ready for 

Generating Digital Signatures

Intended

signatory

OR

another

entity

generates

Intended

signatory

OR 

a TTP

generates

Register the Public Key 

and Identity with a 

TTP

Optional

DSA

and 

ECDSA

Figure 2: Initial Setup by an Intended 

Signatory
12

signature by an entity intending to act as a signatory. 

For the DSA and ECDSA algorithms, the intended signatory shall first obtain appropriate 

domain parameters, either by generating the domain parameters itself, or by obtaining domain 

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

intended signatory shall obtain assurance of the validity of those domain parameters; approved 

methods for obtaining this assurance are provided in SP 800-89. Note that the RSA algorithm 

does not use domain parameters.

Each intended signatory shall obtain a digital signature key pair that is generated as specified for 

the appropriate digital signature algorithm, either by generating the key pair itself or by 

obtaining the key pair from a trusted party. The intended signatory is authorized to use the key 

pair and is the owner of that key pair. Note that if a trusted party generates the key pair, that 

party needs to be trusted not to masquerade as the owner, even though the trusted party knows 

the private key.

After obtaining the key pair, the intended signatory (now the key pair owner) shall obtain (1) 

assurance of the validity of the public key and (2) assurance that he/she actually possesses the 

associated private key. Approved methods for obtaining these assurances are provided in SP 

800-89.

A digital signature verifier requires assurance of the identity of the signatory. Depending on the 

environment in which digital signatures are generated and verified, the key pair owner (i.e., the 

intended signatory) may register the public key and establish proof of identity with a mutually 

trusted party. For example, a certification authority (CA) could sign credentials containing an 

owner’s public key and identity to form a certificate after being provided with proof of the 

owner’s identity. Systems for certifying 

credentials and distributing certificates are 

beyond the scope of this Standard. Other means 

of establishing proof of identity (e.g., by 

providing identity credentials along with the 

public key directly to a prospective verifier) are 

allowed.

3.2 Digital Signature Generation

Figure 3 depicts the steps that are performed by 

an intended signatory (i.e., the entity that 

generates a digital signature). 

Prior to the generation of a digital signature, a 

message digest shall be generated on the 

information to be signed using an appropriate 

approved hash function. 

Generate a Digital Signature

Obtain Additional 

Information for the 

Digital Signature Process

Digital Signature Generation Complete

DSA 

and 

ECDSA

Generate a Message Digest

Optional Verify the Digital Signature

Figure 3: Digital Signature Generation
13

Depending on the digital signature algorithm to be used, additional information shall be 

obtained. For example, a random per-message secret number shall be obtained for DSA and 

ECDSA.

Using the selected digital signature algorithm, the signature private key, the message digest, and 

any other information required by the digital signature process, a digital signature shall be 

generated in accordance with this Standard.

The signatory may optionally verify the digital signature using the signature verification process

and the associated public key. This optional verification serves as a final check to detect 

otherwise undetected signature generation computation errors; this verification may be prudent 

when signing a high-value message, when multiple users are expected to verify the signature, or 

if the verifier will be verifying the signature at a much later time.

3.3 Digital Signature Verification and Validation

Figure 4 depicts the digital signature verification and validation process that are performed by a 

verifier (e.g., the intended recipient of the signed data and associated digital signature). Note that 

the figure depicts a successful verification and validation process (i.e., no errors are detected).

In order to verify a digital signature, the verifier shall obtain the public key of the claimed 

signatory, (usually) based on the claimed identity. If DSA or ECDSA has been used to generate 

the digital signature, the verifier shall also obtain the domain parameters. The public key and 

domain parameters may be obtained, for example, from a certificate created by a trusted party 

(e.g., a CA) or directly from the claimed signatory. A message digest shall be generated on the 

data whose signature is to be verified (i.e., not on the received digital signature) using the same 

hash function that was used during the digital signature generation process. Using the 

appropriate digital signature algorithm, the domain parameters (if appropriate), the public key 

and the newly computed message digest, the received digital signature is verified in accordance 

with this Standard. If the verification process fails, no inference can be made as to whether the 

data is correct, only that in using the specified public key and the specified signature format, the 

digital signature cannot be verified for that data.

Before accepting the verified digital signature as valid, the verifier shall have (1) assurance of 

the signatory’s claimed identity, (2) assurance of the validity of the domain parameters (for DSA 

and ECDSA), (3) assurance of the validity of the public key, and (4) assurance that the claimed 

signatory actually possessed the private key that was used to generate the digital signature at the 

time that the signature was generated. Methods for the verifier to obtain these assurances are 

provided in SP 800-89. Note that assurance of domain parameter validity may have been 

obtained during initial setup (see Section 3.1).

If the verification and assurance processes are successful, the digital signature and signed data 

shall be considered valid. However, if a verification or assurance process fails, the digital 

signature should be considered invalid. An organization’s policy shall govern the action to be 

taken for an invalid digital signature. Such policy is outside the scope of this Standard.
14

Obtain the Domain 

Parameters and Public Key

Get the Claimed 

Signatory’s Identifier

Generate a Message Digest

Verify the Digital Signature

Obtain Assurance of the Claimed 

Signatory’s Identity

Obtain Assurance of Domain 

Parameter Validity

Obtain Assurance of the Validity 

of the Owner’s Public Key

Obtain Assurance that the Owner 

Possesses the Private Key

Digital Signature Validation Complete

Actions Assurances

Figure 4: Digital Signature Verification and Validation
15

4 The Digital Signature Algorithm (DSA)

4.1 DSA Parameters

A DSA digital signature is computed using a set of domain parameters, a private key x, a permessage secret number k, data to be signed, and a hash function. A digital signature is verified 

using the same domain parameters, a public key y that is mathematically associated with the 

private key x used to generate the digital signature, data to be verified, and the same hash 

function that was used during signature generation. These parameters are defined as follows:

p a prime modulus, where 2L–1 < p < 2L, and L is the bit length of p. Values for L are 

provided in Section 4.2.

q a prime divisor of (p – 1), where 2N–1 < q < 2 N, and N is the bit length of q. Values for N

are provided in Section 4.2.

g a generator of a subgroup of order q in the multiplicative group of GF(p), such that 1 < g 

< p.

x the private key that must remain secret; x is a randomly or pseudorandomly generated 

integer, such that 0 < x < q, i.e., x is in the range [1, q–1].

y the public key, where y = gx mod p.

k a secret number that is unique to each message; k is a randomly or pseudorandomly 

generated integer, such that 0 < k < q, i.e., k is in the range [1, q–1].

4.2 Selection of Parameter Sizes and Hash Functions for DSA

This Standard specifies the following choices for the pair L and N (the bit lengths of p and q, 

respectively):

L = 1024, N = 160

L = 2048, N = 224

L = 2048, N = 256

L = 3072, N = 256

Federal Government entities shall generate digital signatures using use one or more of these 

choices.

An approved hash function, as specified in FIPS 180, shall be used during the generation of 

digital signatures. The security strength associated with the DSA digital signature process is no 

greater than the minimum of the security strength of the (L, N) pair and the security strength of 

the hash function that is employed. Both the security strength of the hash function used and the 

security strength of the (L, N) pair shall meet or exceed the security strength required for the 
16

digital signature process. The security strength for each (L, N) pair and hash function is provided 

in SP 800-57. 

SP 800-57 provides information about the selection of the appropriate (L, N) pair in accordance 

with a desired security strength for a given time period for the generation of digital signatures. 

An (L, N) pair shall be chosen that protects the signed information during the entire expected 

lifetime of that information. For example, if a digital signature is generated in 2009 for 

information that needs to be protected for five years, and a particular (L, N) pair is invalid after 

2010, then a larger (L, N) pair shall be used that remains valid for the entire period of time that 

the information needs to be protected.

It is recommended that the security strength of the (L, N) pair and the security strength of the 

hash function used for the generation of digital signatures be the same unless an agreement has 

been made between participating entities to use a stronger hash function. When the length of the 

output of the hash function is greater than N (i.e., the bit length of q), then the leftmost N bits of 

the hash function output block shall be used in any calculation using the hash function output 

during the generation or verification of a digital signature. A hash function that provides a lower 

security strength than the (L, N) pair ordinarily should not be used, since this would reduce the 

security strength of the digital signature process to a level no greater than that provided by the 

hash function.

A Federal Government entity other than a Certification Authority (CA) should use only the first 

three (L, N) pairs (i.e., the (1024, 160), (2048, 224) and (2048, 256) pairs). A CA shall use an (L, 

N) pair that is equal to or greater than the (L, N) pairs used by its subscribers. For example, if 

subscribers are using the (2048, 224) pair, then the CA shall use either the (2048, 224), (2048, 

256) or (3072, 256) pair. Possible exceptions to this rule include cross certification between 

CAs, certifying keys for purposes other than digital signatures and transitioning from one key 

size or algorithm to another. See SP 800-57 for further guidance.

4.3 DSA Domain Parameters

DSA requires that the private/public key pairs used for digital signature generation and 

verification be generated with respect to a particular set of domain parameters. These domain 

parameters may be common to a group of users and may be public. A user of a set of domain 

parameters (i.e., both the signatory and the verifier) shall have assurance of their validity prior to 

using them (see Section 3). Although domain parameters may be public information, they shall

be managed so that the correct correspondence between a given key pair and its set of domain 

parameters is maintained for all parties that use the key pair. A set of domain parameters may

remain fixed for an extended time period. 

The domain parameters for DSA are the integers p, q, and g, and optionally, the 

domain_parameter_seed and counter that were used to generate p and q (i.e., the full set of 

domain parameters is (p, q, g {, domain_parameter_seed, counter})). 
17

4.3.1 Domain Parameter Generation

Domain parameters may be generated by a trusted third party (a TTP, such as a CA) or by an 

entity other than a TTP. Assurance of domain parameter validity shall be obtained prior to key 

pair generation, digital signature generation or digital signature verification (see Section 3).

The integers p and q shall be generated as specified in Appendix A.1. The input to the 

generation process is the selected values of L and N (see Section 4.2); the output of the 

generation process is the values for p and q, and optionally, the values of the 

domain_parameter_seed and counter. 

The generator g shall be generated as specified in Appendix A.2.

The security strength of a hash function used during the generation of the domain parameters

shall meet or exceed the security strength associated with the (L, N) pair. Note that this is more 

restrictive than the hash function that can be used for the digital signature process (see Section 

4.2).

4.3.2 Domain Parameter Management

Each digital signature key pair shall be correctly associated with one specific set of domain 

parameters (e.g., by a public key certificate that identifies the domain parameters associated with 

the public key). The domain parameters shall be protected from unauthorized modification until 

the set is deactivated (if and when the set is no longer needed). The same domain parameters 

may be used for more than one purpose (e.g., the same domain parameters may be used for both 

digital signatures and key establishment). However, using different values for the generator g

reduces the risk that key pairs generated for one purpose could be accidentally used 

(successfully) for another purpose.

4.4 Key Pairs

Each signatory has a key pair: a private key x and a public key y that are mathematically related 

to each other. The private key shall be used for only a fixed period of time (i.e., the private key 

cryptoperiod) in which digital signatures may be generated; the public key may continue to be 

used as long as digital signatures that were generated using the associated private key need to be 

verified (i.e., the public key may continue to be used beyond the cryptoperiod of the associated 

private key). See SP 800-57 for further guidance.

4.4.1 DSA Key Pair Generation

A digital signature key pair x and y is generated for a set of domain parameters (p, q, g {, 

domain_parameter_seed, counter}). Methods for the generation of x and y are provided in 

Appendix B.1.
18

4.4.2 Key Pair Management

Guidance on the protection of key pairs is provided in SP 800-57. The secure use of digital 

signatures depends on the management of an entity’s digital signature key pair as follows:

1. The validity of the domain parameters shall be assured prior to the generation of the key 

pair, or the verification and validation of a digital signature (see Section 3).

2. Each key pair shall be associated with the domain parameters under which the key pair 

was generated. 

3. A key pair shall only be used to generate and verify signatures using the domain 

parameters associated with that key pair.

4. The private key shall be used only for signature generation as specified in this Standard 

and shall be kept secret; the public key shall be used only for signature verification and 

may be made public.

5. An intended signatory shall have assurance of possession of the private key prior to or 

concurrently with using it to generate a digital signature (see Section 3.1).

6. A private key shall be protected from unauthorized access, disclosure and modification.

7. A public key shall be protected from unauthorized modification (including substitution). 

For example, public key certificates that are signed by a CA may provide such protection.

8. A verifier shall be assured of a binding between the public key, its associated domain 

parameters and the key pair owner (see Section 3).

9. A verifier shall obtain public keys in a trusted manner (e.g., from a certificate signed by a 

CA that the entity trusts, or directly from the intended or claimed signatory, provided that 

the entity is trusted by the verifier and can be authenticated as the source of the signed 

information that is to be verified). 

10. Verifiers shall be assured that the claimed signatory is the key pair owner, and that the 

owner possessed the private key that was used to generate the digital signature at the time 

that the signature was generated (i.e., the private key that is associated with the public 

key that will be used to verify the digital signature) (see Section 3.3).

11. A signatory and a verifier shall have assurance of the validity of the public key (see 

Sections 3.1 and 3.3).

4.5 DSA Per-Message Secret Number

A new secret random number k shall be generated prior to the generation of each digital 

signature for use during the signature generation process. This secret number shall be protected 

from unauthorized disclosure and modification.

−1 k is the multiplicative inverse of k with respect to multiplication modulo q; i.e., 0 < −1 k < q
19

and 1 = ( −1 k k) mod q. This inverse is required for the signature generation process (see Section 

4.6). A technique is provided in Appendix C.1 for deriving −1 k from k.

k and −1 k may be pre-computed, since knowledge of the message to be signed is not required for 

the computations. When k and −1 k are pre-computed, their confidentiality and integrity shall be 

protected. 

Methods for the generation of the per-message secret number are provided in Appendix B.2.

4.6 DSA Signature Generation

The intended signatory shall have assurances as specified in Section 3.1.

Let N be the bit length of q. Let min(N, outlen) denote the minimum of the positive integers N

and outlen, where outlen is the bit length of the hash function output block. 

The signature of a message M consists of the pair of numbers r and s that is computed according 

to the following equations:

r = (g

k mod p) mod q.

z = the leftmost min(N, outlen) bits of Hash(M).

s = ( −1 k (z + xr)) mod q.

When computing s, the string z obtained from Hash(M) shall be converted to an integer. The 

conversion rule is provided in Appendix C.2. 

Note that r may be computed whenever k, p, q and g are available, e.g., whenever the domain 

parameters p, q and g are known, and k has been pre-computed (see Section 4.5), r may also be 

pre-computed, since knowledge of the message to be signed is not required for the computation 

of r. Pre-computed k, k

-1 and r values shall be protected in the same manner as the the private 

key x until s has been computed (see SP 800-57).

The values of r and s shall be checked to determine if r = 0 or s = 0. If either r = 0 or s = 0, a 

new value of k shall be generated, and the signature shall be recalculated. It is extremely 

unlikely that r = 0 or s = 0 if signatures are generated properly.

The signature (r, s) may be transmitted along with the message to the verifier.

4.7 DSA Signature Verification and Validation

Signature verification may be performed by any party (i.e., the signatory, the intended recipient 

or any other party) using the signatory’s public key. A signatory may wish to verify that the 

computed signature is correct, perhaps before sending the signed message to the intended 
20

recipient. The intended recipient (or any other party) verifies the signature to determine its 

authenticity.

Prior to verifying the signature of a signed message, the domain parameters, and the claimed 

signatory’s public key and identity shall be made available to the verifier in an authenticated 

manner. The public key may, for example, be obtained in the form of a certificate signed by a 

trusted entity (e.g., a CA) or in a face-to-face meeting with the public key owner. 

Let M ′, r′, and s′ be the received versions of M, r, and s, respectively; let y be the public key of 

the claimed signatory; and let N be the bit length of q. Also, let min(N, outlen) denote the 

minimum of the positive integers N and outlen, where outlen is the bit length of the hash 

function output block.

The signature verification process is as follows:

1. The verifier shall check that 0 < r′ < q and 0 < s′ < q; if either condition is violated, the 

signature shall be rejected as invalid. 

2. If the two conditions in step 1 are satisfied, the verifier computes the following:

w = (s′)

–1 mod q.

z = the leftmost min(N, outlen) bits of Hash(M′ ).

u1 = (zw) mod q.

u2 = ((r′)w) mod q.

v = (((g)

u1 (y)u2

) mod p) mod q.

A technique is provided in Appendix C.1 for deriving (s′ )–1 (i.e., the multiplicative 

inverse of s′ mod q).

The string z obtained from Hash(M′) shall be converted to an integer. The conversion 

rule is provided in Appendix C.2.

3. If v = r′, then the signature is verified. For a proof that v = r′ when M′ = M, r′ = r, and s′

= s, see Appendix E.

4. If v does not equal r′, then the message or the signature may have been modified, there 

may have been an error in the signatory’s generation process, or an imposter (who did not 

know the private key associated with the public key of the claimed signatory) may have 

attempted to forge the signature. The signature shall be considered invalid. No inference 

can be made as to whether the data is valid, only that when using the public key to verify 

the signature, the signature is incorrect for that data. 

5. Prior to accepting the signature as valid, the verifier shall have assurances as specified in 

Section 3.3.
21

An organization’s policy may govern the action to be taken for invalid digital signatures. Such 

policy is outside the scope of this Standard. Guidance about determining the timeliness of 

digitally signed messages is addressed in SP 800-102, Recommendation for Digital Signature 

Timeliness. 
22

5. The RSA Digital Signature Algorithm

The use of the RSA algorithm for digital signature generation and verification is specified in 

American National Standard (ANS) X9.31 and Public Key Cryptography Standard (PKCS) #1. 

While each of these standards uses the RSA algorithm, the format of the ANS X9.31 and PKCS 

#1 data on which the digital signature is generated differs in details that make the algorithms

non-interchangeable.

5.1 RSA Key Pair Generation

An RSA digital signature key pair consists of an RSA private key, which is used to compute a 

digital signature, and an RSA public key, which is used to verify a digital signature. An RSA key 

pair used for digital signatures shall only be used for one digital signature scheme (e.g., ANS 

X9.31, RSASSA-PKCS1 v1.5 or RSASSA-PSS; see Sections 5.4 and 5.5). In addition, an RSA 

digital signature key pair shall not be used for other purposes (e.g., key establishment).

An RSA public key consists of a modulus n, which is the product of two positive prime integers 

p and q (i.e., n = pq), and a public key exponent e. Thus, the RSA public key is the pair of values 

(n, e) and is used to verify digital signatures. The size of an RSA key pair is commonly 

considered to be the length of the modulus n in bits (nlen). 

The corresponding RSA private key consists of the same modulus n and a private key exponent d

that depends on n and the public key exponent e. Thus, the RSA private key is the pair of values 

(n, d) and is used to generate digital signatures. Note that an alternative method for representing 

(n, d) using the Chinese Remainder Theorem (CRT) is allowed.

In order to provide security for the digital signature process, the two integers p and q, and the 

private key exponent d shall be kept secret. The modulus n and the public key exponent e may 

be made known to anyone. Guidance on the protection of these values is provided in SP 800-57.

This Standard specifies three choices for the length of the modulus (i.e., nlen): 1024, 2048 and 

3072 bits. Federal Government entities shall generate digital signatures using one or more of 

these choices. 

An approved hash function, as specified in FIPS 180, shall be used during the generation of key 

pairs and digital signatures. When used during the generation of an RSA key pair (as specified in 

this Standard), the length in bits of the hash function output block shall meet or exceed the 

security strength associated with the bit length of the modulus n (see SP 800-57).

The security strength associated with the RSA digital signature process is no greater than the 

minimum of the security strength associated with the bit length of the modulus and the security 

strength of the hash function that is employed. The security strength for each modulus length and 

hash function used during the digital signature process is provided in SP 800-57. Both the 

security strength of the hash function used and the security strength associated with the bit length 
23

of the modulus n shall meet or exceed the security strength required for the digital signature 

process.

It is recommended that the security strength of the modulus and the security strength of the hash 

function be the same unless an agreement has been made between participating entities to use a 

stronger hash function. A hash function that provides a lower security strength than the security 

strength associated with the bit length of the modulus ordinarily should not be used, since this 

would reduce the security strength of the digital signature process to a level no greater than that 

provided by the hash function. 

Federal Government entities other than CAs should use only the first two choices (i.e., nlen = 

1024 or 2048) during the timeframes indicated in SP 800-57. A CA should use a modulus whose 

length nlen is equal to or greater than the moduli used by its subscribers. For example, if the 

subscribers are using nlen = 2048, then the CA should use nlen ≥ 2048. SP 800-57 provides 

further information about the selection of the bit length of n. Possible exceptions to this rule 

include cross certification between CAs, certifying keys for purposes other than digital 

signatures and transitioning from one key size or algorithm to another.

Criteria for the generation of RSA key pairs are provided in Appendix B.3.1.

When RSA parameters are randomly generated (i.e., the primes p and q, and optionally, the 

public key exponent e), they shall be generated using an approved random bit generator. The 

(pseudo) random bits produced by the random bit generator shall be used as seeds for generating 

RSA parameters (e.g., the (pseudo) random number is used as a prime number generation seed). 

Prime number generation seeds shall be kept secret or destroyed when the modulus n is 

computed. If any prime number generation seed is retained (e.g., to regenerate the RSA modulus 

n, or as evidence that the generated prime factors p and q were generated in compliance with this 

Standard, then the seed shall be kept secret and shall be protected. The strength of this 

protection shall be (at least) equivalent to the protection required for the associated private key.

5.2 Key Pair Management

The secure use of digital signatures depends on the management of an entity’s digital signature 

key pair. Key pair management requirements for RSA are provided in Section 4.4.2, 

requirements 4 – 11. Note that the first three requirements in Section 4.4.2, which address the 

relationship between domain parameters and key pairs, are not applicable to RSA.

5.3 Assurances

The intended signatory shall have assurances as specified in Section 3.1. Prior to accepting a digital 

signature as valid, the verifier shall have assurances as specified in Section 3.3.
24

5.4 ANS X9.31

ANS X9.31, Digital Signatures Using Reversible Public Key Cryptography for the Financial 

Services Industry (rDSA), was developed for the American National Standards Institute by the 

Accredited Standards Committee on Financial Services, X9. See http://www.x9.org for 

information about obtaining copies of ANS X9.31 and any associated errata. The following 

discussions are based on the version of ANS X9.31 that was approved in 1998. 

Methods for the generation of the private prime factors p and q are provided in Appendix B.3.

In ANS X9.31, the length of the modulus n is allowed in increments of 256 bits beyond a

minimum of 1024 bits. Implementations claiming conformance to FIPS 186-4 shall include one 

or more of the modulus sizes specified in Section 5.1.

Two methods for the generation of digital signatures are included in ANS X9.31. When the 

public signature verification exponent e is odd, the digital signature algorithm is commonly 

known as RSA; when the public signature verification exponent e is even, the digital signature 

algorithm is commonly known as Rabin-Williams. This Standard (i.e., FIPS 186-4) adopts the 

use of RSA, but does not adopt the use of Rabin-Williams.

During signature verification, the extraction of the hash value H(M)′ from the data structure IR′

shall be accomplished by either:

• Selecting the hashlen bytes of the data structure IR′ that immediately precedes the two 

bytes of trailer information, where hashlen is the length in bytes of the hash function 

used, regardless of the length of the padding, or

• If the hash value H(M)′ is selected by its location with respect to the last byte of padding

(i.e., 0xBA), including a check that the hash value is followed by only two bytes 

containing the expected trailer value.

ANS X9.31 contains an annex on random number generation. However, implementations of 

ANS X9.31 shall use approved random number generation methods. 

Annexes in ANS X9.31 provide informative discussions of security and implementation 

considerations.

5.5 PKCS #1

Public-Key Cryptography Standard (PKCS) #1, RSA Cryptography Standard, defines 

mechanisms for encrypting and signing data using the RSA algorithm. PKCS #1 v2.1 specifies 

two digital signature processes and corresponding formats: RSASSA-PKCS1-v1.5 and 

RSASSA-PSS. Both signature schemes are approved for use, but additional constraints are 

imposed beyond those specified in PKCS #1 v2.1.

(a) Implementations that generate RSA key pairs shall use the criteria and methods in 

Appendix B.3 to generate those key pairs,
25

(b) Only approved hash functions shall be used. 

(c) Only two prime factors p and q shall be used to form the modulus n.

(d) Random numbers shall be generated using an approved random bit generator.

(e) For RSASSA-PSS: 

• If nlen = 1024 bits (i.e., 128 bytes), and the output length of the approved hash 

function output block is 512 bits (i.e., 64 bytes), then the length (in bytes) of the 

salt (sLen) shall satisfy 0 ≤ sLen ≤ hLen – 2,

• Otherwise, the length (in bytes) of the salt (sLen) shall satisfy 0 ≤ sLen ≤ hLen,

where hLen is the length of the hash function output block (in bytes).

(f) For RSASSA-PKCS-v1.5, when the hash value is recovered from the encoded message 

EM during the verification of the digital signature1

, the extraction of the hash value shall

be accomplished by either:

• Selecting the rightmost (least significant) bits of EM, based on the size of the hash 

function used, regardless of the length of the padding, or

• If the hash value is selected by its location with respect to the last byte of padding, 

including a check that the hash value is located in the rightmost (least significant)

bytes of EM (i.e., no other information follows the hash value in the encoded 

message).

Note: PKCS #1 was initially developed by RSA Laboratories in 1991 and has been revised as 

multiple versions. At the time of the approval of FIPS 186-4, three versions of PKSC #1 were 

available: version 1.5, version 2.0 and version 2.1. This Standard references only version 2.1.

 

1 PKCS #1, v2.1 provides two methods for comparing the hash values: by comparing the encoded messages EM and 

EM′, and by extracting the hash value from the decoding of the encoded message (see the Note in PKCS #1, v2.1). 

Step (f) above applies to the latter case. 
26

6. The Elliptic Curve Digital Signature Algorithm (ECDSA)

ANS X9.62, Public Key Cryptography for the Financial Services Industry: The Elliptic Curve 

Digital Signature Standard (ECDSA), was developed for the American National Standards 

Institute by the Accredited Standards Committee on Financial Services, X9. Information about 

obtaining copies of ANS X9.62 is available at http://www.x9.org. The following discussions are 

based on the version of ANS X9.62 that was approved in 2005. This version of ANS X9.62 shall

be used, subject to the transition period referenced in the implementation schedule of this 

Standard.

ANS X9.62 defines methods for digital signature generation and verification using the Elliptic 

Curve Digital Signature Algorithm (ECDSA). Specifications for the generation of the domain 

parameters used during the generation and verification of digital signatures are also included in 

ANS X9.62. ECDSA is the elliptic curve analog of DSA. ECDSA keys shall not be used for any 

other purpose (e.g., key establishment).

6.1 ECDSA Domain Parameters

ECDSA requires that the private/public key pairs used for digital signature generation and 

verification be generated with respect to a particular set of domain parameters. These domain 

parameters may be common to a group of users and may be public. Domain parameters may 

remain fixed for an extended time period.

Domain parameters for ECDSA are of the form (q, FR, a, b {, domain_parameter_seed}, G, n, 

h), where q is the field size; FR is an indication of the basis used; a and b are two field elements 

that define the equation of the curve; domain_parameter_seed is the domain parameter seed and 

is an optional bit string that is present if the elliptic curve was randomly generated in a verifiable 

fashion, G is a base point of prime order on the curve (i.e., G = (xG, yG)), n is the order of the 

point G, and h is the cofactor (which is equal to the order of the curve divided by n). 

6.1.1 Domain Parameter Generation

This Standard specifies five ranges for n (see Table 1). For each range, a maximum cofactor size

is also specified. Note that the specification of a cofactor h in a set of domain parameters is 

optional in ANS X9.62, whereas implementations conforming to this Standard (i.e., FIPS 186-4) 

shall specify the cofactor h in the set of domain parameters. Table 1 provides the maximum sizes 

for the cofactor h.
27

Table 1: ECDSA Security Parameters

Bit length of n Maximum 

Cofactor (h)

160 - 223 210

224 - 255 214

256 - 383 216

384 - 511 224

≥ 512 232

ECDSA is defined for two arithmetic fields: the finite field GFp and the finite fieldGF m2 . For 

the field GFp , p is required to be an odd prime. 

NIST Recommended curves are provided in Appendix D of this Standard (i.e., FIPS 186-4). 

Three types of curves are provided:

1. Curves over prime fields, which are identified as P-xxx,

2. Curves over Binary fields, which are identified as B-xxx, and

3. Koblitz curves, which are identified as K-xxx,

where xxx indicates the bit length of the field size.

Alternatively, ECDSA domain parameters may be generated as specified in ANS X9.62; when 

ECDSA domain parameters are generated (i.e., the NIST Recommended curves are not used), 

the value of G should be generated canonically (verifiably random). An approved hash function, 

as specified in FIPS 180, shall be used during the generation of ECDSA domain parameters. 

When generating these domain parameters, the security strength of a hash function used shall

meet or exceed the security strength associated with the bit length of n (see footnote 2 below).

An approved hash function, as specified in FIPS 180, is required during the generation of domain

parameters. The security strength of the hash function used shall meet or exceed the security 

strength associated with the bit length of n. The security strengths for the ranges of n and the 

hash functions are provided in SP 800-57. It is recommended that the security strength 

associated with the bit length of n and the security strength of the hash function be the same 

 

2 The NIST Recommended curves were generated prior to the formulation of this guidance and using SHA-1, which 

was the only approved hash function available at that time. Since SHA-1 was considered secure at the time of 

generation, the curves were made public, and SHA-1 will only be used to validate those curves, the NIST 

Recommended curves are still considered secure and appropriate for Federal government use.
28

unless an agreement has been made between participating entities to use a stronger hash 

function; a hash function that provides a lower security strength than is associated with the bit 

length of n shall not be used. If the length of the output of the hash function is greater than the 

bit length of n, then the leftmost n bits of the hash function output block shall be used in any 

calculation using the hash function output during the generation or verification of a digital 

signature.

Normally, a CA should use a bit length of n whose assessed security strength is equal to or 

greater than the assessed security strength associated with the bit length of n used by its 

subscribers. For example, if subscribers are using a bit length of n with an assessed security 

strength of 112 bits, then CAs should use a bit length of n whose assessed security strength is 

equal to or greater than 112 bits. SP 800-57 provides further information about the selection of a 

bit length of n. Possible exceptions to this rule include cross certification between CAs, 

certifying keys for purposes other than digital signatures and transitioning from one key size or 

algorithm to another. However, these exceptions require further analysis.

6.1.2 Domain Parameter Management

Each ECDSA key pair shall be correctly associated with one specific set of domain parameters 

(e.g., by a public key certificate that identifies the domain parameters associated with the public 

key). The domain parameters shall be protected from unauthorized modification until the set is 

deactivated (if and when the set is no longer needed). The same domain parameters may be used 

for more than one purpose (e.g., the same domain parameters may be used for both digital 

signatures and key establishment). However, using different domain parameters reduces the risk 

that key pairs generated for one purpose could be accidentally used (successfully) for another 

purpose.

6.2 Private/Public Keys

An ECDSA key pair consists of a private key d and a public key Q that is associated with a 

specific set of ECDSA domain parameters; d, Q and the domain parameters are mathematically 

related to each other. The private key is normally used for a period of time (i.e., the 

cryptoperiod); the public key may continue to be used as long as digital signatures that have been

generated using the associated private key need to be verified (i.e., the public key may continue 

to be used beyond the cryptoperiod of the associated private key). See SP 800-57 for further 

guidance.

ECDSA keys shall only be used for the generation and verification of ECDSA digital signatures.
29

6.2.1 Key Pair Generation

A digital signature key pair d and Q is generated for a set of domain parameters (q, FR, a, b {,

domain_parameter_seed}, G, n, h). Methods for the generation of d and Q are provided in 

Appendix B.4.

6.2.2 Key Pair Management

The secure use of digital signatures depends on the management of an entity’s digital signature 

key pair as specified in Section 4.4.2. 

6.3 Secret Number Generation

A new secret random number k shall be generated prior to the generation of each digital 

signature for use during the signature generation process. This secret number shall be protected 

from unauthorized disclosure and modification. Methods for the generation of the per-message 

secret number are provided in Appendix B.5.

−1 k is the multiplicative inverse of k with respect to multiplication modulo n; i.e., 0 < −1 k < n

and 1 = ( −1 k k) mod n. This inverse is required for the signature generation process. A technique 

is provided in Appendix C.1 for deriving −1 k from k.

k and −1 k may be pre-computed, since knowledge of the message to be signed is not required for 

the computations. When k and −1 k are pre-computed, their confidentiality and integrity shall be 

protected.

6.4 ECDSA Digital Signature Generation and Verification

An ECDSA digital signature (r, s) shall be generated as specified in ANS X9.62, using:

1. Domain parameters that are generated in accordance with Section 6.1.1, 

2. A private key that is generated as specified in Section 6.2.1,

3. A per-message secret number that is generated as specified in Section 6.3,

4. An approved hash function as discussed below, and

5. An approved random bit generator.

An ECDSA digital signature shall be verified as specified in ANS X9.62, using the same domain 

parameters and hash function that were used during signature generation.

An approved hash function, as specified in FIPS 180, shall be used during the generation of 

digital signatures. The security strength associated with the ECDSA digital signature process is 

no greater than the minimum of the security strength associated with the bit length of n and the 

security strength of the hash function that is employed. Both the security strength of the hash 
30

function used and the security strength associated with the bit length of n shall meet or exceed 

the security strength required for the digital signature process. The security strengths for the 

ranges of the bit lengths of n and for each hash function is provided in SP 800-57. 

It is recommended that the security strength associated with the bit length of n and the security 

strength of the hash function be the same unless an agreement has been made between 

participating entities to use a stronger hash function. When the length of the output of the hash 

function is greater than the bit length of n, then the leftmost n bits of the hash function output 

block shall be used in any calculation using the hash function output during the generation or 

verification of a digital signature. A hash function that provides a lower security strength than

the security strength associated with the bit length of n ordinarily should not be used, since this 

would reduce the security strength of the digital signature process to a level no greater than that 

provided by the hash function.

6.5 Assurances

The intended signatory shall have assurances as specified in Section 3.1. Prior to accepting a 

signature as valid, the verifier shall have assurances as specified in Section 3.3.
31

APPENDIX A: Generation and Validation of FFC Domain Parameters

Finite field cryptography (FFC) is a method of implementing discrete logarithm cryptography 

using finite field mathematics. DSA, as specified in this Standard, is an example of FFC. The 

Diffie-Hellman and MQV key establishment algorithms specified in SP 800-56A can also be 

implemented as FFC. 

The domain parameters for FFC consist of the set of values (p, q, g {, domain_parameter_seed, 

counter}). This appendix specifies techniques for the generation of the FFC domain parameters 

p, q and g and performing an explicit domain parameter validation. During the generation 

process, the values for domain_parameter_seed and counter are obtained.

A.1 Generation of the FFC Primes p and q

This section provides methods for generating the primes p and q that fulfill the criteria specified 

in Sections 4.1 and 4.2. One of these methods shall be used when generating these primes. A 

method is provided in Appendix A.1.1 to generate random candidate integers and then test them 

for primality using a probabilistic algorithm. A second method is provided in Appendix A.1.2 

that constructs integers from smaller integers so that the constructed integer is guaranteed to be 

prime. 

During the generation, validation and testing processes, conversions between bit strings and 

integers are required. Appendix C.2 provides methods for these conversions.

A.1.1 Generation and Validation of Probable Primes

Previous versions of this Standard contained a method for the generation of the domain 

parameters p and q using SHA-1 and probabilistic methods. This method is no longer approved

for domain parameter generation; however, the validation process for this method is provided in 

Appendix A.1.1.1 to validate previously generated domain parameters.

A method for the generation and validation of the primes p and q using probabilistic methods is 

provided in Appendix A.1.1.2 and is based on the use of an approved hash function; this method 

shall be used for generating probable primes. The validation process for this method is provided 

in Appendix A.1.1.3. 

The probabilistic methods use a hash function and an arbitrary seed (domain_parameter_seed). 

Arbitrary seeds could be anything, e.g., a user’s favorite number or a random or pseudorandom 

number output by an approved random number generator. The domain_parameter_seed

determines a sequence of candidates for p and q in the required intervals that are then tested for 

primality using a probabilistic primality test (see Appendix C.3). The test determines that the 

candidate is either not a prime (i.e., it is a composite integer) or is “probably a prime” (i.e., there 

is a very small probability that a composite integer will be declared to be a prime). p and q shall

be the first candidate set that passes the primality tests. Note that the domain_parameter_seed
32

shall be unique for every unique set of domain parameters that are generated using the same 

method.

A.1.1.1 Validation of the Probable Primes p and q that were Generated Using SHA-1 as 

Specified in Prior Versions of this Standard

This prime validation algorithm is used to validate that the primes p and q that were generated by 

the prime generation algorithm specified in previous versions of this Standard. The algorithm 

requires the values of p, q, domain_parameter_seed and counter, which were output from the 

prime generation algorithm. 

Let SHA1( ) be the SHA-1 hash function specified in FIPS 180. The following process or its 

equivalent shall be used to validate p and q for this method. 

Input:

1. p, q The generated primes p and q. 

2. domain_parameter_seed A seed that was used to generate p and q.

3. counter A count value that was determined during generation.

Output:

1. status The status returned from the validation procedure, where 

status is either VALID or INVALID.

Process:

1. If (len (p) ≠ 1024) or (len (q) ≠ 160), then return INVALID. 

2. If (counter > 4095), then return INVALID.

3. seedlen = len (domain_parameter_seed).

4. If (seedlen < 160), then return INVALID.

5. computed_q = SHA1(domain_parameter_seed) ⊕ SHA1((domain_parameter_seed + 

1) mod 2seedlen).

6. Set the first and last bits of computed_q equal to 1 (i.e., the 159th and 0th bits).

7. Test whether or not computed_q is prime as specified in Appendix C.3. If 

(computed_q ≠ q) or (computed_q is not prime), then return INVALID.

8. offset = 2.

9. For i = 0 to counter do

9.1 For j = 0 to 6 do

Vj = SHA1((domain_parameter_seed + offset + j) mod 2seedlen).

9.2 W = V0 + (V1 ∗ 2160) + (V2 ∗ 2320) + (V3 ∗ 2480) + (V4 ∗ 2640) + (V5 ∗ 2800) + 
33

((V6 mod 263) ∗ 2960).

9.3 X = W + 21023. Comment: 0 ≤ W < 2L–1.

9.4 c = X mod 2q.

9.5 computed_p = X – (c – 1).Comment: computed_p ≡ 1 (mod 2q).

9.6 If (computed_p < 21023), then go to step 9.8.

9.7 Test whether or not computed_p is prime as specified in Appendix C.3. If 

computed_p is determined to be prime, then go to step 10.

9.8 offset = offset + 7.

10. If ((i ≠ counter) or (computed_p ≠ p) or (computed_p is not prime)), then return

INVALID.

11. Return VALID.

A.1.1.2 Generation of the Probable Primes p and q Using an Approved Hash Function

This method uses an approved hash function and may be used for the generation of the primes p

and q for any application (e.g., digital signatures or key establishment). The security strength of 

the hash function shall be equal to or greater than the security strength associated with the (L, N) 

pair.

An arbitrary domain_parameter_seed of seedlen bits is also used, where seedlen shall be equal 

to or greater than N.

The generation process returns a set of integers p and q that have a very high probability of being 

prime. For another entity to validate that the primes were generated correctly using the validation 

process in Appendix A.1.1.3, the value of the domain_parameter_seed and the counter used to 

generate the primes must also be returned and made available to the validating entity; the 

domain_parameter_seed and counter need not be kept secret. Let Hash( ) be the selected hash 

function, and let outlen be the bit length of the output block, where outlen shall be equal to or 

greater than N.

The following process or its equivalent shall be used to generate p and q for this method. 

Input: 

1. L The desired length of the prime p (in bits).

2. N The desired length of the prime q (in bits). 

3. seedlen The desired length of the domain parameter seed; seedlen shall be 

equal to or greater than N.

Output:

1. status The status returned from the generation procedure, where status is 
34

either VALID or INVALID. If INVALID is returned as the status, 

either no values for the other output parameters shall be returned, or 

invalid values shall be returned (e.g., zeros or Null strings).

2. p, q The generated primes p and q. 

3. domain_parameter_seed

(Optional) A seed that was used to generate p and q.

4. counter (Optional) A count value that was determined during generation.

Process:

1. Check that the (L, N) pair is in the list of acceptable (L, N pairs) (see Section 4.2). If 

the pair is not in the list, then return INVALID.

2. If (seedlen < N), then return INVALID.

3. n = L ⁄ outlen – 1.

4. b = L – 1 – (n ∗ outlen).

5. Get an arbitrary sequence of seedlen bits as the domain_parameter_seed.

6. U = Hash (domain_parameter_seed) mod 2N–1

.

7. q = 2N–1 + U + 1 – ( U mod 2).

8. Test whether or not q is prime as specified in Appendix C.3.

9. If q is not a prime, then go to step 5.

10. offset = 1.

11. For counter = 0 to (4L – 1) do

11.1 For j = 0 to n do

Vj = Hash ((domain_parameter_seed + offset + j) mod 2seedlen).

11.2 W = V0 + (V1 ∗ 2outlen) + ...+ (Vn–1 ∗ 2(n–1) ∗ outlen) + ((Vn mod 2b

) ∗ 2n ∗ outlen).

11.3 X = W + 2L–1

. Comment: 0 ≤ W < 2L–1; hence, 2L–1 ≤ X < 2L.

11.4 c = X mod 2q.

11.5 p = X – (c – 1). Comment: p ≡ 1 (mod 2q).

11.6 If (p < 2L–1

), then go to step 11.9.

11.7 Test whether or not p is prime as specified in Appendix C.3.

11.8 If p is determined to be prime, then return VALID and the values of p, q and 

(optionally) the values of domain_parameter_seed and counter.
35

11.9 offset = offset + n + 1. Comment: Increment offset; then, as part of 

the loop in step 11, increment counter; if 

counter < 4L, repeat steps 11.1 through 11.8.

12. Go to step 5.

A.1.1.3 Validation of the Probable Primes p and q that were Generated Using an 

Approved Hash Function

This prime validation algorithm is used to validate that the integers p and q were generated by 

the prime generation algorithm given in Appendix A.1.1.2. The validation algorithm requires the 

values of p, q, domain_parameter_seed and counter, which were output from the prime 

generation algorithm. Let Hash( ) be the hash function used to generate p and q, and let outlen

be its output block length.

The following process or its equivalent shall be used to validate p and q for this method. 

Input:

1. p, q The generated primes p and q. 

3. domain_parameter_seed The domain parameter seed that was used to generate p and 

q.

4. counter A count value that was determined during generation.

Output:

1. status The status returned from the validation procedure, where 

status is either VALID or INVALID.

Process:

1. L = len (p).

2. N = len (q).

3. Check that the (L, N) pair is in the list of acceptable (L, N) pairs (see Section 4.2). If the 

pair is not in the list, return INVALID.

4. If (counter > (4L – 1)), then return INVALID.

5. seedlen = len (domain_parameter_seed).

6. If (seedlen < N), then return INVALID.

7. U = Hash(domain_parameter_seed) mod 2N–1.

8. computed_q = 2N–1 + U + 1 – ( U mod 2).

9. Test whether or not computed_q is prime as specified in Appendix C.3. If (computed_q ≠

q) or (computed_q is not prime), then return INVALID.
36

10. n = L ⁄ outlen – 1.

11. b = L – 1 – (n ∗ outlen).

12. offset = 1.

13. For i = 0 to counter do

13.1 For j = 0 to n do

Vj = Hash((domain_parameter_seed + offset + j) mod 2seedlen).

13.2 W = V0 + (V1 ∗ 2outlen) + ...+ (Vn–1 ∗ 2(n–1) ∗ outlen) + ((Vn mod 2b

) ∗ 2n ∗ outlen).

13.3 X = W + 2L–1

.

13.4 c = X mod 2q.

13.5 computed_p = X – (c – 1).

13.6 If (computed_p < 2L–1

), then go to step 13.9

13.7 Test whether or not computed_p is prime as specified in Appendix C.3.

13.8 If computed_p is determined to be a prime, then go to step 14.

13.9 offset = offset + n + 1.

14. If ((i ≠ counter) or (computed_p ≠ p) or (computed_p is not a prime)), then return 

INVALID.

15. Return VALID.

A.1.2 Construction and Validation of the Provable Primes p and q

Primes can be generated so that they are guaranteed to be prime. The following algorithm for 

generating p and q uses an approved hash function and an arbitrary seed (firstseed) to construct p

and q in the required intervals. The security strength of the hash function shall be equal to or 

greater than the security strength associated with the (L, N) pair.

Arbitrary seeds can be anything, e.g., a user’s favorite number or a random or pseudorandom 

number that is output from a random number generator. Note that the firstseed must be unique to 

produce a unique set of domain parameters. Candidate primes are tested for primality using a 

deterministic primality test that proves whether or not the candidate is prime. The resulting p and 

q are guaranteed to be primes.

A.1.2.1 Construction of the Primes p and q Using the Shawe-Taylor Algorithm

For each set of domain parameters generated, an arbitrary initial seed (firstseed) of at least 

seedlen bits shall be determined, where seedlen shall be ≥ N.

The generation process returns a set of integers p and q that are guaranteed to be prime. For 
37

another entity to validate that the primes were generated correctly (using the validation process 

in Appendix A.1.2.2), the value of the firstseed, the two computed seeds (pseed and qseed) and 

the counters used to generate the primes (pgen_counter and qgen_counter) must be made 

available to the validating entity; the seeds and the counters need not be kept secret. The domain 

parameters for DSA are identified in Section 4.3 as (p, q, g {, domain_parameter_seed, 

counter}). When using the Shawe-Taylor algorithm for generating p and q, the 

domain_parameter_seed consists of three seed values (firstseed, pseed, and qseed), and the 

counter consists of the pair of counter values (pgen_counter and qgen_counter).

Let Hash( ) be the selected hash function (see Appendix A.1.2), and let outlen be the bit length 

of the output block of that hash function. 

A.1.2.1.1 Get the First Seed

The following process or its equivalent shall be used to generate firstseed for this constructive 

method.

Input:

1. N The length of q in bits.

2. seedlen The length of firstseed, where seedlen ≥ N.

Output:

1. status The status returned from the generation procedure, where status is 

either SUCCESS or FAILURE. If FAILURE is returned, then 

either no firstseed value shall be provided or an invalid value shall 

be returned.

2. firstseed The first seed generated.

Process:

1. firstseed = 0.

2. Check that N is in the list of acceptable (L, N) pairs (see Section 4.2). If not, then 

return FAILURE.

3. If (seedlen < N), then return FAILURE.

4. While firstseed < 2N–1

.

Get an arbitrary sequence of seedlen bits as firstseed. 

5. Return SUCCESS and the value of firstseed.

Note: This routine could be incorporated into the beginning of the constructive prime generation 

procedure in Appendix A.1.2.1.2. However, this was not done in this specification so that the 
38

validation process in Appendix A.1.2.2 could also call the constructive prime generation 

procedure and provide the value of firstseed as input.

A.1.2.1.2 Constructive Prime Generation

The following process or its equivalent shall be used to generate p and q for this constructive 

method. 

Input:

1. L The requested length for p (in bits).

2. N The requested length for q (in bits).

3. firstseed The first seed to be used. This was obtained as specified in 

Appendix A.1.2.1.1.

Output:

1. status The status returned from the generation procedure, where status is 

either SUCCESS or FAILURE. If FAILURE is returned, then 

either no other values shall be returned, or invalid values shall be 

returned.

2. p, q The requested primes.

3. pseed, qseed (Optional) Computed seed values that were used to generate p and 

q. The entire seed for the generation of p and q consists of 

firstseed, pseed and qseed.

4. pgen_counter, qgen_counter

(Optional) The count values that were determined during generation.

Process:

1. Check that the (L, N) pair is in the list of acceptable (L, N) pairs (see Section 4.2). If 

the pair is not in the list, return FAILURE.

Comment: Use the Shawe-Taylor random

prime routine in Appendix C.6 to generate 

random primes.

2. Using N as the length and firstseed as the input_seed, use the random prime 

generation routine in Appendix C.6 to obtain q, qseed and qgen_counter. If 

FAILURE is returned, then return FAILURE.

3. Using L / 2 + 1 as the length and qseed as the input_seed, use the random prime 

routine in Appendix C.6 to obtain p0 , pseed, and pgen_counter. If FAILURE is 

returned, then return FAILURE.
39

4. iterations = L / outlen –1.

5. old_counter = pgen_counter.

Comment: Generate a (pseudo) random x in 

the interval [2L−1, 2L]. 

6. x = 0.

7. For i = 0 to iterations do

x = x + (Hash(pseed + i) ∗ 2 i * outlen).

8. pseed = pseed + iterations + 1. 

9. x = 2L−1 + (x mod 2L–1).

Comment: Generate p, a candidate for the 

prime, in the interval [2L−1

, 2L].

10. t = x / (2q p0).

11. If (2tq p0 + 1) > 2L

, then t = 2L−1 / (2q p0). 

12. p = 2tq p0 + 1.

13. pgen_counter = pgen_counter + 1. 

Comment: Test p for primality; choose an 

integer a in the interval [2, p–2].

14. a = 0

15. For i = 0 to iterations do 

a = a + (Hash(pseed + i) ∗ 2 i * outlen).

16. pseed = pseed + iterations + 1.

17. a = 2 + (a mod (p–3)).

18. z = a

2tq mod p.

19. If ((1 = GCD(z–1, p) ) and ( 1 = z p0 mod p )), then return SUCCESS and the values 

of p, q and (optionally) pseed, qseed, pgen_counter, and qgen_counter.

20. If (pgen_counter > (4L + old_counter)), then return FAILURE.

21. t = t + 1.

22. Go to step 11.
40

A.1.2.2 Validation of the DSA Primes p and q that were Constructed Using the ShaweTaylor Algorithm

The validation of the primes p and q that were generated by the method described in Appendix 

A.1.2.1.2 may be performed if the values of firstseed, pseed, qseed, pgen_counter and 

qgen_counter were saved and are provided for use in the following algorithm. 

The following process or its equivalent shall be used to validate p and q for this constructive 

method.

Input:

1. p, q The primes to be validated.

2. firstseed, pseed, qseed Seed values that were used to generate p and q. 

3. pgen_counter, qgen_counter

The count values that were determined during generation.

Output:

1. status The status returned from the validation procedure, where 

status is either SUCCESS or FAILURE.

Process:

1. L = len (p).

2. N = len (q).

3. Check that the (L, N) pair is in the list of acceptable (L, N) pairs (see Section 4.2). If 

the pair is not in the list, then return FAILURE.

4. If (firstseed < 2N–1), then return FAILURE.

5. If (2N ≤ q), then return FAILURE).

6. If (2L ≤ p), then return FAILURE.

7. If ((p – 1) mod q ≠ 0), then return FAILURE.

8. Using L, N and firstseed, perform the constructive prime generation procedure in 

Appendix A.1.2.1.2 to obtain p_val, q_val, pseed_val, qseed_val, pgen_counter_val, 

and qgen_counter_val. If FAILURE is returned, or if (q_val ≠ q) or (qseed_val ≠
41

qseed) or (qgen_counter_val ≠ qgen_counter) or (p_val ≠ p) or (pseed_val ≠ pseed) 

or (pgen_counter_val ≠ pgen_counter), then return FAILURE.

9. Return SUCCESS.

A.2 Generation of the Generator g

The generator g depends on the values of p and q. Two methods for determining the generator g

are provided; one of these methods shall be used. The first method, discussed in Appendix 

A.2.1, may be used when complete validation of the generator g is not required; it is 

recommended that this method be used only when the party generating g is trusted to not 

deliberately generate a g that has a potentially exploitable relationship to another generator g′. 

For example, it must be hard to determine an exponential relationship between the generators 

such that g = (g′)

x mod p for a known value of x. (Note: Read (g′)x as g prime to the x.)

Appendix A.2.2 provides a method for partial validation when the method of generation in 

Appendix A.2.1 is used. The second method for generating g, discussed in Appendix A.2.3, shall

be used when validation of the generator g is required; the method for the validation of a 

generator determined using the method in Appendix A.2.3 is provided in Appendix A.2.4.

A.2.1 Unverifiable Generation of the Generator g

This method is used to determine a value for g, based on the values of p and q. It may be used 

when validation of the generator g is not required. The correct generation of g cannot be 

completely validated (see Appendix A.2.2). Note that this generation method for g was also 

specified in previous versions of this Standard.

The following process or its equivalent shall be used to generate the generator g for this method.

Input:

1. p, q The generated primes.

Output:

1. g The requested value of g.

Process:

1. e = (p – 1)/q.

2. Set h = any integer satisfying 1 < h < ( p – 1), such that h differs from any value 

previously tried. Note that h could be obtained from a random number generator or

from a counter that changes after each use. 

3. g = he mod p.
42

4. If (g = 1), then go to step 2.

5. Return g.

A.2.2 Assurance of the Validity of the Generator g

The order of the generator g that was generated using Appendix A.2.1 can be partially validated 

by checking the range and order, thereby performing a partial validation of g.

The following process or its equivalent shall be used when partial validation of the generator g is 

required:

Input:

1. p, q, g The domain parameters. 

Output:

1. status The status returned from the generation routine, where status is either 

PARTIALLY VALID or INVALID.

Process:

1. Verify that 2 ≤ g ≤ (p–1). If not true, return INVALID.

2. If (gq = 1 mod p), then return PARTIALLY VALID.

3. Return INVALID.

The non-existence of a potentially exploitable relationship of g to another generator g′ (that is 

known to the entity that generated g, but may not be known by other entities) cannot be checked. 

In this sense, the correct generation of g cannot be completely validated.

A.2.3 Verifiable Canonical Generation of the Generator g

The generation of g is based on the values of p, q and domain_parameter_seed (which are 

outputs of the generation processes in Appendix A.1). When p and q were generated using the 

method in Appendix A.1.1.2, the domain_parameter_seed value must have been returned from 

the generation routine. When p and q were generated using the method in Appendix A.1.2.1, the 

firstseed, pseed, and qseed values must have been returned from the generation routine; in this 

case, domain_parameter_seed = firstseed || pseed || qseed shall be used in the following process.

This method of generating a generator g can be validated (see Appendix A.2.4). 

This generation method supports the generation of multiple values of g for specific values of p

and q. The use of different values of g for the same p and q may be used to support key 

separation; for example, using the g that is generated with index = 1 for digital signatures and 

with index = 2 for key establishment.
43

Let Hash( ) be the hash function used to generate p and q (see Appendix A.1). The following 

process or its equivalent shall be used to generate the generator g.

Input:

1. p, q The primes.

2. domain_parameter_seed The seed used during the generation of p and q.

3. index The index to be used for generating g. index is a bit string 

of length 8 that represents an unsigned integer.

Output:

1. status The status returned from the generation routine, where status is either 

VALID or INVALID.

2. g The value of g that was generated.

Process: Note: count is an unsigned 16-bit integer.

Comment: Check that a valid value of the index has 

been provided (see above).

1. If (index is incorrect), then return INVALID.

2. N = len(q).

3. e = (p – 1)/q.

4. count = 0.

5. count = count + 1.

Comment: Check that count does not wrap around 

to 0.

6. If (count = 0), then return INVALID.

Comment: the length of the 

domain_parameter_seed has already been checked.

“ggen” is the bit string 0x6767656E. 

7. U = domain_parameter_seed || “ggen” || index || count.

8. W = Hash(U).

9. g = We mod p.

10. If (g < 2), then go to step 5. Comment: If a generator has not been found.

11. Return VALID and the value of g.
44

A.2.4 Validation Routine when the Canonical Generation of the Generator g 

Routine Was Used

This algorithm shall be used to validate the value of g that was generated using the process in 

Appendix A.2.3, based on the values of p, q, domain_parameter_seed, and the appropriate value 

of index. It is assumed that the values of p and q have been previously validated according to 

Appendix A.1. Note that the method specified in Appendix A.2.3 for the generation of g was not 

included in previous versions of this Standard; therefore, this validation method is not 

appropriate for that case.

The domain_parameter_seed is an output from the generation of p and q. When p and q were 

generated using the method in Appendix A.1.1.2, the domain_parameter_seed must have been 

returned from the generation routine and made available to the validating party. When p and q

were generated using the method in Appendix A.1.2.1, the firstseed, pseed, and qseed values 

must have been returned from the generation routine and made available; firstseed, pseed, and 

qseed shall be concatenated to form the domain_parameter_seed used in the following process. 

Let Hash( ) be the hash function used to generate g (i.e., the hash function also used to generate 

p and q).

The input index is the index number for the generator g. See Appendix A.2.3 for more details.

The following process or its equivalent shall be used to validate the generator g for this method.

Input:

1. p, q The primes.

2. domain_parameter_seed The seed used to generate p and q.

3. index The index used in Appendix A.2.3 to generate x. index is 

a bit string of length 8 that represents an unsigned 

integer.

4. g The value of g to be validated.

Output:

1. status The status returned from the generation routine, where 

status is either VALID or INVALID.

Process: Note: count is an unsigned 16-bit integer.

Comment: Check that a valid value of the index has been 

provided (see above).

1. If (index is incorrect), then return INVALID.

2. Verify that 2 ≤ g ≤ (p–1). If not true, return INVALID.

3. If (gq ≠ 1 mod p), then return INVALID.
45

4. N = len(q).

5. e = (p – 1)/q.

6. count = 0.

7. count = count + 1.

Comment: Check that count does not wrap around 

to 0.

8. If (count = 0), then return INVALID.

Comment: “ggen” is the bit string 0x6767656E.

9. U = domain_parameter_seed || “ggen” || index || count.

10. W = Hash(U).

11. computed_g = We mod p.

12. If (computed_g < 2), then go to step 7. Comment: If a generator has not been found.

13. If (computed_g = g), then return VALID, else return INVALID.
46

APPENDIX B: Key Pair Generation

Discrete logarithm cryptography (DLC) is divided into finite field cryptography (FFC) and 

elliptic curve cryptography (ECC); the difference between the two is the type of math that is 

used. DSA is an example of FFC; ECDSA is an example of ECC. Other examples of DLC are 

the Diffie-Hellman and MQV key agreement algorithms, which have both FFC and ECC forms.

The most common example of integer factorization cryptography (IFC) is RSA.

This appendix specifies methods for the generation of FFC and ECC key pairs and secret 

numbers, and the generation of IFC key pairs. All generation methods require the use of an 

approved, properly instantiated random bit generator (RBG); the RBG shall have a security 

strength equal to or greater than the security strength associated with the key pairs and secret 

numbers to be generated. See SP 800-57 for guidance on security strengths and key sizes.

This appendix does not indicate the required conversions between bit strings and integers. When 

required by a process in this appendix, the conversion shall be accomplished as specified in 

Appendix C.2. 

B.1 FFC Key Pair Generation

An FFC key pair (x, y) is generated for a set of domain parameters (p, q, g {, 

domain_parameter_seed, counter}). Two methods are provided for the generation of the FFC 

private key x and public key y; one of these two methods shall be used. Prior to generating DSA 

key pairs, assurance of the validity of the domain parameters (p, q and g) shall have been 

obtained as specified in Section 3.1.

For DSA, the valid values of L and N are provided in Section 4.2.

B.1.1 Key Pair Generation Using Extra Random Bits

In this method, 64 more bits are requested from the RBG than are needed for x so that bias 

produced by the mod function in step 6 is negligible. 

The following process or its equivalent may be used to generate an FFC key pair.

Input: 

(p, q, g) The subset of the domain parameters that are used for this process. p, q

and g shall either be provided as integers during input, or shall be 

converted to integers prior to use.
47

Output:

1. status The status returned from the key pair generation process. The status will 

indicate SUCCESS or an ERROR.

2. (x, y) The generated private and public keys. If an error is encountered during 

the generation process, invalid values for x and y should be returned, as 

represented by Invalid_x and Invalid_y in the following specification. x

and y are returned as integers. The generated private key x is in the range 

[1, q–1], and the public key is in the range [1, p–1].

Process:

1. N = len(q); L = len(p).

Comment: Check that the (L, N) pair is specified in 

Section 4.2.

2. If the (L, N) pair is invalid, then return an ERROR indicator, Invalid_x, and 

Invalid_y.

3. requested_security_strength = the security strength associated with the (L, N) pair; 

see SP 800-57.

4. Obtain a string of N+64 returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_x, and Invalid_y.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. x = (c mod (q–1)) + 1. Comment: 0 ≤ c mod (q–1) ≤ q–2 and implies that 

1 ≤ x ≤ q–1.

7. y = gx mod p.

8. Return SUCCESS, x, and y.

B.1.2 Key Pair Generation by Testing Candidates

In this method, a random number is obtained and tested to determine that it will produce a value 

of x in the correct range. If x is out-of-range, another random number is obtained (i.e., the 

process is iterated until an acceptable value of x is obtained. 

The following process or its equivalent may be used to generate an FFC key pair.

Input: 

(p, q, g) The subset of the domain parameters that are used for this process. p, q

and g shall either be provided as integers during input, or shall be 

converted to integers prior to use.
48

Output:

1. status The status returned from the key pair generation process. The status will 

indicate SUCCESS or an ERROR.

2. (x, y) The generated private and public keys. If an error is encountered during 

the generation process, invalid values for x and y should be returned, as 

represented by Invalid_x and Invalid_y in the following specification. x

and y are returned as integers. The generated private key x is in the range 

[1, q–1], and the public key is in the range [1, p–1].

Process:

1. N = len(q); L = len(p).

Comment: Check that the (L, N) pair is specified in 

Section 4.2.

2. If the (L, N) pair is invalid, then return an ERROR indication, Invalid_x, and 

Invalid_y.

3. requested_security_strength = the security strength associated with the (L, N) pair; 

see SP 800-57.

4. Obtain a string of N returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_x, and Invalid_y.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. If (c > q–2), then go to step 4.

7. x = c + 1.

8. y = gx mod p.

9. Return SUCCESS, x, and y.

B.2 FFC Per-Message Secret Number Generation

DSA requires the generation of a new random number k for each message to be signed. Two 

methods are provided for the generation of k; one of these two methods or another approved

method shall be used. 

The valid values of N are provided in Section 4.2. Let inverse(k, q) be a function that computes 

the inverse of a (non-negative) integer k with respect to multiplication modulo the prime number 

q. A technique for computing the inverse is provided in Appendix C.1.
49

B.2.1 Per-Message Secret Number Generation Using Extra Random Bits

In this method, 64 more bits are requested from the RBG than are needed for k so that bias 

produced by the mod function in step 6 is not readily apparent.

The following process or its equivalent may be used to generate a per-message secret number. 

Input: 

(p, q, g) DSA domain parameters that are generated as specified in Section 4.3.1.

Output:

1. status The status returned from the secret number generation process. The status 

will indicate SUCCESS or an ERROR.

2. (k, −1 k ) The per-message secret number k and its mod q inverse, −1 k . If an error is 

encountered during the generation process, invalid values for k and −1 k

should be returned, as represented by Invalid_k and Invalid_k_inverse in 

the following specification. k and −1 k are in the range [1, q–1].

Process:

1. N = len(q); L = len(p).

Comment: Check that the (L, N) pair is specified in 

Section 4.2.

2. If the (L, N) pair is invalid, then return an ERROR indication, Invalid_k, and 

Invalid_k_inverse.

3. requested_security_strength = the security strength associated with the (L, N) pair; 

see SP 800-57.

4. Obtain a string of N+64 returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_k, and Invalid_k_inverse.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. k = (c mod (q–1)) + 1.

7. (status, k

–1

) = inverse (k, q).

8. Return status, k, and −1 k .

B.2.2 Per-Message Secret Number Generation by Testing Candidates

In this method, a random number is obtained and tested to determine that it will produce a value 

of k in the correct range. If k is out-of-range, another random number is obtained (i.e., the 

process is iterated until an acceptable value of k is obtained.
50

The following process or its equivalent may be used to generate a per-message secret number. 

Input: 

(p, q, g) DSA domain parameters that are generated as specified in Section 4.3.1.

Output:

1. status The status returned from the secret number generation process. The status 

will indicate SUCCESS or an ERROR.

2. (k, −1 k ) The per-message secret number k and its inverse, −1 k . If an error is 

encountered during the generation process, invalid values for k and −1 k should be returned, as represented by Invalid_k and Invalid_k_inverse 

in the following specification. k and −1 k are in the range [1, q–1].

Process:

1. N = len(q); L = len(p).

Comment: Check that the (L, N) pair is specified in 

Section 4.2).

2. If the (L, N) pair is invalid, then return an ERROR indication, Invalid_k, and 

Invalid_k_inverse.

3. requested_security_strength = the security strength associated with the (L, N) pair; 

see SP 800-57.

4. Obtain a string of N returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_k, and Invalid_k_inverse.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. If (c > q–2), then go to step 4.

7. k = c + 1.

8. (status, −1 k ) = inverse(k, q).

9. Return status, k, and −1 k .

B.3 IFC Key Pair Generation

B.3.1 Criteria for IFC Key Pairs

Key pairs for IFC consist of a public key (n, e), and a private key (n, d), where n is the modulus 

and is the product of two prime numbers p and q. The security of IFC depends on the quality and 

secrecy of these primes and the private exponent d. The primes p and q shall be generated using 
51

one of the following methods:

A. Both p and q are randomly generated prime numbers (Random Primes), where p and q

shall both be either :

1. Provable primes (see Appendix B.3.2), or

2. Probable primes (see Appendix B.3.3). 

Using methods 1 and 2, p and q with lengths of 1024 or 1536 bits may be generated; p 

and q with lengths of 512 bits shall not be generated using these methods. Instead, p 

and q with lengths of 512 bits shall be generated using the conditions based on auxiliary 

primes (see Appendices B.3.4, B.3.5, or B.3.6).

B. Both p and q are randomly generated prime numbers that satisfy the following additional 

conditions (Primes with Conditions):

• (p–1) has a prime factor p1

• (p+1) has a prime factor p2

• (q–1) has a prime factor q1

• (q+1) has a prime factor q2

where p1, p2, q1 and q2 are called auxiliary primes of p and q. 

Using this method, one of the following cases shall apply:

1. The primes p1, p2, q1, q2, p and q shall all be provable primes (see Appendix 

B.3.4), 

2. The primes p1, p2, q1 and q2 shall be provable primes, and the primes p and q

shall be probable primes (see Appendix B.3.5), or

3 The primes p1, p2, q1, q2, p and q shall all be probable primes (see Appendix 

B.3.6).

The minimum lengths for each of the auxiliary primes p1, p2, q1 and q2 are dependent on 

nlen, where nlen is the length of the modulus n in bits. Note that nlen is also called the 

key size. The lengths of the auxiliary primes may be fixed or randomly chosen, subject to 

the restrictions in Table B.1. The maximum length is determined by nlen (the sum of the 

length of each auxiliary prime pair) and whether the primes p and q are probable primes 

or provable primes (e.g., for the auxiliary prime pair p1 and p2, len(p1) + len(p2) shall be 

less than a value determined by nlen, whether p1 and p2 are generated to be probable or 

provable primes)3. 

 

3 For the probable primes p and q: len(p1) + len(p2) < len(p) – log2(len(p)) – 6; similarly for len(q1) + len(q2) and 

len(q). For the provable primes p and q: len(p1) + len(p2) < len(p)/2 – log2(len(p)) – 7; similarly for len(q1) + len(q2) 
52

Table B.1. Minimum and maximum lengths of p1, p2, q1 and q2

nlen Min. length of 

auxiliary primes 

p1, p2, q1 and q2

Max. length of len(p1) + len(p2) and 

len(q1) + len(q2)

p, q Probable primes p, q Provable primes

1024 > 100 bits < 496 bits < 239 bits

2048 > 140 bits < 1007 bits < 494 bits

3072 > 170 bits < 1518 bits < 750 bits

For different values of nlen (i.e., different key sizes), the methods allowed for the generation of 

p and q are specified in Table B.2.

Table B.2. Allowable Prime Generation Methods

nlen Random Primes Primes with Conditions

1024 No Yes

2048 Yes Yes

3072 Yes Yes

In addition, all IFC keys shall meet the following criteria in order to conform to FIPS 186-4:

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

(a) The public verification exponent e shall be selected prior to generating the primes 

p and q, and the private signature exponent d.

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

216 < e < 2256.

Note that the value of e may be any value that meets constraint 1(b), i.e., e may be 

either a fixed value or a random value.

2. The primes p and q shall be selected with the following constraints:

(a) (p–1) and (q–1) shall be relatively prime to the public exponent e.

(b) The private prime factor p shall be selected randomly and shall satisfy

 

and len(q). In each case, len(p) = len(q) = nlen/2.
53

( 2 )(2(nlen / 2) – 1) ≤ p ≤ (2nlen / 2– 1), where nlen is the appropriate length for the 

desired security_strength. 

(c) The private prime factor q shall be selected randomly and shall satisfy 

( 2 )(2(nlen / 2) – 1

) ≤ q ≤ (2nlen / 2– 1), where nlen is the appropriate length for the 

desired security_strength. 

(d) |p – q| > 2(nlen / 2) – 100.

3. The private signature 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 

2nlen/ 2 < d < LCM(p–1, q–1), and

(b) d = e

–1 mod (LCM(p–1, q–1)).

That is, the inequality in (a) holds, and 1 ≡ (ed) (mod LCM(p–1, q–1)).

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

determined. A different value of e may be used, although this is not required.

Any hash function used during the generation of the key pair shall be approved (i.e., specified in 

FIPS 180).

B.3.2 Generation of Random Primes that are Provably Prime

An approved method that satisfies the constraints of Appendix B.3.1 shall be used for the 

generation of IFC random primes p and q that are provably prime (see case A.1). One such 

method is provided in Appendix B.3.2.1 and B.3.2.2. For this method, a random seed is initially 

required (see Appendix B.3.2.1); the length of the seed is equal to twice the security strength 

associated with the modulus n. After the seed is obtained, the primes can be generated (see 

Appendix B.3.2.2). 

B.3.2.1 Get the Seed

The following process or its equivalent shall be used to generate the seed for this method.

Input:

nlen The intended bit length of the modulus n.

Output:

status The status to be returned, where status is either SUCCESS or FAILURE.

seed The seed. If status = FAILURE, a value of zero is returned as the seed.

Process:

1. If nlen is not valid (see Section 5.1), then Return (FAILURE, 0).
54

2. Let security_strength be the security strength associated with nlen, as specified in SP 

800-57, Part 1.

3. Obtain a string seed of (2 * security_strength) bits from an RBG that supports the 

security_strength.

4. Return (SUCCESS, seed).

B.3.2.2 Construction of the Provable Primes p and q

The following process or its equivalent shall be used to construct the random primes p and q (to 

be used as factors of the RSA modulus n) that are provably prime:

Input:

nlen The intended bit length of the modulus n.

e The public verification exponent.

seed The seed obtained using the method in Appendix B.3.2.1.

Output:

status The status of the generation process, where status is either SUCCESS or 

FAILURE. When FAILURE is returned, zero values shall be returned as the 

other parameters.

p and q The private prime factors of n.

Process:

1. If nlen is neither 2048 nor 3072, then return (FAILURE, 0, 0).

2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).

3. Set the value of security_strength in accordance with the value of nlen, as specified in SP 

800-57, Part 1. 

4. If (len(seed) ≠ 2 * security_strength), then return (FAILURE, 0, 0).

5. working_seed = seed.

6. Generate p:

6.1 Using L = nlen/2, N1 = 1, N2 = 1, first_seed = working_seed and e, use the provable 

prime construction method in Appendix C.10 to obtain p and pseed. If FAILURE

is returned, then return (FAILURE, 0, 0).

6.2 working_seed = pseed.
55

7. Generate q:

7.1 Using L = nlen/2, N1 = 1, N2 = 1, first_seed = working_seed and e, use the provable 

prime construction method in Appendix C.10 to obtain q and qseed. If FAILURE

is returned, then return (FAILURE, 0, 0).

7.2 working_seed = qseed.

8. If ( |p – q| ≤ 2nlen/2 – 100), then go to step 7.

9. Zeroize the internally generated seeds:

9.1 pseed = 0; 

9.2 qseed = 0; 

9.3 working_seed = 0.

10. Return (SUCCESS, p, q).

B.3.3 Generation of Random Primes that are Probably Prime

An approved method that satisfies the constraints of Appendix B.3.1 shall be used for the 

generation of IFC random primes p and q that are probably prime (see case A.2). 

The following process or its equivalent shall be used to construct the random probable primes p

and q (to be used as factors of the RSA modulus n):

Input:

nlen The intended bit length of the modulus n.

e The public verification exponent.

Output:

status The status of the generation process, where status is either SUCCESS or 

FAILURE.

p and q The private prime factors of n. When FAILURE is returned, zero values shall be 

returned as p and q.

Process:

1. If nlen is neither 2048 nor 3072, return (FAILURE, 0, 0).

2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).

3. Set the value of security_strength in accordance with the value of nlen, as specified in SP 

800-57, Part 1. 

4. Generate p:

4.1 i = 0.
56

4.2 Obtain a string p of (nlen/2) bits from an RBG that supports the security_strength. 

4.3 If (p is not odd), then p = p + 1.

4.4 If ((p < ( 2 )(2(nlen / 2) – 1)), then go to step 4.2.

4.5 If (GCD(p−1, e) = 1), then

4.5.1 Test p for primality as specified in Appendix C.3, using an appropriate 

value from Table C-2 or C-3 in Appendix C.3 as the number of iterations. 

4.5.2 If p is PROBABLY PRIME, then go to step 5.

4.6 i= i + 1.

4.7 If (i ≥ 5(nlen/2)), then return (FAILURE, 0, 0)

Else go to step 4.2.

5. Generate q:

5.1 i = 0.

5.2 Obtain a string q of (nlen/2) bits from an RBG that supports the security_strength

5.3 If (q is not odd), then q = q + 1.

5.4 If (|p – q| ≤ 2nlen/2 – 100), then go to step 5.2.

5.5 If ((q < ( 2 )(2(nlen / 2) – 1)), then go to step 5.2.

5.6 If (GCD(q−1, e) = 1) then

5.6.1 Test q for primality as specified in Appendix C.3, using an appropriate 

value from Table C-2 or C-3 in Appendix C.3 as the number of iterations. 

5.6.2 If q is PROBABLY PRIME, then return (SUCCESS, p, q).

5.7 i = i + 1.

5.8 If (i ≥ 5(nlen/2)), then return (FAILURE, 0, 0)

Else go to step 5.2.

B.3.4 Generation of Provable Primes with Conditions Based on Auxiliary 

Provable Primes

This section specifies an approved method for the generation of the IFC primes p and q with the 

additional conditions specified in Appendix B.3.1, case B.1, where p, p1, p2, q, q1 and q2 are all 

provable primes. For this method, a random seed is initially required (see Appendix B.3.2.1); the 

length of the seed is equal to twice the security strength associated with the modulus n. After the 

first seed is obtained, the primes can be generated.
57

Let bitlen1, bitlen2, bitlen3, and bitlen4 be the bit lengths for p1, p2, q1 and q2, respectively, in 

accordance with Table B.1. The following process or its equivalent shall be used to generate the 

provable primes:

Input:

nlen The intended bit length of the modulus n.

e The public verification exponent.

seed The seed obtained using the method in Appendix B.3.2.1.

Output:

status The status of the generation process, where status is either SUCCESS or 

FAILURE. If FAILURE is returned then zeros shall be returned as the values 

for p and q.

p and q The private prime factors of n.

Process:

1. If nlen is neither 1024, 2048, nor 3072, then return (FAILURE, 0, 0).

2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).

3. Set the value of security_strength in accordance with the value of nlen, as specified in SP 

800-57, Part 1. 

4. If (len(seed) ≠ 2 * security_strength), then return (FAILURE, 0, 0).

5. working_seed = seed.

6. Generate p: 

6.1 Using L = nlen/2, N1 = bitlen1, N2 = bitlen2, firstseed = working_seed and e, use the 

provable prime construction method in Appendix C.10 to obtain p, p1, p2 and pseed. 

If FAILURE is returned, return (FAILURE, 0, 0).

6.2 working_seed = pseed.

7. Generate q: 

7.1 Using L = nlen/2, N1 = bitlen3, N2 = bitlen4 and firstseed = working_seed and e, use 

the provable prime construction method in Appendix C.10 to obtain q, q1, q2 and 

qseed. If FAILURE is returned, return (FAILURE, 0, 0).

7.2 working_seed = qseed.

8. If ( |p – q| ≤ 2nlen/2 – 100), then go to step 7.

9. Zeroize the internally generated seeds:

9.1 pseed = 0. 
58

9.2 qseed = 0. 

9.3 working_seed = 0.

10. Return (SUCCESS, p, q).

B.3.5 Generation of Probable Primes with Conditions Based on Auxiliary 

Provable Primes 

This section specifies an approved method for the generation of the IFC primes p and q with the 

additional conditions specified in Appendix B.3.1, case B.2, where p1, p2, q1 and q2 are provably 

prime, and p and q are probably prime. For this method, a random seed is initially required (see 

Appendix B.3.2.1); the length of the seed is equal to twice the security strength associated with 

the modulus n. After the first seed is obtained, the primes can be generated.

Let bitlen1, bitlen2, bitlen3, and bitlen4 be the bit lengths for p1, p2, q1 and q2, respectively in 

accordance with Table B.1. The following process or its equivalent shall be used to construct p

and q.

Input:

nlen The intended bit length of the modulus n.

e The public verification exponent.

seed The seed obtained using the method in Appendix B.3.2.1.

Output:

status The status of the generation process, where status is either SUCCESS or 

FAILURE. If FAILURE is returned then zeros shall be returned as the 

values for p and q.

p and q The private prime factors of n.

Process:

1. If nlen is neither 1024, 2048, nor 3072, then return (FAILURE, 0, 0).

2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).

3. Set the value of security_strength in accordance with the value of nlen, as specified in 

SP 800-57, Part 1. 

4. If (len(seed) ≠ 2 * security_strength), then return (FAILURE, 0, 0).

Comment: Generate four primes p1, p2, q1 and q2

that are provably prime.
59

5. Generate p:

5.1 Using bitlen1 as the length, and seed as the input_seed, use the random prime 

generation routine in Appendix C.6 to obtain p1 and prime_seed. If FAILURE

is returned, the return (FAILURE, 0, 0).

5.2 Using bitlen2 as the length, and prime_seed as the input_seed, use the random 

prime generation routine in Appendix C.6 to obtain p2 and a new value for 

prime_seed. If FAILURE is returned, the return (FAILURE, 0, 0).

5.3 Generate a prime p using the routine in Appendix C.9 with inputs of p1, p2, 

nlen, e and security_strength, also obtaining Xp. If FAILURE is returned, 

return (FAILURE, 0, 0).

6. Generate q:

6.1. Using bitlen3 as the length, and prime_seed as the input_seed, use the random 

prime generation routine in Appendix C.6 to obtain q1 and a new value for 

prime_seed. If FAILURE is returned, the return (FAILURE, 0, 0).

6.2 Using bitlen4 as the length, and prime_seed as the input_seed, use the random 

prime generation routine in Appendix C.6 to obtain q2 and a new value for 

prime_seed. If FAILURE is returned, the return (FAILURE, 0, 0).

6.3 Generate a prime q using the routine in Appendix C.9 with inputs of q1, q2, 

nlen, e and security_strength, also obtaining Xq. If FAILURE is returned, 

return (FAILURE, 0, 0).

7. If ((|p – q| ≤ 2nlen/2 –100) OR (|Xp – Xq| ≤ 2nlen/2 – 100)), then go to step 6.

8. Zeroize the internally generated that are not returned:

8.1 Xp = 0.

8.2 Xq = 0.

8.3 prime_seed = 0.

8.4 p1 = 0.

8.5 p2 = 0.

8.6 q1 = 0.

8.7 q2 = 0.

9. Return (SUCCESS, p, q).
60

B.3.6 Generation of Probable Primes with Conditions Based on Auxiliary 

Probable Primes 

An approved method that satisfies the constraints of Appendix B.3.1 shall be used for the 

generation of IFC primes p and q that are probably prime and meet the additional constraints of 

Appendix B.3.1 (see case B.3). For this case, the prime factors p1, p2, q1 and q2 are also 

probably prime.

Four random numbers Xp1, Xp2, Xq1 and Xq2 are generated, from which the prime factors p1, p2, 

q1 and q2 are determined. p1 and p2, and an additional random number Xp are then used to 

determine p, and q1 and q2 and a random number Xq are used to obtain q. Let bitlen1, bitlen2, 

bitlen3, and bitlen4 be the bit lengths for p1, p2, q1 and q2, respectively chosen in accordance 

with Table B.1.

The following process or its equivalent shall be used to generate p and q:

Input:

nlen The intended bit length of the modulus n.

e The public verification exponent.

Output:

status The status of the generation process, where status is either SUCCESS or 

FAILURE. If FAILURE is returned then zeros shall be returned as the 

values for p and q.

p and q The private prime factors of n.

Process:

1. If nlen is neither 1024, 2048, nor 3072, then return (FAILURE, 0, 0).

2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).

3. Set the value of security_strength in accordance with the value of nlen, as specified in 

SP 800-57, Part 1. 

4. Generate p:

4.1 Generate an odd integer Xp1 of length bitlen1 bits, and a second odd integer Xp2

of length bitlen2 bits, using an approved random number generator that supports 

the security_strength.

4.2 Sequentially search successive odd integers, starting at Xp1 until the first 

probable prime p1 is found. Candidate integers shall be tested for primality as 

specified in Appendix C.3. Repeat the process to find p2, starting at Xp2. The 

probable primes p1 and p2 shall be the first integers that pass the primality test.
61

4.3 Generate a prime p using the routine in Appendix C.9 with inputs of p1, p2, 

nlen, e and security_ strength, also obtaining Xp. If FAILURE is returned, 

return (FAILURE, 0, 0).

5. Generate q:

5.1 Generate an odd integer Xq1 of length bitlen3 bits, and a second odd integer Xq2

of length bitlen4 bits, using an approved random number generator that supports 

the security_strength.

5.2 Sequentially search successive odd integers, starting at Xq1 until the first 

probable prime q1 is found. Candidate integers shall be tested for primality as 

specified in Appendix C.3. Repeat the process to find q2, starting at Xq2. The 

probable primes q1 and q2 shall be the first integers that pass the primality test.

5.3 Generate a prime q using the routine in Appendix C.9 with inputs of q1, q2, 

nlen, e and security_ strength, also obtaining Xq. If FAILURE is returned, 

return (FAILURE, 0, 0).

6. If ((|Xp – Xq| ≤ 2nlen/2 –100) OR (|p – q| ≤ 2nlen/2 – 100)

)), then go to step 5.

7. Zeroize the internally generated values that are not returned:

7.1 Xp = 0.

7.2 Xq = 0.

7.3 Xp1 = 0.

7.4 Xp2 = 0.

7.5 Xq1 = 0.

7.6 Xq2 = 0.

7.7 p1 = 0.

7.8 p2 = 0.

7.9 q1 = 0.

7.10 q2 = 0.

8. Return (SUCCESS, p, q).

B.4 ECC Key Pair Generation

An ECC key pair d and Q is generated for a set of domain parameters (q, FR, a, b {, 

domain_parameter_seed}, G, n, h). Two methods are provided for the generation of the ECC 

private key d and public key Q; one of these two methods shall be used to generate d and Q. 
62

Prior to generating ECDSA key pairs, assurance of the validity of the domain parameters (q, FR, 

a, b {, domain_parameter_seed}, G, n, h) shall have been obtained as specified in Section 3.1.

For ECDSA, the valid bit-lengths of n are provided in Section 6.1.1. See ANS X9.62 for 

definitions of the elliptic curve math and the conversion routines.

B.4.1 Key Pair Generation Using Extra Random Bits

In this method, 64 more bits are requested from the RBG than are needed for d so that bias 

produced by the mod function in step 6 is negligible. 

The following process or its equivalent may be used to generate an ECC key pair.

Input: 

1. (q, FR, a, b {, domain_parameter_seed}, G, n, h)

The domain parameters that are used for this process. n is a prime number,

and G is a point on the elliptic curve.

Output:

1. status The status returned from the key pair generation procedure. The status will 

indicate SUCCESS or an ERROR.

2. (d, Q) The generated private and public keys. If an error is encountered during 

the generation process, invalid values for d and Q should be returned, as 

represented by Invalid_d and Invalid_Q in the following specification. d is 

an integer, and Q is an elliptic curve point. The generated private key d is 

in the range [1, n–1].

Process:

1. N = len(n).

Comment: Check that N is included in Table 1 of 

Section 6.1.1.

2. If N is invalid, then return an ERROR indication, Invalid_d, and Invalid_Q.

3. requested_security_strength = the security strength associated with N; see SP 800-57, 

Part 1.

4. Obtain a string of N+64 returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_d, and Invalid_Q.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. d = (c mod (n–1)) + 1.
63

7. Q = dG.

8. Return SUCCESS, d, and Q.

B.4.2 Key Pair Generation by Testing Candidates

In this method, a random number is obtained and tested to determine that it will produce a value 

of d in the correct range. If d is out-of-range, another random number is obtained (i.e., the 

process is iterated until an acceptable value of d is obtained. 

The following process or its equivalent may be used to generate an ECC key pair.

Input: 

1. (q, FR, a, b {, domain_parameter_seed}, G, n, h)

The domain parameters that are used for this process. n is a prime number, 

and G is a point on the elliptic curve.

Output:

1. status The status returned from the key pair generation procedure. The status will 

indicate SUCCESS or an ERROR.

2. (d, Q) The generated private and public keys. If an error is encountered during 

the generation process, invalid values for d and Q should be returned, as 

represented by Invalid_d and Invalid_Q in the following specification. d is 

an integer, and Q is an elliptic curve point. The generated private key d is 

in the range [1, n–1].

Process:

1. N = len(n).

Comment: Check that N is included in Table 1 of 

Section 6.1.1.

2. If N is invalid, then return an ERROR indication, Invalid_d, and Invalid_Q.

3. requested_security_strength = the security strength associated with N; see SP 800-57, 

Part 1.

4. Obtain a string of N returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_d, and Invalid_Q.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. If (c > n–2), then go to step 4.

7. d = c + 1.
64

8. Q = dG.

9. Return SUCCESS, d, and Q.

B.5 ECC Per-Message Secret Number Generation

ECDSA requires the generation of a new random number k for each message to be signed. Two 

methods are provided for the generation of k; one of these two methods or another approved

method shall be used. 

The valid values of n are provided in Section 6.1.1. See ANS X9.62 for definitions of the elliptic 

curve math and the conversion routines.

Let inverse(k, n) be a function that computes the inverse of a (non-negative) integer k with 

respect to multiplication modulo the prime number n. A technique for computing the inverse is 

provided in Appendix C.1.

B.5.1 Per-Message Secret Number Generation Using Extra Random Bits

In this method, 64 more bits are requested from the RBG than are needed for k so that bias 

produced by the mod function in step 6 is not readily apparent.

The following process or its equivalent may be used to generate a per-message secret number. 

Input: 

1. (q, FR, a, b {, domain_parameter_seed}, G, n, h)

The domain parameters that are used for this process. n is a prime 

number, and G is a point on the elliptic curve.

Output:

1. status The status returned from the key pair generation procedure. The status will 

indicate SUCCESS or an ERROR.

2. (k, −1 k ) The generated secret number k and its inverse k

–1

. If an error is 

encountered during the generation process, invalid values for k and −1 k should be returned, as represented by Invalid_k and Invalid_k_inverse

in the following specification. k and −1 k are integers in the range [1, n–1].

Process:

1. N = len(n).

Comment: Check that N is included in Table 1 of 

Section 6.1.1.

2. If N is invalid, then return an ERROR indication, Invalid_k, and Invalid_k_inverse.
65

3. requested_security_strength = the security strength associated with N; see SP 800-57, 

Part 1.

4. Obtain a string of N+64 returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_k, and Invalid_k_inverse.

5. Convert returned_bits to the non-negative integer c (see Appendix C.2.1).

6. k = (c mod (n–1)) + 1.

7. (status, −1 k ) = inverse(k, n).

8. Return status, k, and −1 k .

B.5.2 Per-Message Secret Number Generation by Testing Candidates

In this method, a random number is obtained and tested to determine that it will produce a value 

of k in the correct range. If k is out-of-range, another random number is obtained (i.e., the 

process is iterated until an acceptable value of k is obtained.

The following process or its equivalent may b used to generate a per-message secret number. 

Input: 

1. (q, FR, a, b {, domain_parameter_seed}, G, n, h) 

The domain parameters that are used for this process. n is a prime number, 

and G is a point on the elliptic curve.

Output:

1. status The status returned from the key pair generation procedure. The status will 

indicate SUCCESS or an ERROR.

2. (k, −1 k ) The generated secret number k and its inverse k

–1

. If an error is 

encountered during the generation process, invalid values for k and −1 k

should be returned, as represented by Invalid_k and Invalid_k_inverse in 

the following specification. k and −1 k are integers in the range [1, n–1].

Process:

1. N = len(n).

Comment: Check that N is included in Table 1 of 

Section 6.1.1.

2. If N is not included in Table 1, then return an ERROR indication, Invalid_k, and 

Invalid_k_inverse.
66

3. requested_security_strength = the security strength associated with N; see SP 800-57, 

Part 1.

4. Obtain a string of N returned_bits from an RBG with a security strength of 

requested_security_strength or more. If an ERROR indication is returned, then 

return an ERROR indication, Invalid_k, and Invalid_k_inverse.

5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).

6. If (c > n–2), then go to step 4.

7. k = c + 1.

8. (status, k

–1

) = inverse(k, n).

9. Return status, k, and −1 k .
67

Appendix C: Generation of Other Quantities

This appendix contains routines for supplementary processes required for the implementation of 

this Standard. Appendix C.1 is needed to produce the inverse of the per-message secret k (see 

Section 4.5, and Appendices B.2.1, B.2.2, B.5.1 and B.5.2) and the inverse of the signature 

portion s that is used during signature verification (see Section 4.7). The routines in Appendix 

C.2 are required to convert between bit strings and integers where required in implementing this 

Standard. Appendix C.3 contains probabilistic primality tests to be used during the generation of 

DSA domain parameters and RSA key pairs. Appendices C.4 and C.5 contain algorithms 

required during the Lucas probabilistic primality test of Appendix C.3.3 to check for a perfect 

square and to compute the Jacobi symbol. Appendix C.6 contains the Shawe-Taylor algorithm 

for the construction of primes. Appendix C.7 provides a process to perform trial division, as 

required by the random prime generation routine in Appendix C.6. The sieve procedure in 

Appendix C.8 is needed by the trial division routine in Appendix C.7. The trial division process 

in Appendix C.7 and the sieve procedure in Appendix C.8 have been extracted from ANS X9.80, 

Prime Number Generation, Primality Testing, and Primality Certificates. Appendix C.9 is 

required during the generation of RSA key pairs. Appendix C.10 provides a method for 

constructing provable primes for RSA (see Appendix B.3.2.2 and B.3.4).

C.1 Computation of the Inverse Value

This algorithm or an algorithm that produces an equivalent result shall be used to compute the 

multiplicative inverse z–1 mod a, where 0 < z < a, 0 < z–1 < a, and a is a prime number. In this 

Standard, z is either k or s, and a is either q or n.

Input:

1. z The value to be inverted mod a (i.e., either k or s).

2. a The domain parameter and (prime) modulus (i.e., either q or n).

Output:

1. status The status returned from this function, where the status is either 

SUCCESS or ERROR.

2. z

–1 The multiplicative inverse of z mod a, if it exists.

Process:

1. Verify that a and z are positive integers such that z < a; if not, return an ERROR

indication.

2. Set i = a, j = z, y2 = 0, and y1 = 1.

3. quotient =  i/j.
68

4. remainder = i –( j * quotient).

5. y = y2 –(y1 * quotient).

6. Set i = j, j = remainder, y2 = y1, and y1 = y.

7. If (j > 0), then go to step 3.

8. If (i ≠ 1), then return an ERROR indication.

9. Return SUCCESS and y2 mod a.

C.2 Conversion Between Bit Strings and Integers

C.2.1 Conversion of a Bit String to an Integer

An n-long sequence of bits { x1, ..., xn } is converted to an integer by the rule

{ x1, ... , xn } → (x1

∗ 2

n–1

) + (x2∗ 2

n–2

) + ...+ (n1∗ 2) + xn .

Note that the first bit of a sequence corresponds to the most significant bit of the corresponding 

integer, and the last bit corresponds to the least significant bit. 

Input:

1. b1, b2,... , bn The bit string to be converted.

Output:

1. C The requested integer representation of the bit string.

Process:

1. Let (b1, b2,... , bn) be the bits of b from leftmost to rightmost.

2. ∑=

− =

n

i

i

n i C b

1

( ) 2

3. Return C.

In this Standard, the binary length of an integer C is defined as the smallest integer n satisfying C

< 2n

. 

C.2.2 Conversion of an Integer to a Bit String

An integer x in the range 0 ≤ x < 2n may be converted to an n-long sequence of bits by using its 

binary expansion as shown below:
69

x = (x1 ∗ 2n–1) + (x2 ∗ 2n–2) + ...+ (xn–1 ∗ 2) + xn → {x1, ..., xn}

Note that the first bit of a sequence corresponds to the most significant bit of the corresponding 

integer, and the last bit corresponds to the least significant bit. 

Input:

1. C The non-negative integer to be converted.

Output:

1. b1, b2, ..., bn The bit string representation of the integer C.

Process:

1. Let (b1, b2, ..., bn) represent the bit string, where bi = 0 or 1, and b1 is the most 

significant bit, while bn is the least significant bit.

2. For any integer n that satisfies C < 2n, the bits bi shall satisfy:

∑

=

− = n

i

i

( n i ) C b

1

2

3. Return b1, b2, ..., bn.

In this Standard, the binary length of the integer C is defined as the smallest integer n that 

satisfies C < 2n.

C.3 Probabilistic Primality Tests

A probabilistic primality test may be required during the generation and validation of prime 

numbers. An approved robust probabilistic primality test shall be selected and used.

There are several probabilistic algorithms available. The Miller-Rabin probabilistic primality 

tests described in Appendices C.3.1 and C.3.2 are versions of a procedure due to M.O. Rabin, 

based in part on ideas of Gary L. Miller; one of these versions shall be used as the Miller-Rabin 

test discussed below. For more information, see [4]. For these tests, let RBG be an approved 

random bit generator.

There are several Lucas probabilistic primality tests available; the version provided in [5] is 

specified in Appendix C.3.3.

This Standard allows two alternatives for testing primality: either using several iterations of only 

the Miller-Rabin test, or using the iterated Miller-Rabin test, followed by a single Lucas test. The 

value of iterations (as used in Appendices C.3.1 and C.3.2) depends on the algorithm being used, 

the security strength, the error probability used, the length (in bits) of the candidate prime and 
70

the type of tests to be performed. Tables C.1, C.2 and C.3 list the minimum number of iterations

of the Miller-Rabin tests that shall be performed.

As stated in Appendix F, if the definition of the error probability that led to the values of the 

number of Miller-Rabin tests for p and q in Tables C.1, C.2 and C.3 is not conservative enough, 

the prescribed number of Miller-Rabin tests can be followed by a single Lucas test. Since there 

are no known non-prime values that pass the two test combination (i.e., the indicated number of 

rounds of the Miller-Rabin test with randomly selected bases, followed by one round of the 

Lucas test), the two test combination may provide additional assurance of primality over the use 

of only the Miller-Rabin test. For DSA, the two-test combination may provide better 

performance. However, the Lucas test is not required when testing the p1, p2, q1 and q2 values 

for primality when generating RSA primes. See Appendix F for further information.

Table C.1. Minimum number of Miller-Rabin iterations for DSA

Parameters M-R Tests Only M-R Tests when followed 

by One Lucas test

 p: 1024 bits

 q: 160 bits

Error probability = 80 2−

For p and q: 40 For p: 3

For q: 19

 p: 2048 bits

 q: 224 bits

Error probability = 112 2−

For p and q: 56 For p: 3

For q: 24

 p: 2048 bits

 q: 256 bits

Error probability = 112 2−

For p and q: 56 For p: 3

For q: 27

 p: 3072 bits

 q: 256 bits

 Error probability = 128 2−

For p and q: 64 For p: 2

For q: 27
71

Table C.2. Minimum number of rounds of M-R testing when generating primes for use in 

RSA Digital Signatures

Parameters M-R Tests Only

1 p , 2 p , q1 and 2 q > 100 bits

p and q: 512 bits

Error probability = 80 2−

For 1 p , 2 p , q1 and 2 q : 28

For p and q: 5

1 p , 2 p , q1 and 2 q > 140 bits

p and q: 1024 bits

Error probability = 112 2−

For 1 p , p2 , q1 and 2 q : 38

For p and q: 5

1 p , p2 , q1 and 2 q > 170 bits

p and q: 1536 bits

Error probability = 2 –128

For 1 p , p2 , q1 and, 2 q : 41

For p and q: 4

Table C.3. Minimum number of rounds of M-R testing when generating primes for use in 

RSA Digital Signatures using an error probability of 2–100

Parameters M-R Tests Only

p1 , p2 , q1 and q2 > 100 bits

p and q: 512

For p1 , p2 , q1 and q2 : 38

For p and q: 7

p1 , p2 , q1 and q2 > 140 bits

p and q: 1024 bits

For 1 p , 2 p , q1 and 2 q : 32

For p and q: 4 

p1 , p2 , q1 and q2 > 170 bits

p and q: 1536 bits

For p1 , p2 , q1 and q2 : 27

For p and q: 3

C.3.1 Miller-Rabin Probabilistic Primality Test

Let RBG be an approved random bit generator.

Input:

1. w The odd integer to be tested for primality. This will be either p or 

q, or one of the auxiliary primes p1, p2, q1 or q2.
72

2. iterations The number of iterations of the test to be performed; the value 

shall be consistent with Table C.1, C.2 or C.3.

Output:

1. status The status returned from the validation procedure, where status is 

either PROBABLY PRIME or COMPOSITE.

Process:

1. Let a be the largest integer such that 2a divides w−1.

2. m = (w−1) / 2a.

3. wlen = len (w).

4. For i = 1 to iterations do

4.1 Obtain a string b of wlen bits from an RBG. 

Comment: Ensure that 1 < b < w−1.

4.2 If ((b ≤ 1) or (b ≥ w−1)), then go to step 4.1.

4.3 z = bm mod w.

4.4 If ((z = 1) or (z = w − 1)), then go to step 4.7.

4.5 For j = 1 to a − 1 do.

4.5.1 z = z

2 mod w.

4.5.2 If (z = w−1), then go to step 4.7.

4.5.3 If (z = 1), then go to step 4.6.

4.6 Return COMPOSITE.

4.7 Continue. Comment: Increment i for the do-loop in 

step 4.

5. Return PROBABLY PRIME.

C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test

This method provides additional information when an error is encountered that may be useful 

when generating or validating RSA moduli. Let RBG be an approved random bit generator.

Input:

1. w The odd integer to be tested for primality. This will be either p or 

q, or one of the auxiliary primes p1, p2, q1 or q2.
73

2. iterations The number of iterations of the test to be performed; the value 

shall be consistent with Table C.1, C.2 or C.3.

Output:

1. status The status returned from the validation procedure, where status is 

either PROBABLY PRIME, PROVABLY COMPOSITE 

WITH FACTOR (returned with the factor), and PROVABLY 

COMPOSITE AND NOT A POWER OF A PRIME.

Process:

1. Let a be the largest integer such that 2a divides w–1.

2. m = (w–1) / 2a.

3. wlen = len (w).

4. For i = 1 to iterations do

4.1 Obtain a string b of wlen bits from an RBG.

Comment: Ensure that 1 < b < w–1.

4.2 If ((b ≤ 1) or (b ≥ w–1)), then go to step 4.1.

4.3 g = GCD(b, w).

4.4 If (g > 1), then return PROVABLY COMPOSITE WITH FACTOR and the 

value of g.

4.5 z = bm mod w.

4.6 If ((z = 1) or (z = w – 1)), then go to step 4.15.

4.7 For j = 1 to a – 1 do.

4.7.1 x = z. Comment: x ≠ 1 and x ≠ w–1. 

4.7.2 z = x

2 mod w.

4.7.3 If (z = w–1), then go to step 4.15.

4.7.4 If (z = 1), then go to step 4.12.

4.8 x = z. Comment: x = b(w–1)/2 mod w and x ≠ w–1.

4.9 z = x

2 mod w.

4.10 If (z = 1), then go to step 4.12.

4.11 x = z. Comment: x = b(w–1) mod w and x ≠ 1.
74

4.12 g = GCD(x–1, w).

4.13 If (g > 1), then return PROVABLY COMPOSITE WITH FACTOR and the 

value of g.

4.14 Return PROVABLY COMPOSITE AND NOT A POWER OF A PRIME.

4.15 Continue. Comment: Increment i for the do-loop in 

step 4.

5. Return PROBABLY PRIME.

C.3.3 (General) Lucas Probabilistic Primality Test

The following process or its equivalent shall be used as the Lucas test. 

Input: 

C The candidate odd integer to be tested for primality.

Output:

status Where status is either PROBABLY PRIME or COMPOSITE.

Process:

1. Test whether C is a perfect square (see Appendix C.4). If so, return (COMPOSITE).

2. Find the first D in the sequence {5, –7, 9, –11, 13, –15, 17, ...} for which the Jacobi 

symbol ( ) C

D = –1. See Appendix C.5 for an approved method to compute the Jacobi 

Symbol. If ( ) C

D = 0 for any D in the sequence, return (COMPOSITE).

3. K = C+1.

4. Let Kr Kr – 1 ... K0 be the binary expansion of K, with Kr = 1.

5. Set Ur = 1 and Vr = 1.

6. For i = r–1 to 0, do

6.1 Utemp = Ui+1 Vi+1 mod C.

6.2 Vtemp = mod . 2

2

1

2

1 C

Vi+ + DUi+

6.3 If (Ki = 1), then Comment: If Ki = 1, then do steps 6.3.1 and 6.3.2; 

otherwise, do steps 6.3.3 and 6.3.4.

6.3.1 Ui = 

2

Utemp +Vtemp mod C.
75

6.3.2 Vi = 

2

Vtemp + DUtemp mod C.

Else

6.3.3 Ui = Utemp.

6.3.4 Vi = Vtemp.

7. If (U0 = 0), then return (PROBABLY PRIME). Otherwise, return (COMPOSITE).

Steps 6.2, 6.3.1 and 6.3.2 contain expressions of the form A/2 mod C, where A is an integer, and 

C is an odd integer. If A/2 is not an integer (i.e., A is odd), then A/2 mod C may be calculated as 

(A+C)/2 mod C. Alternatively, A/2 mod C = A·(C+1)/2 mod C, for any integer A, without regard 

to A being odd or even.

C.4 Checking for a Perfect Square

The following algorithm may be used to determine whether an n-bit positive integer C is a 

perfect square:

Input:

C The integer to be checked.

Output:

status Where status is either PERFECT SQUARE or NOT A PERFECT SQUARE.

Process:

1. Set n, such that 2n > C ≥ 2(n−1).

2. m = n/2.

3. i = 0.

4. Select X0, such that 2m > X0 ≥ 2(m−1).

5. Repeat

5.1 i = i + 1.

5.2 Xi = ((Xi–1)

2 + C)/(2Xi–1).

Until (Xi)

2 < 2m + C.

6. If C =  Xi  2

, then

status = PERFECT SQUARE.

Else

status = NOT A PERFECT SQUARE.
76

7. Return status.

Notes:

1. By starting with X0 > (1/2) Sqrt(C), X0 − Sqrt(C)is guaranteed to be less than X0 . 

This inequality is maintained in step 5; i.e., Xi − Sqrt(C)< Xi for all i.

2. For i ≥ 1, 0 ≤ Xi − Sqrt(C) = (Xi–1 − Sqrt(C))2 / (2 Xi–1) < X0/2i.

In particular, 0 ≤ Xm − Sqrt(C) < 1. If Sqrt(C) were an integer, then it would

be equal to the floor of Xm .

3. In general, the inequality Xi − Sqrt(C) < 1 will occur for values of i that are much less 

than m. To detect this, the fact that 2(m−1) ≤ Sqrt(C) < Xi for all i ≥ 1 can be used,

 Xi − Sqrt(C) = ((Xi)

2 − C)/( Xi + Sqrt(C)) 

≤ ((Xi)

2 − C)/( 2 Sqrt(C)) 

≤ ((Xi)

2 − C)/(2m)

Thus, the condition (Xi)

2 < 2m + C implies that Xi − Sqrt(C) < 1.

C.5 Jacobi Symbol Algorithm

This routine computes the Jacobi symbol 



 





n

a .

Jacobi( ):

Input:

a Any integer. For this Standard, the initial value is in the sequence {5, –7, 9, –11, 

13, –15, 17, ...}, as determined by Appendix C.3.3. 

n Any integer. For this Standard, the initial value is the candidate being tested, as 

determined by Appendix C.3.3.

Output:

result The calculated Jacobi symbol.

Process:

1. a = a mod n. Comment: a will be in the range 0 ≤ a < n.

2. If a = 1, or n = 1, then return (1).

3. If a = 0, then return (0).

4. Define e and a1 such that a = 2ea1, where a1 is odd.
77

5. If e is even, then s = 1.

Else if ((n ≡ 1 (mod 8)) or (n ≡ 7 (mod 8))), then s = 1.

Else if ((n ≡ 3 (mod 8)) or (n ≡ 5 (mod 8)), then s = –1.

6. If ((n ≡ 3 (mod 4)) and (a1 ≡ 3 (mod 4))), then s = –s.

7. n1 = n mod a1.

8. Return (s * Jacobi (n1, a1)). Comment: Call this routine recursively.

Example: Compute the Jacobi symbol for a = 5 and n = 3439601197:

1. n is not 1, and a is not 1, so proceed to Step 2.

2. a is not 0, so proceed to Step 3.

3. 5 = 20 * 5, so e = 0, and a1 = 5.

4. e is even, so s = 1.

5. a1 is not congruent to 3 mod 4, so do not change s.

6. n1 = 2 = n mod 5.

7. Compute and return (1 * Jacobi(2, 5)). This calls Jacobi recursively. Compute the Jacobi 

symbol for a = 2 and n = 5:

7.1 n is not 1, and a is not 1, so proceed to Step 7.2.

7.2 a is not 0, so proceed to Step 7.3.

7.3 2 = 21 * 1, so e = 1, and a1 = 1.

7.4 e is odd, and n ≡ 5 (mod 8), so set s = –1.

7.5 n is not 3 mod 4, and a1 is not 3 mod 4, so proceed to step 7.6.

7.6 n1 = 0 = n mod 1.

7.7 Return (–1 * Jacobi(0, 1) = –1). This calls Jacobi recursively. Compute the Jacobi 

symbol for a = 0 and n = 1:

7.7.1 n = 1, so return 1.

Thus, Jacobi (0,1) = 1, so Jacobi (2,5) = –1*(1) = –1, and Jacobi (5, 3439601197) = 1* (–1) = –1.

C.6 Shawe-Taylor Random_Prime Routine

This routine is recursive and may be used to construct a provable prime number using a hash 

function.
78

Let Hash( ) be the selected hash function, and let outlen be the bit length of the hash function 

output block. The following process or its equivalent shall be used to generate a prime number 

for this constructive method.

ST_Random_Prime ( ):

Input:

1. length The length of the prime to be generated.

2. input_seed The seed to be used for the generation of the requested prime.

Output:

1. status The status returned from the generation routine, where status is 

either SUCCESS or FAILURE. If FAILURE is returned, then 

zeros are returned as the other output values.

2. prime The requested prime.

3 prime_seed A seed determined during generation.

4. prime_gen_counter (Optional) A counter determined during the generation of the 

prime.

Process:

1. If (length < 2), then return (FAILURE, 0, 0 {, 0}).

2. If (length ≥ 33), then go to step 14.

3. prime_seed = input_seed.

4. prime_gen_counter = 0.

Comment: Generate a pseudorandom integer 

c of length bits.

5. c = Hash(prime_seed) ⊕ Hash(prime_seed + 1).

6. c = 2length – 1 + (c mod 2length – 1

).

7. c = (2 ∗ c / 2 ) + 1.

Comment: Set prime to the least odd 

integer greater than or equal to c.

8. prime_gen_counter = prime_gen_counter + 1.

9. prime_seed = prime_seed + 2.
79

10. Perform a deterministic primality test on c. For example, since c is small, its primality 

can be tested by trial division. See Appendix C.7.

11. If (c is a prime number), then 

11.1 prime = c.

11.2 Return (SUCCESS, prime, prime_seed {, prime_gen_counter}).

12. If (prime_gen_counter > (4 ∗ length)), then return (FAILURE, 0, 0 {, 0}).

13. Go to step 5.

14. (status, c0, prime_seed, prime_gen_counter) = (ST_Random_Prime (( length / 2

+ 1), input_seed).

15. If FAILURE is returned, return (FAILURE, 0, 0 {, 0}).

16. iterations = length / outlen – 1.

17. old_counter = prime_gen_counter.

Comment: Generate a pseudorandom integer 

x in the interval [2length – 1

, 2length].

18. x = 0.

19. For i = 0 to iterations do

x = x + (Hash(prime_seed + i) ∗ 2i × outlen).

20. prime_seed = prime_seed + iterations + 1.

21. x = 2length – 1 + (x mod 2length – 1).

Comment: Generate a candidate prime c in 

the interval [2length – 1, 2length].

22. t = x / (2c0).

23. If (2tc0 + 1 > 2length), then t = 2length – 1 / (2c0).

24. c = 2tc0 + 1.

25. prime_gen_counter = prime_gen_counter + 1.

Comment: Test the candidate prime c for 

primality; first pick an integer a between 2 

and c – 2.

26. a = 0.

27. For i = 0 to iterations do

a = a + (Hash(prime_seed + i) ∗ 2 i * outlen).
80

28. prime_seed = prime_seed + iterations + 1.

29. a = 2 + (a mod (c – 3)).

30. z = a

2t

mod c.

31. If ((1 = GCD(z – 1, c)) and (1 = 0c z mod c)), then 

31.1 prime = c.

31.2 Return (SUCCESS, prime, prime_seed {, prime_gen_counter}).

32. If (prime_gen_counter ≥ ((4 ∗ length) + old_counter)), then return (FAILURE, 0, 0 

{, 0}).

33. t = t + 1.

34. Go to step 23.

C.7 Trial Division

An integer is proven to be prime by showing that it has no prime factors less than or equal to its 

square root. This procedure is not recommended for testing any integers longer than 10 digits.

To prove that c is prime: 

1. Prepare a table of primes less than c . This can be done by applying the sieve procedure in 

Appendix C.8.

2. Divide c by every prime in the table. If c is divisible by one of the primes, then declare that c

is composite and exit. If convenient, c may be divided by composite numbers. For example, 

rather than preparing a table of primes, it might be more convenient to divide by all integers 

except those divisible by 3 or 5.

3. Otherwise, declare that c is prime and exit.

C.8 Sieve Procedure

A sieve procedure is described as follows: Given a sequence of integers Y0, Y0 + 1,..., Y0 + J, a 

sieve will identify the integers in the sequence that are divisible by primes up to some selected 

limit.

Note that the definitions of the mathematical symbols in this process (e.g., h, L, M, p) are 

internal to this process only, and should not be confused with their use elsewhere in this 

Standard.

Start by selecting a factor base of all the primes pj, from 2 up to some selected limit L. The value 

of L is arbitrary and may be determined by computer limitations. A good, typical value of L

would be anywhere from 103 to 105. 
81

1. Compute Sj =Y0 mod pj for all pj in the factor base. 

2. Initialize an array of length J + 1 to zero. 

3. Starting at Y0 – Sj + pj , let every pj

th element of the array be set to 1. Do this for the 

entire length of the array and for every j.

4. When finished, every location in the array that has the value 1 is divisible by some small 

prime, and is therefore a composite.

The array can be either a bit array for compactness when memory is small, or a byte array for 

speed when memory is readily available. There is no need to sieve the entire sieve interval at 

once. The array can be partitioned into suitably small pieces, sieving each piece before going on 

to the next piece. When finished, every location with the value 0 is a candidate for prime testing.

The amount of work for this procedure is approximately M log log L, where M is the length of 

the sieve interval; this is a very efficient procedure for removing composite candidates for 

primality testing. If L = 105

, the sieve will remove about 96% of all composites.

In some cases, rather than having a set of consecutive integers to sieve, the set of integers to be 

tested consists of integers lying in an arithmetic progression Y0, Y0 + h, Y0 + 2h, ..., Y0 + Jh, 

where h is large and not divisible by any primes in the factor base.

1. Select a factor base and initialize an array of length J + 1 to 0. 

2. Compute Sj =Y0 mod pj for all pj in the factor base. 

3. Compute Tj = h mod pj and r = – Sj Tj

– 1 mod pj. 

4. Starting at Y0 + r, let every pj

th element of the array be set to 1. Do this for the entire 

length of the array and for every j. Note that the position Y0 + r in the array actually 

denotes the number Y0 + rh.

5. When finished, every location in the array that has the value 1 is divisible by some small 

prime and is therefore composite.

Note: The prime “2” takes the longest amount of time (M/2) to sieve, since it touches the most 

locations in the sieve array. An easy optimization is to combine the initialization of the sieve 

array with the sieving of the prime “2”. It is also possible to sieve the prime “3” during 

initialization. These optimizations can save about 1/3 of the total sieve time.

C.9 Compute a Probable Prime Factor Based on Auxiliary Primes

This routine constructs a probable prime (a candidate for p or q) using two auxiliary prime 

numbers and the Chinese Remainder Theorem (CRT).

Input:

r1 and r2 Two odd prime numbers satisfying 

log2(r1r2) ≤ (nlen/2) – log2(nlen/2) – 6.
82

nlen The desired length of n, the RSA modulus.

e The public verification exponent.

security_strength The minimum security strength required for random number 

generation.

Output:

status The status returned from the generation procedure, where status is 

either SUCCESS or FAILURE. If FAILURE is returned, then 

zeros are returned as the other output values.

private_prime_factor The prime factor of n.

X The random number used during the generation of the 

private_prime_factor.

Process:

1. If (GCD(2r1, r2) ≠ 1), then return (FAILURE, 0, 0).

2. R = ((r2

–1 mod 2r1) * r2) – (((2r1)

–1 mod r2) * 2r1).

Comment: Apply the CRT, so that R ≡ 1 (mod 2r1)

and R ≡ –1 (mod r2).

3. Generate a random number X using an approved random number generator that 

supports the security_ strength, such that ( 2)(2 ) (2 1) 2 1 2 ≤ ≤ − nlen / − nlen / X .

4. Y = X + ((R – X) mod 2r1r2). Comment: Y is the first odd integer ≥ X, such that r1

is a prime factor of Y–1, and r2 is a prime factor of 

Y+1.

Comment: Determine the requested prime number 

by constructing candidates from a sequence and 

performing primality tests.

5. i = 0.

6. If (Y ≥ 2nlen/2), then go to step 3.

7. If (GCD(Y–1, e) = 1), then

7.1 Check the primality of Y as specified in Appendix C.3. If PROBABLY PRIME

is not returned, go to step 8.

7.2 private_prime_factor = Y.

7.3 Return (SUCCESS, private_prime_factor, X).

8. i = i + 1.
83

9. If (i ≥ 5(nlen/2)), then return (FAILURE, 0, 0).

10. Y = Y + (2r1r2).

11. Go to step 6.

C.10 Construct a Provable Prime (possibly with Conditions), Based on 

Contemporaneously Constructed Auxiliary Provable Primes

The following process (or its equivalent) shall be used to generate an L-bit provable prime p (a 

candidate for one of the prime factors of an RSA modulus). Note that the use of p in this 

specification is used generically; both RSA prime factors p and q may be generated using this 

method.

If a so-called “strong prime” is required, this process can generate primes p1 and p2 (of specified 

bit-lengths N1 and N2) that divide p−1 and p+1, respectively. The resulting prime p will satisfy 

the conditions traditionally required of a strong prime, provided that the requested bit-lengths for 

p1 and p2 have appropriate sizes. 

Regardless of the bit-lengths selected for p1 and p2, the quantity p−1 will have a prime divisor p0

whose bit-length is slightly more than half that of p. In addition, the quantity 

p0 −1 will have a prime divisor whose bit-length is slightly more than half that of p0.

This algorithm requires that N1 + N2 ≤ L – L/2 – 4. Values for N1 and N2 should be chosen 

such that N1 + N2 ≤ (L/2) – log2(L) – 7, to ensure that the algorithm can generate as many as 5L

distinct candidates for p.

Let Hash be the selected hash function to be used, and let outlen be the bit length of the hash 

function output block.

Provable_Prime_Construction():

Input:

1. L A positive integer equal to the requested bit-length for p. Note that 

acceptable values for L= nlen/2 are computed as specified in 

Appendix B.3.1, criteria 2(b) and (c), with nlen assuming a value 

specified in Table B.1. 

2. N1 A positive integer equal to the requested bit-length for p1. If N1 ≥

2, then p1 is an odd prime of N1 bits; otherwise, p1 = 1. Acceptable 

values for N1 ≥ 2 are provided in Table B.1

3. N2 A positive integer equal to the requested bit-length for p2. If N2 ≥

2, then p2 is an odd prime of N2 bits; otherwise, p2 = 1.

Acceptable values for N2 ≥ 2 are provided in Table B.1

4. firstseed A bit string equal to the first seed to be used.
84

5. e The public verification exponent.

Output:

1. status The status returned from the generation procedure, where status is 

either SUCCESS or FAILURE. If FAILURE is returned, then 

zeros are returned as the other output values.

2. p, p1, p2 The required prime p, along with p1 and p2 having the property 

that p1 divides p−1 and p2 divides p+1.

3. pseed A seed determined during generation.

Process:

1. If L, N1, and N2 are not acceptable, then, return (FAILURE, 0, 0, 0, 0).

Comment: Generate p1 and p2, as well as the prime

p0.

2. If N1 = 1, then 

2.1 p1 = 1.

2.2 p2seed = firstseed. 

3. If N1 ≥ 2, then 

3.1 Using N1 as the length and firstseed as the input_seed, use the random prime 

generation routine in Appendix C.6 to obtain p1 and p2seed. 

3.2 If FAILURE is returned, then return (FAILURE, 0, 0, 0, 0).

4. If N2 = 1, then 

4.1 p2 = 1.

4.2 p0seed = p2seed. 

5. If N2 ≥ 2, then

5.1 Using N2 as the length and p2seed as the input_seed, use the random prime 

generation routine in Appendix C.6 to obtain p2 and p0seed. 

5.2 If FAILURE is returned, then return (FAILURE, 0, 0, 0, 0).

6. Using L / 2 + 1 as the length and p0seed as the input_seed, use the random prime 

generation routine in Appendix C.6 to obtain p0 and pseed. If FAILURE is returned, 

then return (FAILURE, 0, 0, 0, 0).

Comment: Generate a (strong) prime p in the 

interval [( 2 )(2L−1), 2L −1].
85

7. iterations = L / outlen −1.

8. pgen_counter = 0.

Comment: Generate pseudo-random x in the 

interval [( 2 )(2L−1)−1, 2L −1]. 

9. x = 0.

10. For i = 0 to iterations do

x = x + (Hash(pseed + i))∗ 2 i * outlen.

11. pseed = pseed + iterations + 1. 

12. x = ( 2 )(2L−1

) + ( x mod (2L − ( 2 )(2L−1) ) ).

Comment: Generate a candidate for the prime p.

13. If (GCD(p0p1, p2) ≠ 1), then return (FAILURE, 0, 0, 0, 0).

14. Compute y in the interval [1, p2] such that 0 = ( y p0 p1–1) mod p2.

15. t = ((2 y p0 p1) + x)/(2 p0 p1 p2).

16. If ((2(t p2 − y) p0 p1 + 1) > 2L), then 

t = ( (2 y p0 p1) + ( 2 )(2L−1) ) / (2 p0 p1 p2). 

Comment: p satisfies

0 = ( p–1) mod (2p0 p1) and 

0 = ( p+1) mod p2.

17. p = 2(t p2 − y) p0 p1 + 1.

18. pgen_counter = pgen_counter + 1. 

19. If (GCD(p–1, e) = 1), then

Comment: Choose an integer a in the interval [2, p–

2].

19.1 a = 0

19.2 For i = 0 to iterations do 

a = a + (Hash(pseed + i))∗ 2 i * outlen.

19.3 pseed = pseed + iterations + 1.

19.4 a = 2 + (a mod (p–3)).

Comment: Test p for primality:

19.5 z = a

2(t p2 − y) p1 mod p.
86

19.6 If ((1 = GCD(z–1, p)) and (1 = (z p0 mod p)), then return (SUCCESS, p, p1, 

p2, pseed).

20. If (pgen_counter ≥ 5L), then return (FAILURE, 0, 0, 0, 0).

21. t = t + 1.

22. Go to step 16.
87

Appendix D: Recommended Elliptic Curves for Federal 

Government Use

This collection of elliptic curves is recommended for Federal government use and contains 

choices for the private key length and underlying fields. These curves were generated using 

SHA-1 and the method given in the ANS X9.62 and IEEE Standard 1363-2000 standards. This 

appendix describes the process that was used. Note that these curves are the same as those 

included in the previous version of this Standard.

D.1 NIST Recommended Elliptic Curves

D.1.1 Choices

D.1.1.1 Choice of Key Lengths 

The principal parameters for elliptic curve cryptography are the elliptic curve E and a designated 

point G on E called the base point. The base point has order n, which is a large prime. The 

number of points on the curve is hn for some integer h (the cofactor), which is not divisible by n. 

For efficiency reasons, it is desirable to have the cofactor be as small as possible.

All of the curves given below have cofactors 1, 2, or 4. As a result, the private and public keys 

for a curve are approximately the same length. 

D.1.1.2 Choice of Underlying Fields 

For each key length, two kinds of fields are provided.

• A prime field is the field GF(p), which contains a prime number p of elements. The 

elements of this field are the integers modulo p, and the field arithmetic is implemented 

in terms of the arithmetic of integers modulo p. 

• A binary field is the field GF(2m), which contains 2m elements for some m (called the 

degree of the field). The elements of this field are the bit strings of length m, and the field 

arithmetic is implemented in terms of operations on the bits. 

The security strengths for five ranges of the bit length of n is provided in SP 800-57. For the 

field GF(p), the security strength is dependent on the length of the binary expansion of p. For the 

field GF(2m), the security strength is dependent on the value of m. Table E-1 provides the bit 

lengths of the various underlying fields of the curves provided in this appendix. Column 1 lists 

the ranges for the bit length of n (also see Table 1 in Section 6.1.1). Column 2 identifies the 

value of p used for the curves over prime fields, where len(p) is the length of the binary 

expansion of the integer p. Column 3 provides the value of m for the curves over binary fields.
88

Table D-1: Bit Lengths of the Underlying Fields of the Recommended Curves

Bit Length of n Prime Field Binary Field

161 – 223 len(p) = 192 m = 163

224 – 255 len(p) = 224 m = 233

256 – 383 len(p) = 256 m = 283

384 – 511 len(p) = 384 m = 409

≥ 512 len(p) = 521 m = 571

D.1.1.3 Choice of Basis for Binary Fields

To describe the arithmetic of a binary field, it is first necessary to specify how a bit string is to be 

interpreted. This is referred to as choosing a basis for the field. There are two common types of 

bases: a polynomial basis and a normal basis. 

• A polynomial basis is specified by an irreducible polynomial modulo 2, called the field 

polynomial. The bit string (am–1 ... a2 a1 a0) is taken to represent the polynomial

am–1 t m–1 +...+ a2 t

2 + a1 t + a0

over GF(2). The field arithmetic is implemented as polynomial arithmetic modulo p(t), 

where p(t) is the field polynomial.

• A normal basis is specified by an element θ of a particular kind. The bit string (a0 a1 a2

... am–1) is taken to represent the element

a0θ + a1θ2 + a2θ2 2

+ ...+ am–1θ2 m–1.

Normal basis field arithmetic is not easy to describe or efficient to implement in general, 

except for a special class called Type T low-complexity normal bases. For a given field 

degree m, the choice of T specifies the basis and the field arithmetic (see Appendix D.3). 

There are many polynomial bases and normal bases from which to choose. The following 

procedures are commonly used to select a basis representation. 

• Polynomial Basis: If an irreducible trinomial t

m + tk + 1 exists over GF (2), then the field 

polynomial p(t) is chosen to be the irreducible trinomial with the lowest-degree middle 

term t

k . If no irreducible trinomial exists, then a pentanomial t m + t a + t b + tc + 1 is 

selected. The particular pentanomial chosen has the following properties: the second term 

t

a has the lowest degree m; the third term tb has the lowest degree among all irreducible 

pentanomials of degree m and second term t

a

; and the fourth term t

c has the lowest degree 

among all irreducible pentanomials of degree m, second term t

a

, and third term t

b

.
89

• Normal Basis: Choose the Type T low-complexity normal basis with the smallest T. 

For each binary field, the parameters are given for the above basis representations. 

D.1.1.4 Choice of Curves 

Two kinds of curves are given: 

• Pseudo-random curves are those whose coefficients are generated from the output of a 

seeded cryptographic hash function. If the domain parameter seed value is given along 

with the coefficients, it can be easily verified that the coefficients were generated by that 

method. 

• Special curves are those whose coefficients and underlying field have been selected to 

optimize the efficiency of the elliptic curve operations. 

For each curve size range, the following curves are given: 

→ A pseudo-random curve over GF(p).

→ A pseudo-random curve over GF(2m).

→ A special curve over GF(2m) called a Koblitz curve or anomalous binary curve. 

The pseudo-random curves were generated as specified in ANS X9.62 using SHA-1.

D.1.1.5 Choice of Base Points 

Any point of order n can serve as the base point. Each curve is supplied with a sample base 

point G = (Gx , Gy ). Users may want to generate their own base points to ensure cryptographic 

separation of networks. See ANS X9.62 or IEEE Standard 1363-2000.

D.1.2 Curves over Prime Fields

For each prime p, a pseudo-random curve 

E : y

2 ≡ x3 – 3x +b (mod p)

of prime order n is listed4

. (Thus, for these curves, the cofactor is always h = 1.) The following 

parameters are given: 

• The prime modulus p

• The order n

• The 160-bit input seed SEED to the SHA-1 based algorithm (i.e., the domain parameter 

seed)

• The output c of the SHA-1 based algorithm 

 

4 The selection a ≡ -3 for the coefficient of x was made for reasons of efficiency; see IEEE Std 1363-2000.
90

• The coefficient b (satisfying b2c ≡ –27 (mod p)) 

• The base point x coordinate Gx

• The base point y coordinate Gy

The integers p and n are given in decimal form; bit strings and field elements are given in 

hexadecimal. 

D.1.2.1 Curve P-192

p = 6277101735386680763835789423207666416083908700390324961279

n = 6277101735386680763835789423176059013767194773182842284081 

SEED = 3045ae6f c8422f64 ed579528 d38120ea e12196d5 

c = 3099d2bb bfcb2538 542dcd5f b078b6ef 5f3d6fe2 c745de65 

b = 64210519 e59c80e7 0fa7e9ab 72243049 feb8deec c146b9b1

G x = 188da80e b03090f6 7cbf20eb 43a18800 f4ff0afd 82ff1012

G y = 07192b95 ffc8da78 631011ed 6b24cdd5 73f977a1 1e794811

D.1.2.2 Curve P-224

p = 2695994666715063979466701508701963067355791626002630814351

0066298881 

n = 2695994666715063979466701508701962594045780771442439172168

2722368061

SEED = bd713447 99d5c7fc dc45b59f a3b9ab8f 6a948bc5

c = 5b056c7e 11dd68f4 0469ee7f 3c7a7d74 f7d12111 6506d031 

218291fb

b = b4050a85 0c04b3ab f5413256 5044b0b7 d7bfd8ba 270b3943 

2355ffb4

G x = b70e0cbd 6bb4bf7f 321390b9 4a03c1d3 56c21122 343280d6 

115c1d21

G y = bd376388 b5f723fb 4c22dfe6 cd4375a0 5a074764 44d58199 

85007e34
91

D.1.2.3 Curve P-256

p = 1157920892103562487626974469494075735300861434152903141955

33631308867097853951 

n = 115792089210356248762697446949407573529996955224135760342

422259061068512044369 

SEED = c49d3608 86e70493 6a6678e1 139d26b7 819f7e90 

c = 7efba166 2985be94 03cb055c 75d4f7e0 ce8d84a9 c5114abc 

af317768 0104fa0d

b = 5ac635d8 aa3a93e7 b3ebbd55 769886bc 651d06b0 cc53b0f6 

3bce3c3e 27d2604b

G x = 6b17d1f2 e12c4247 f8bce6e5 63a440f2 77037d81 2deb33a0 

f4a13945 d898c296

G y = 4fe342e2 fe1a7f9b 8ee7eb4a 7c0f9e16 2bce3357 6b315ece 

cbb64068 37bf51f5

D.1.2.4 Curve P-384

p = 3940200619639447921227904010014361380507973927046544666794

8293404245721771496870329047266088258938001861606973112319

n = 3940200619639447921227904010014361380507973927046544666794

6905279627659399113263569398956308152294913554433653942643

SEED = a335926a a319a27a 1d00896a 6773a482 7acdac73

c = 79d1e655 f868f02f ff48dcde e14151dd b80643c1 406d0ca1 

0dfe6fc5 2009540a 495e8042 ea5f744f 6e184667 cc722483

b = b3312fa7 e23ee7e4 988e056b e3f82d19 181d9c6e fe814112 

0314088f 5013875a c656398d 8a2ed19d 2a85c8ed d3ec2aef

G x = aa87ca22 be8b0537 8eb1c71e f320ad74 6e1d3b62 8ba79b98 

59f741e0 82542a38 5502f25d bf55296c 3a545e38 72760ab7

G y = 3617de4a 96262c6f 5d9e98bf 9292dc29 f8f41dbd 289a147c 

e9da3113 b5f0b8c0 0a60b1ce 1d7e819d 7a431d7c 90ea0e5f 
92

D.1.2.5 Curve P-521

p = 686479766013060971498190079908139321726943530014330540939

446345918554318339765605212255964066145455497729631139148

0858037121987999716643812574028291115057151 

n = 686479766013060971498190079908139321726943530014330540939

446345918554318339765539424505774633321719753296399637136

3321113864768612440380340372808892707005449 

SEED = d09e8800 291cb853 96cc6717 393284aa a0da64ba

c = 0b4 8bfa5f42 0a349495 39d2bdfc 264eeeeb 077688e4 

4fbf0ad8 f6d0edb3 7bd6b533 28100051 8e19f1b9 ffbe0fe9 

ed8a3c22 00b8f875 e523868c 70c1e5bf 55bad637 

b = 051 953eb961 8e1c9a1f 929a21a0 b68540ee a2da725b 

99b315f3 b8b48991 8ef109e1 56193951 ec7e937b 1652c0bd 

3bb1bf07 3573df88 3d2c34f1 ef451fd4 6b503f00

G x = c6 858e06b7 0404e9cd 9e3ecb66 2395b442 9c648139 

053fb521 f828af60 6b4d3dba a14b5e77 efe75928 fe1dc127 

a2ffa8de 3348b3c1 856a429b f97e7e31 c2e5bd66

G y = 118 39296a78 9a3bc004 5c8a5fb4 2c7d1bd9 98f54449 

579b4468 17afbd17 273e662c 97ee7299 5ef42640 c550b901 

3fad0761 353c7086 a272c240 88be9476 9fd16650

D.1.3 Curves over Binary Fields 

For each field degree m, a pseudo-random curve is given, along with a Koblitz curve. The 

pseudo-random curve has the form 

E: y

2 + x y = x3 + x2 + b,

and the Koblitz curve has the form 

Ea: y

2 + x y = x 3 + ax 2 + 1,

where a = 0 or 1. 

For each pseudorandom curve, the cofactor is h = 2. The cofactor of each Koblitz curve is h = 2 

if a = 1, and h = 4 if a = 0. 
93

The coefficients of the pseudo-random curves, and the coordinates of the base points of both 

kinds of curves, are given in terms of both the polynomial and normal basis representations 

discussed in Appendix D.1.1.3. 

For each m, the following parameters are given: 

Field Representation: 

• The normal basis type T

• The field polynomial (a trinomial or pentanomial) 

Koblitz Curve: 

• The coefficient a

• The base point order n

• The base point x coordinate G x

• The base point y coordinate G y

Pseudo-random curve: 

• The base point order n

Pseudo-random curve (Polynomial Basis representation): 

• The coefficient b

• The base point x coordinate G x

• The base point y coordinate G y

Pseudo-random curve (Normal Basis representation): 

• The 160-bit input seed SEED to the SHA-1 based algorithm (i.e., the domain parameter 

seed)

• The coefficient b (i.e., the output of the SHA-1 based algorithm) 

• The base point x coordinate G x

• The base point y coordinate G y

Integers (such as T, m, and n) are given in decimal form; bit strings and field elements are given 

in hexadecimal. 

D.1.3.1 Degree 163 Binary Field

T = 4 

p(t) = t 163 + t 7 + t 6 + t 3 + 1 
94

D.1.3.1.1 Curve K-163

a = 1 

n = 5846006549323611672814741753598448348329118574063

Polynomial Basis:

G x = 2 fe13c053 7bbc11ac aa07d793 de4e6d5e 5c94eee8

G y = 2 89070fb0 5d38ff58 321f2e80 0536d538 ccdaa3d9

Normal Basis:

G x = 0 5679b353 caa46825 fea2d371 3ba450da 0c2a4541

G y = 2 35b7c671 00506899 06bac3d9 dec76a83 5591edb2

D.1.3.1.2 Curve B-163

n = 5846006549323611672814742442876390689256843201587 

Polynomial Basis: 

b = 2 0a601907 b8c953ca 1481eb10 512f7874 4a3205fd

G x = 3 f0eba162 86a2d57e a0991168 d4994637 e8343e36

G y = 0 d51fbc6c 71a0094f a2cdd545 b11c5c0c 797324f1

Normal Basis: 

SEED = 85e25bfe 5c86226c db12016f 7553f9d0 e693a268 

b = 6 645f3cac f1638e13 9c6cd13e f61734fb c9e3d9fb

G x = 0 311103c1 7167564a ce77ccb0 9c681f88 6ba54ee8 

G y = 3 33ac13c6 447f2e67 613bf700 9daf98c8 7bb50c7f

D.1.3.2 Degree 233 Binary Field

T = 2 

p(t) = t 233 + t 74 + 1 
95

D.1.3.2.1 Curve K-233

a = 0

n = 345087317339528189371737793113851276057094098886225212\

6328087024741343

Polynomial Basis:

G x = 172 32ba853a 7e731af1 29f22ff4 149563a4 19c26bf5 

0a4c9d6e efad6126

G y = 1db 537dece8 19b7f70f 555a67c4 27a8cd9b f18aeb9b 

56e0c110 56fae6a3

Normal Basis:

G x = 0fd e76d9dcd 26e643ac 26f1aa90 1aa12978 4b71fc07 

22b2d056 14d650b3

G y = 064 3e317633 155c9e04 47ba8020 a3c43177 450ee036 

d6335014 34cac978

D.1.3.2.2 Curve B-233

n = 690174634679056378743475586227702555583981273734501355\

5379383634485463

Polynomial Basis: 

b = 066 647ede6c 332c7f8c 0923bb58 213b333b 20e9ce42 

81fe115f 7d8f90ad

G x = 0fa c9dfcbac 8313bb21 39f1bb75 5fef65bc 391f8b36 

f8f8eb73 71fd558b

G y = 100 6a08a419 03350678 e58528be bf8a0bef f867a7ca 

36716f7e 01f81052
96

Normal Basis: 

SEED = 74d59ff0 7f6b413d 0ea14b34 4b20a2db 049b50c3

b = 1a0 03e0962d 4f9a8e40 7c904a95 38163adb 82521260 

0c7752ad 52233279 

G x = 18b 863524b3 cdfefb94 f2784e0b 116faac5 4404bc91 

62a363ba b84a14c5

G y = 049 25df77bd 8b8ff1a5 ff519417 822bfedf 2bbd7526 

44292c98 c7af6e02 

D.1.3.3 Degree 283 Binary Field

T = 6 

p(t) = t 283 + t 12 + t 7 + t 5 + 1 

D.1.3.3.1 Curve K-283

a = 0 

n = 3885337784451458141838923813647037813284811733793061324

295874997529815829704422603873 

Polynomial Basis:

G x = 503213f 78ca4488 3f1a3b81 62f188e5 53cd265f 23c1567a 

16876913 b0c2ac24 58492836

G y = 1ccda38 0f1c9e31 8d90f95d 07e5426f e87e45c0 e8184698 

e4596236 4e341161 77dd2259

Normal Basis:

G x = 3ab9593 f8db09fc 188f1d7c 4ac9fcc3 e57fcd3b db15024b 

212c7022 9de5fcd9 2eb0ea60

G y = 2118c47 55e7345c d8f603ef 93b98b10 6fe8854f feb9a3b3 

04634cc8 3a0e759f 0c2686b1
97

D.1.3.3.2 Curve B-283

n = 7770675568902916283677847627294075626569625924376904889

109196526770044277787378692871 

Polynomial Basis: 

b = 27b680a c8b8596d a5a4af8a 19a0303f ca97fd76 45309fa2 

a581485a f6263e31 3b79a2f5

G x = 5f93925 8db7dd90 e1934f8c 70b0dfec 2eed25b8 557eac9c 

80e2e198 f8cdbecd 86b12053

G y = 3676854 fe24141c b98fe6d4 b20d02b4 516ff702 350eddb0 

826779c8 13f0df45 be8112f4

Normal Basis: 

SEED = 77e2b073 70eb0f83 2a6dd5b6 2dfc88cd 06bb84be

b = 157261b 894739fb 5a13503f 55f0b3f1 0c560116 66331022 

01138cc1 80c0206b dafbc951

G x = 749468e 464ee468 634b21f7 f61cb700 701817e6 bc36a236 

4cb8906e 940948ea a463c35d

G y = 62968bd 3b489ac5 c9b859da 68475c31 5bafcdc4 ccd0dc90 

5b70f624 46f49c05 2f49c08c

D.1.3.4 Degree 409 Binary Field

T = 4

p(t) = t 409 + t 87 + 1 

D.1.3.4.1 Curve K-409

a = 0 

n = 33052798439512429947595765401638551991420234148214060964\

232439502288071128924919105067325845777745801409636659061

7731358671 
98

Polynomial Basis:

G x = 060f05f 658f49c1 ad3ab189 0f718421 0efd0987 e307c84c 

27accfb8 f9f67cc2 c460189e b5aaaa62 ee222eb1 b35540cf 

e9023746

Gy = 1e36905 0b7c4e42 acba1dac bf04299c 3460782f 918ea427 

e6325165 e9ea10e3 da5f6c42 e9c55215 aa9ca27a 5863ec48 

d8e0286b

Normal Basis:

G x = 1b559c7 cba2422e 3affe133 43e808b5 5e012d72 6ca0b7e6 

a63aeafb c1e3a98e 10ca0fcf 98350c3b 7f89a975 4a8e1dc0 

713cec4a

G y = 16d8c42 052f07e7 713e7490 eff318ba 1abd6fef 8a5433c8 

94b24f5c 817aeb79 852496fb ee803a47 bc8a2038 78ebf1c4 

99afd7d6

D.1.3.4.2 Curve B-409

n = 6610559687902485989519153080327710398284046829642812192

84648798304157774827374805208143723762179110965979867288

366567526771 

Polynomial Basis: 

b = 021a5c2 c8ee9feb 5c4b9a75 3b7b476b 7fd6422e f1f3dd67 

4761fa99 d6ac27c8 a9a197b2 72822f6c d57a55aa 4f50ae31 

7b13545f

G x = 15d4860 d088ddb3 496b0c60 64756260 441cde4a f1771d4d 

b01ffe5b 34e59703 dc255a86 8a118051 5603aeab 60794e54 

bb7996a7
99

G y = 061b1cf ab6be5f3 2bbfa783 24ed106a 7636b9c5 a7bd198d 

0158aa4f 5488d08f 38514f1f df4b4f40 d2181b36 81c364ba 

0273c706

Normal Basis: 

SEED = 4099b5a4 57f9d69f 79213d09 4c4bcd4d 4262210b

b = 124d065 1c3d3772 f7f5a1fe 6e715559 e2129bdf a04d52f7 

b6ac7c53 2cf0ed06 f610072d 88ad2fdc c50c6fde 72843670 

f8b3742a

G x = 0ceacbc 9f475767 d8e69f3b 5dfab398 13685262 bcacf22b 

84c7b6dd 981899e7 318c96f0 761f77c6 02c016ce d7c548de 

830d708f

G y = 199d64b a8f089c6 db0e0b61 e80bb959 34afd0ca f2e8be76 

d1c5e9af fc7476df 49142691 ad303902 88aa09bc c59c1573 

aa3c009a

D.1.3.5 Degree 571 Binary Field

T = 10 

p(t) = t 571 + t 10 + t 5 + t 2 + 1

D.1.3.5.1 Curve K-571

a = 0 

n = 1932268761508629172347675945465993672149463664853217499

32861762572575957114478021226813397852270671183470671280

08253514612736749740666173119296824216170925035557336852

76673 

Polynomial Basis:

G x = 26eb7a8 59923fbc 82189631 f8103fe4 ac9ca297 0012d5d4 

60248048 01841ca4 43709584 93b205e6 47da304d b4ceb08c 
100

bbd1ba39 494776fb 988b4717 4dca88c7 e2945283 a01c8972

G y = 349dc80 7f4fbf37 4f4aeade 3bca9531 4dd58cec 9f307a54 

ffc61efc 006d8a2c 9d4979c0 ac44aea7 4fbebbb9 f772aedc 

b620b01a 7ba7af1b 320430c8 591984f6 01cd4c14 3ef1c7a3

Normal Basis:

G x = 04bb2db a418d0db 107adae0 03427e5d 7cc139ac b465e593 

4f0bea2a b2f3622b c29b3d5b 9aa7a1fd fd5d8be6 6057c100 

8e71e484 bcd98f22 bf847642 37673674 29ef2ec5 bc3ebcf7

G y = 44cbb57 de20788d 2c952d7b 56cf39bd 3e89b189 84bd124e 

751ceff4 369dd8da c6a59e6e 745df44d 8220ce22 aa2c852c 

fcbbef49 ebaa98bd 2483e331 80e04286 feaa2530 50caff60

D.1.3.5.2 Curve B-571

n = 3864537523017258344695351890931987344298927329706434998

65723525145151914228956042453614399938941577308313388112

19269444862468724628168130702345282883033324113931911052

85703 

Polynomial Basis: 

b = 2f40e7e 2221f295 de297117 b7f3d62f 5c6a97ff cb8ceff1 

cd6ba8ce 4a9a18ad 84ffabbd 8efa5933 2be7ad67 56a66e29 

4afd185a 78ff12aa 520e4de7 39baca0c 7ffeff7f 2955727a

G x = 303001d 34b85629 6c16c0d4 0d3cd775 0a93d1d2 955fa80a 

a5f40fc8 db7b2abd bde53950 f4c0d293 cdd711a3 5b67fb14 

99ae6003 8614f139 4abfa3b4 c850d927 e1e7769c 8eec2d19

G y = 37bf273 42da639b 6dccfffe b73d69d7 8c6c27a6 009cbbca 

1980f853 3921e8a6 84423e43 bab08a57 6291af8f 461bb2a8 

b3531d2f 0485c19b 16e2f151 6e23dd3c 1a4827af 1b8ac15b
101

Normal Basis: 

SEED = 2aa058f7 3a0e33ab 486b0f61 0410c53a 7f132310

b = 3762d0d 47116006 179da356 88eeaccf 591a5cde a7500011 

8d9608c5 9132d434 26101a1d fb377411 5f586623 f75f0000 

1ce61198 3c1275fa 31f5bc9f 4be1a0f4 67f01ca8 85c74777

G x = 0735e03 5def5925 cc33173e b2a8ce77 67522b46 6d278b65 

0a291612 7dfea9d2 d361089f 0a7a0247 a184e1c7 0d417866 

e0fe0feb 0ff8f2f3 f9176418 f97d117e 624e2015 df1662a8 

G y = 04a3642 0572616c df7e606f ccadaecf c3b76dab 0eb1248d 

d03fbdfc 9cd3242c 4726be57 9855e812 de7ec5c5 00b4576a 

24628048 b6a72d88 0062eed0 dd34b109 6d3acbb6 b01a4a97

D.2 Implementation of Modular Arithmetic

The prime moduli in the above examples are of a special type (called generalized Mersenne 

numbers) for which modular multiplication can be carried out more efficiently than in general. 

This section provides the rules for implementing this faster arithmetic for each of the prime 

moduli appearing in the examples. 

The usual way to multiply two integers (mod m) is to take the integer product and reduce it (mod 

m). One therefore has the following problem: given an integer A less than m 

2

, compute 

B = A mod m.

In general, one must obtain B as the remainder of an integer division. If m is a generalized 

Mersenne number, however, then B can be expressed as a sum or difference (mod m) of a small 

number of terms. To compute this expression, the integer sum or difference can be evaluated and

the result reduced modulo m. The latter reduction can be accomplished by adding or subtracting 

a few copies of m. 

The prime modulus p for each of the five example curves is a generalized Mersenne number. 

D.2.1 Curve P-192 

The modulus for this curve is p = 2 192 – 2 64 – 1. Every integer A less than p

2 can be written as

0

64

1

128

2

192

3

256

4

320

5 A = A ⋅ 2 + A ⋅ 2 + A ⋅ 2 + A ⋅ 2 + A ⋅ 2 + A ,

where each Ai is a 64-bit integer. As a concatenation of 64-bit words, this can be denoted by 

A = (A5 || A4 || A3 || A2 || A0).

The expression for B is 

B = T + S1 + S2 + S3 mod p,
102

where the 192-bit terms are given by 

T = (A2 || A1 || A0)

S1 = (A3 || A3)

S2 = (A4 || A4 || 0)

S3 = (A5 || A5 || A5).

D.2.2 Curve P-224 

The modulus for this curve is 2 2 1. 224 96 p = − + Every integer A less than p

2 can be written as:

2 2 2 2 2 ,

2 2 2 2 2 2 2 2

0

32

1

64

2

96

3

128

4

160

5

192

6

224

7

256

8

288

9

320

10

352

11

384

12

416

13

A A A A A A

A A A A A A A A A

⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

where each Ai is a 32-bit integer. As a concatenation of 32-bit words, this can be denoted by:

A = ( A13 || A12 || ...|| A0 ).

The expression for B is:

B = T + S 1 + S 2 – D1 – D2 mod p,

where the 224-bit terms are given by:

T = ( A6 || A5 || A4 || A3 || A2 || A1 || A0 )

S1 = ( A10 || A9 || A8 || A7 || 0 || 0 || 0 )

S2 = ( 0 || A13 || A12 || A11 || 0 || 0 || 0 )

D1 = ( A13 || A12 || A11 || A10 || A9 || A8 || A7 )

D2 = ( 0 || 0 || 0 || 0 || A13 || A12 || A11 ).

D.2.3 Curve P-256 

The modulus for this curve is p = 2256 – 2224 + 2192 + 296 – 1. Every integer A less than p

2 can be 

written as:

2 2 2 2 2 2 2 ,

2 2 2 2 2 2 2 2

0

32

1

64

2

96

3

128

4

160

5

192

6

224

7

256

8

288

9

320

10

352

11

384

12

416

13

448

14

480

15

A A A A A A A A

A A A A A A A A A

⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

where each A i is a 32-bit integer. As a concatenation of 32-bit words, this can be denoted by 

A = (A15 || A14 || ⋅ ⋅ ⋅ || A0 ).

The expression for B is:
103

B = T + 2S1 + 2S2 + S3 + S4 – D1 – D2 – D3 – D4 mod p,

where the 256-bit terms are given by:

T = ( A7 || A6 || A5  || A4 || A3 || A2 || A1 || A0 )

S1 = ( A15 || A14 || A13 || A12 || A11 || 0 || 0 || 0 )

S2 = ( 0 || A15 || A14 || A13 || A12 || 0 || 0 || 0 )

S3 = ( A15 || A14 || 0 || 0 || 0 || A10 || A9 || A8 )

S4 = ( A8 || A13 || A15 || A14 || A13 || A11 || A10 || A9 )

D1 = ( A10 || A8 || 0 || 0 || 0 || A13 || A12 || A11 )

D2 = ( A11 || A9 || 0 || 0 || A15 || A14 || A13 || A12)

D3 = ( A12 || 0 || A10 || A9 || A8 || A15 || A14 || A13 )

D4 = ( A13 || 0 || A11 || A10 || A9 || 0 || A15 || A14 )

D.2.4 Curve P-384 

The modulus for this curve is p = 2 384 – 2 128 – 2 96 + 2 32 – 1. Every integer A less than p

2

can 

be written as:

2 2 2 2 2 2 2 ,

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

0

32

1

64

2

96

3

128

4

160

5

192

6

224

7

256

8

288

9

320

10

352

11

384

12

416

13

448

14

480

15

512

16

544

17

576

18

608

19

640

20

672

21

704

22

736

23

A A A A A A A A

A A A A A A A A

A A A A A A A A A

⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ +

where each A i is a 32-bit integer. As a concatenation of 32-bit words, this can be denoted by 

A = (A23 || A22 || ⋅ ⋅ ⋅ || A0 ).

The expression for B is: 

B = T + 2S1 + S2 + S3 + S4 + S5 + S6 – D1 – D2 – D3 mod p,

where the 384-bit terms are given by:

T = (A11 || A10 || A9 || A8 || A7 || A6 || A5 || A4 || A3 || A2 || A1 || A0 )

S1 = ( 0 || 0 || 0 || 0 || 0 || A23 || A22 || A21 || 0 || 0 || 0 || 0 )

S2 = (A23 || A22 || A21 || A20 || A19 || A18 || A17 || A16 || A15 || A14 || A13 || A12)

S3 = (A20 || A19 || A18 || A17 || A16 || A15 || A14 || A13 || A12 || A23|| A22|| A21)

S4 = ( A19 || A18 || A17 || A16 || A15 || A14 || A13 || A12 || A20 || 0 || A23 || 0 )

S5 = ( 0 || 0 || 0 || 0 || A23 || A22 || A21 || A20 || 0 || 0 || 0 || 0 )
104

S6 = ( 0 || 0 || 0 || 0 || 0 || 0 || A23 || A22 || A21 || 0 || 0 || A20 )

D1 = (A22 || A21 || A20 || A19 || A18 || A17 || A16 || A15 || A14 || A13 || A12 || A23 )

D2 = ( 0 || 0 || 0 || 0 || 0 || 0 || 0 || A23 || A22 || A21 || A20 || 0 )

D3 = ( 0 || 0 || 0 || 0 || 0 || 0 || 0 || A23 || A23 || 0 || 0 || 0 ).

D.2.5 Curve P-521

The modulus for this curve is p = 2 521 – 1. Every integer A less than p

2 can be written

A = A1 ⋅ 2521 + A0,

where each A i is a 521-bit integer. As a concatenation of 521-bit words, this can be denoted by 

A = (A1 || A0).

The expression for B is:

B = (A0 + A1) mod p.

D.3 Normal Bases

The elements of GF(2m) are expressed in terms of the type T normal basis5 B for GF(2m), for 

some T. Each element has a unique representation as a bit string: 

( a0 a1 ... am–1 ).

The arithmetic operations are performed as follows.

Addition: addition of two elements is implemented by bit-wise addition modulo 2. Thus, for 

example, 

(1100111) + (1010010) = (0110101).

Squaring: if 

α = ( a0 a1 ... am–1 )

then

α2 = (am–1 a0 a1 ... am–2 ).

Multiplication: to perform multiplication, a function F(u,v) is constructed on inputs

 

5 It is assumed in this section that m is odd and T is even, since this is the only case considered in this Standard.
105

u = ( u0 u1 ... um–1 ) and v = ( v0 v1 ... vm–1 )

as follows.

1. Set p ← Tm + 1.

2. Let u be an integer having order T modulo p.

3. Compute the sequence F (1), F (2),... ,F (p–1) as follows:

3.1 Set w← 1.

3.2 For j from 0 to T–1 do

3.2.1 Set n ← w.

3.2.2 For i = 0 to m–1 do

3.2.2.1 Set F(n) ← i.

3.2.2.2 Set n ← 2n mod p.

3.2.3 Set w ← uw mod p.

4. Output the formula:

∑

−

=

= + −

2

1

( 1) ( ) ( , ) : .

p

k

F k F p k F u v u v

This computation need only be performed once per basis.

Given the function F for B, the product

( c0 c1 ... cm–1 ) = ( a0 a1 ... am–1 ) * ( b0 b1 ... bm–1 )

is computed as follows:

1. Set ( u0 u1 ... um–1 ) ← ( a0 a1 . . . am–1 ).

2. Set ( v0 v1 ... vm–1 ) ← ( b0 b1 . . . bm–1 ).

3. For k = 0 to m – 1 do

3.1 Compute

ck = F(u, v).

3.2 Set u ← LeftShift (u) and v ← LeftShift (v), where LeftShift denotes the circular 

left shift operation.

4. Output c = ( c0 c1 ... cm–1 ).
106

Example: For the type 4 normal basis for GF(27), p = 29 and u = 12 or 17. Thus, the values of F

are given by:

F (1) = 0 F (8) = 3 F (15) = 6 F (22) = 5 

F (2) = 1 F (9) = 3 F (16) = 4 F (23) = 6 

F (3) = 5 F (10) = 2 F (17) = 0 F (24) = 1 

F (4) = 2 F (11) = 4 F (18) = 4 F (25) = 2 

F (5) = 1 F (12) = 0 F (19) = 2 F (26) = 5 

F (6) = 6 F (13) = 4 F (20) = 3 F (27) = 1 

F (7) = 5 F (14) = 6 F (21) = 3 F (28) = 0 

Therefore,

F (u, v) = u0 v1 + u1 (v0 + v2 + v5 + v6 ) + u2 (v1 + v3 + v4 + v5 ) + u3 (v2 + v5 ) +

u4 (v2 + v6 ) + u5 (v1 + v2 + v3 + v6 ) + u6 (v1 + v4 + v5 + v6 ).

Thus, if 

a = (1 0 1 0 1 1 1) and b = (1 1 0 0 0 0 1),

then 

c0 = F ( (1 0 1 0 1 1 1), (1 1 0 0 0 0 1) ) = 1,

c1 = F ( (0 1 0 1 1 1 1), (1 0 0 0 0 1 1) ) = 0,



c6 = F ( (1 1 0 1 0 1 1), (1 1 1 0 0 0 0) ) = 1,

so that c = ab = (1 0 1 1 0 0 1).

D.4 Scalar Multiplication on Koblitz Curves

This section describes a particularly efficient method of computing the scalar multiple nP on the 

Koblitz curve Ea over GF(2m).

The operation τ is defined by:

τ (x, y) = (x

2

, y

2

).

When the normal basis representation is used, then the operation τ is implemented by performing 

right circular shifts on the bit strings representing x and y. 

Given m and a, define the following parameters: 

• C is some integer greater than 5.
107

• μ = (–1)1–a.

• For i = 0 and i = 1, define the sequence si(m) by:

si(0) = 0, si(1) = 1 – i,

si(m) = μ • si(m – 1) – 2 si(m – 2) + (–1)i

• Define the sequence V(m)

V(0) = 2, V(1) = μ

V(m) = μ • v(m –1) – 2V(m – 2).

For the example curves, the quantities si(m) and V(m) are as follows.

Curve K-163:

s0(163) = 2579386439110731650419537

s1(163) = –755360064476226375461594

V(163) = –4845466632539410776804317

Curve K-233:

s0(233) = –27859711741434429761757834964435883

s1(233) = –44192136247082304936052160908934886

V(233) = –137381546011108235394987299651366779

Curve K-283:

s0(283) = –665981532109049041108795536001591469280025

s1(283) = 1155860054909136775192281072591609913945968

V(283) = 7777244870872830999287791970962823977569917

Curve K-409:

s0(409) = –18307510456002382137810317198756461378590542487556869338419259

s1(409) = –8893048526138304097196653241844212679626566100996606444816790

V(409)= 10457288737315625927447685387048320737638796957687575791173829

Curve K-571:

s0(571) = –3737319446876463692429385892476115567147293964596131024123406420\

235241916729983261305

s1(571) = –3191857706446416099583814595948959674131968912148564658610565117\

58982848515832612248752
108

V(571)= –1483809269816914138996191402970514903645425741804939362329123395\

34208516828973111459843

The following algorithm computes the scalar multiple nP on the Koblitz curve Ea over GF(2m). 

The average number of elliptic additions and subtractions is at most ∼ 1 + (m/3), and is at most ∼

m/3 with probability at least 1 – 25–C

.

1. For i = 0 to 1 do

1.1 n′ ←  n / 2a–C + (m–9) / 2.

1.2 g′ ← si(m) · n′.

1.3 h′ ←  g′ / 2m .

1.4 j′ ← V(m) · h′.

1.5 l′ ← Round((g′ + j′) / 2(m+5) / 2).

1.6 λi ← l′ / 2C.

1.7 fi ← Round(λi).

1.8 ηi ← λi – fi..

1.9 hi ← 0.

2. η ← 2 η0 + μ η1.

3. If (η ≥ 1),

then

if (ηo – 3 μη1 < –1)

then set h1 ← μ

else set h0 ← 1.

else

if (η0 + 4 μ η1 ≥ 2)

then set h1 ← μ.

4. If (η < –1)

then

if (η0 – 3 μ η1 ≥ 1)

then set h1 ← – μ

else set h0 ← –1.
109

else

if (η0 + 4 μ η1 < –2)

then set h1 ← – μ.

5. q0 ← f0 + h0.

6. q1 ← f1 + h1.

7. r0 ← n – (s0 + μ s1) q0 – 2s1 q1.

8. r1 ← s1 q0 – s0 q1.

9. Set Q ← O.

10. P0 ← P.

11. While ((r0 ≠ 0) or (r1 ≠ 0))

11.1 If (r0 odd), then

11.1.1 set u ← 2 – (r0 – 2 r1 mod 4).

11.1.2 set r0 ← r0 – u.

11.1.3 if (u = 1), then set Q ← Q + P0.

11.1.4 if (u = –1), then set Q ← Q – P0.

11.2 Set P0 ← τP0.

11.3 Set (r0 , r1) ← (r1 + μr0 /2, – r0 /2).

Endwhile

12. Output Q.

D.5 Generation of Pseudo-Random Curves (Prime Case)

Let l be the bit length of p, and define 

v =  ( l – 1) /160

w = l – 160v – 1.

1. Choose an arbitrary 160-bit string s as the domain parameter seed.

2. Compute h = SHA-1(s).

3. Let h0 be the bit string obtained by taking the w rightmost bits of h. 

4. Let z be the integer whose binary expansion is given by the 160-bit string s. 

5. For i from 1 to v do: 
110

5.1 Define the 160-bit string si to be binary expansion of the integer 

 (z + i) mod (2 160). 

5.2 Compute hi = SHA-1(si).

6. Let h be the bit string obtained by the concatenation of h0 , h1, ..., hv as follows: 

h = h0 || h1 || ...|| hv.

7. Let c be the integer whose binary expansion is given by the bit string h. 

8. If ((c = 0 or 4c + 27 ≡ 0 (mod p))), then go to Step 1. 

9. Choose integers a, b ∈GF(p) such that 

c b2 ≡ a

3 (mod p).

(The simplest choice is a = c and b = c. However, one may want to choose differently for 

performance reasons.) 

10. Check that the elliptic curve E over GF(p) given by y 

2

= x

3 + ax + b has suitable order. If 

not, go to Step 1. 

D.6 Verification of Curve Pseudo-Randomness (Prime Case)

Given the 160-bit domain parameter seed value s, verify that the coefficient b was obtained from 

s via the cryptographic hash function SHA-1 as follows. 

Let l be the bit length of p, and define 

v =  ( l – 1) /160

w = l – 160v – 1.

1. Compute h = SHA-1(s).

2. Let h0 be the bit string obtained by taking the w rightmost bits of h. 

3. Let z be the integer whose binary expansion is given by the 160-bit string s. 

4. For i = 1 to v do

4.1 Define the 160-bit string si to be binary expansion of the integer 

 (z + i) mod (2160 ). 

4.2 Compute hi = SHA-1(si).

5. Let h be the bit string obtained by the concatenation of h0 , h1, ..., hv as follows: 

h = h0 || h1 || ...|| hv.
111

6. Let c be the integer whose binary expansion is given by the bit string h. 

7. Verify that b2

c ≡ –27 (mod p). 

D.7 Generation of Pseudo-Random Curves (Binary Case)

Let: 

v =  (m – 1) /B

w = m – Bv.

1. Choose an arbitrary 160-bit string s as the domain parameter seed.

2. Compute h = SHA-1(s).

3. Let h0 be the bit string obtained by taking the w rightmost bits of h. 

4. Let z be the integer whose binary expansion is given by the 160-bit string s. 

5. For i from 1 to v do: 

5.1 Define the 160-bit string si to be binary expansion of the integer 

 (z + i) mod (2160 ). 

5.2 Compute hi = SHA-1(si). 

6. Let h be the bit string obtained by the concatenation of h0 , h1, ..., hv as follows: 

h = h0 || h1 || ...|| hv.

7. Let b be the element of GF(2m) which binary expansion is given by the bit string h.

8. Choose an element a of GF(2m).

9. Check that the elliptic curve E over GF(2m) given by y

2 + xy = x3 + ax2 + b has suitable 

order. If not, go to Step 1. 

D.8 Verification of Curve Pseudo-Randomness (Binary Case)

Given the 160-bit domain parameter seed value s, verify that the coefficient b was obtained from 

s via the cryptographic hash function SHA-1 as follows. 

Define 

v =  (m – 1) /160

w= m – 160v

1. Compute h = SHA-1(s).

2. Let h0 be the bit string obtained by taking the w rightmost bits of h. 
112

3. Let z be the integer whose binary expansion is given by the 160-bit string s. 

4. For i = 1 to v do 

4.1 Define the 160-bit string si to be binary expansion of the integer (z + i) mod (2160 ).

4.2 Compute hi = SHA-1(si ). 

5. Let h be the bit string obtained by the concatenation of h0 , h1, ..., hv as follows: 

h = h0 || h1 || ...|| hv.

6. Let c be the element of GF(2m ) which is represented by the bit string h. 

7. Verify that c = b. 

D.9 Polynomial Basis to Normal Basis Conversion

Suppose that α is an element of the field GF(2m). Let p be the bit string representing α with 

respect to a given polynomial basis. It is desired to compute n, the bit string representing α with 

respect to a given normal basis. This is done via the matrix computation 

p Γ = n,

where Γ is an m-by-m matrix with entries in GF(2). The matrix Γ, which depends only on the 

bases, can be computed easily given its second-to-last row. The second-to-last row for each 

conversion is given the below.

Degree 163:

3 e173bfaf 3a86434d 883a2918 a489ddbd 69fe84e1

Degree 233: 

 0be 19b89595 28bbc490 038f4bc4 da8bdfc1 ca36bb05 853fd0ed 

0ae200ce 

Degree 283: 

3347f17 521fdabc 62ec1551 acf156fb 0bceb855 f174d4c1 7807511c 

9f745382 add53bc3 

Degree 409: 

0eb00f2 ea95fd6c 64024e7f 0b68b81f 5ff8a467 acc2b4c3 b9372843 

6265c7ff a06d896c ae3a7e31 e295ec30 3eb9f769 de78bef5 

Degree 571: 

7940ffa ef996513 4d59dcbf e5bf239b e4fe4b41 05959c5d 4d942ffd 

46ea35f3 e3cdb0e1 04a2aa01 cef30a3a 49478011 196bfb43 c55091b6 

1174d7c0 8d0cdd61 3bf6748a bad972a4 
113

Given the second-to-last row r of Γ, the rest of the matrix is computed as follows. Let β be the 

element of GF(2m) whose representation with respect to the normal basis is r. Then the rows of 

Γ, from top to bottom, are the bit strings representing the elements

βm–1, βm–2, ..., β2, β, 1

with respect to the normal basis. (Note that the element 1 is represented by the all-1 bit string.)

Alternatively, the matrix is the inverse of the matrix described in Appendix D.10.

More details of these computations can be found in Annex A.7 of the IEEE Standard 1363-2000

standard.

D.10 Normal Basis to Polynomial Basis Conversion

Suppose that α is an element of the field GF(2m). Let n be the bit string representing α with 

respect to a given normal basis. It is desired to compute p, the bit string representing α with 

respect to a given polynomial basis. This is done via the matrix computation 

n Γ = p,

where Γ is an m-by-m matrix with entries in GF(2). The matrix Γ, which depends only on the 

bases, can be computed easily given its top row. The top row for each conversion is given 

below.

Degree 163: 

 7 15169c10 9c612e39 0d347c74 8342bcd3 b02a0bef 

Degree 233: 

 149 9e398ac5 d79e3685 59b35ca4 9bb7305d a6c0390b cf9e2300 

253203c9 

Degree 283: 

 31e0ed7 91c3282d c5624a72 0818049d 053e8c7a b8663792 bc1d792e 

ba9867fc 7b317a99 

Degree 409: 

0dfa06b e206aa97 b7a41fff b9b0c55f 8f048062 fbe8381b 4248adf9 

2912ccc8 e3f91a24 e1cfb395 0532b988 971c2304 2e85708d 

Degree 571: 

452186b bf5840a0 bcf8c9f0 2a54efa0 4e813b43 c3d41496 06c4d27b 

487bf107 393c8907 f79d9778 beb35ee8 7467d328 8274caeb da6ce05a 

eb4ca5cf 3c3044bd 4372232f 2c1a27c4 
114

Given the top row r of Γ, the rest of the matrix is computed as follows. Let β be the element of 

GF(2 m) whose representation with respect to the polynomial basis is r. Then the rows of Γ, from 

top to bottom, are the bit strings representing the elements

β, β2, β22,... , β2m–1

with respect to the polynomial basis.

Alternatively, the matrix is the inverse of the matrix described in Appendix D.9.

More details of these computations can be found in Annex A.7 of the IEEE Std 1363-2000

standard.
115

Appendix E: A Proof that v = r in the DSA

(Informative)

The purpose of this appendix is to show that if M′ = M, r′ = r and s′ = s in the signature 

verification, then v = r′. Let Hash be an approved hash function. The following result is needed.

Lemma: Let p and q be primes such that q divides (p – 1), let h be a positive integer less 

than p, and let g = (h

(p–1)/q

mod p). Then (gq mod p) = 1, and if (m mod q) = (n 

mod q ), then (g

m

mod p) = (g

n

mod p).

Proof:

gqmod p = (h(p–1) / q mod p)

q mod p

= h(p – 1) mod p

= 1

by Fermat’s Little Theorem. Now let (m mod q) = (n mod q), i.e., m = (n + kq) for some integer 

k. Then

gm mod p = gn + kq mod p

= (gn gkq) mod p

= ((gn mod p) (gq mod p)

k

) mod p

= gn mod p,

since (gq mod p) = 1.

Proof of the main result:

Theorem: If M′ = M, r′ = r, and s′ = s in the signature verification, then v = r′.

Proof:

w = (s′)

–1 mod q = s–1 mod q

u1 = ((Hash(M ′))w) mod q = ((Hash(M))w) mod q

u2 = ((r′)w) mod q = (rw) mod q.

Now y = (g

x

mod p), so that by the lemma,

v = ((gu1y

u2

) mod p) mod q

= ((g

Hash(M)w yrw) mod p) mod q

= ((g

Hash(M)w gxrw) mod p) mod q

= ((g(Hash(M) + xr)w) mod p) mod q.
116

Also:

s = (k

–1 (Hash(M) + xr)) mod q.

Hence:

w = (k (Hash(M) + xr)

–1

) mod q

(Hash(M) + xr)w mod q = k mod q.

Thus, by the lemma:

v = (g

k mod p) mod q = r
117

Appendix F: Calculating the Required Number of Rounds of Testing 

Using the Miller-Rabin Probabilistic Primality Test

(Informative)

F.1 The Required Number of Rounds of the Miller-Rabin Primality Tests

The ideas of paper [1] were applied to estimate p k,t , the probability that an odd k-bit integer that 

passes t rounds of Miller-Rabin (M-R) testing is actually composite. The probability pk ,t is 

understood as the ratio of the number of odd composite numbers of a binary length k that can be 

expected to pass t rounds of M-R testing (with randomly generated bases) to the sum of that 

value and the number of odd prime integers of binary length k. This is equivalent to assuming 

that candidates selected for testing will be chosen uniformly at random from the entire set of odd 

k-bit integers. Following Pomerance, et al., pk,t can be (over) estimated by the ratio of the 

expected number of odd composite numbers of binary length k that will pass t rounds of M-R 

testing (with randomly generated bases) to the total number of odd primes of binary length k. 

From the perspective of a party charged with the responsibility of generating a k-bit prime, the 

objective is to determine a value of t such that k t p , is no greater than an acceptably small target 

value t et p arg .

Using [1], it is possible to compute an upper bound for pk ,t as a function of k and t. From this, 

an upper bound can be computed for t as a function of k and ptarget, the maximum allowed 

probability of accidentally generating a composite number. The following is an algorithm for 

computing t: 

1. For t = 1, 2 ... –log2(ptarget)/2

1.1 For M = 3, 4 ... 2 k −1 −1 (1)

1.1.1 Compute k t p , as in (2).

1.1.2 If pk,t ≤ ptarget

1.1.2.1 Accept t.

1.1.2.2 Stop.

In (1), k is the bit length of the candidate primes and (2) is as follows:



















 − = ⋅ ⋅ ⋅ + ∑ ∑ = = 













 − +

− − − − − − M

m

m

j j

k j

k k Mt k m m t

k t p k

3 2

( 1)

2 ( 1)

2

2

,

2

1 2 2

3

8( 6) 2.00743 ln(2) 2 2 π . (2)
118

Using this expression for t, the following methodologies are used for testing the DSA and RSA 

candidate primes.

F.2 Generating DSA Primes

For DSA, the maximum possible care must be taken when generating the primes p and q that are 

used for the domain parameters. The same primes p and q are used by many parties. This means 

that any weakness that these numbers may possess would affect multiple users. It also means 

that the primes are not generated very often; typically, an entire system uses the same set of 

domain parameters for an extended period of time. Therefore, in this case, some additional care 

is called for.

With this in mind, it may be too optimistic to simply subject candidate primes to t rounds of M-R 

testing, where the minimal acceptable value for t is determined according to (1) and (2) in 

Appendix F.1. This might be the case, for example, if there is a reason to doubt that the 

assumptions made in [1] have been satisfied during the process of selecting candidates for 

primality testing. One may gain more confidence in the process by performing some additional 

(different) primality test(s) on the candidates that survive the M-R testing. As another option, 

one could, of course, perform additional rounds of M-R testing. These considerations lead to the 

following alternatives: either (A) use the number of rounds of M-R testing determined according 

to (1) and (2) in Appendix F.1, and follow that with a single Lucas test (as recommended in ANS 

X9.31), or (B) use a (much) more conservative approach when determining t (e.g., as described 

below) and subject candidate primes to additional rounds of M-R testing.

One approach for strategy (B) would be to adopt the viewpoint of the majority of system users, 

who have no part in generating the (supposed) prime, but who must rely upon its primality for 

their security. Such parties may be concerned that the candidates for M-R testing have been 

selected in a fashion that deviates significantly from the uniform distribution – which was 

assumed when determining t according to (1) and (2) in Appendix F.1. In cases where the 

selection process could be unusually biased in some way, it is important to minimize the 

probability that a composite number will survive testing. It can be shown that for any k-bit odd 

composite number (regardless of how it was selected), the probability that it will pass t rounds of 

M-R testing with randomly chosen bases is less than −t 4 (although this is not a particularly tight 

bound). Selecting t such that 4–t≤ ptarget is equivalent to choosing t ≥ −log2(ptarget)/2. To ensure 

that a composite number has a probability no greater than ptarget of surviving the M-R tests, the 

number of rounds can be set at t = –log2(ptarget)/2. Even if the method of selecting candidates 

were so biased that it offered nothing but composite numbers for testing, it is reasonable to

expect that it would take at least 1/ ptarget attempts (which is greater than 4t ) before a composite 

number would slip through the t-round M-R testing process.

WARNING: As the discussion above illustrates, care must be taken when using the phrase 

“error probability” in connection with the recommended number of rounds of M-R testing. The 

probability that a composite number survives t rounds of Miller-Rabin testing is not the same as 
119

p k,t , which is the probability that a number surviving t rounds of Miller-Rabin testing is 

composite. Ordinarily, the latter probability is the one that should be of most interest to a party 

responsible for generating primes, while the former may be more important to a party 

responsible for validating the primality of a number generated by someone else. However, for 

sufficiently large k (e.g., k ≥ 51), it can be shown that p k,t ≤ 4–t under the same assumptions 

concerning the selection of candidates as those made to obtain formula (2) in Appendix F.1 (see 

[1].) In such cases, t = –log2(ptarget)/2 rounds of Miller-Rabin testing can be used both in 

generating and validating primes, with ptarget serving as an upper bound on both the probability 

that the generation process yields a composite number and the probability that a composite 

number would survive an attempt to validate its primality.

Table C.1 in Appendix C.3 identifies the minimum values for t when generating the primes p and 

q for DSA using either strategy (A) or (B) above. To obtain the t values shown in the column 

titled “M-R Tests Only”, the conservative strategy (B) was followed; those t values are sufficient 

to validate the primality of p and q. The t values shown in the column titled “M-R Tests when 

followed by One Lucas Test” result from following strategy (A) using computations (1) and (2)

in Appendix F.1.

F.3 Generating Primes for RSA Signatures 

When generating primes for the RSA signature algorithm, it is still very important to reduce the 

probability of errors in the M-R testing procedure. However, since the (probable) primes are 

used to generate a user’s key pair, if a composite number survives the testing process, the 

consequences of the error may be less dramatic than in the case of generating DSA domain 

parameters; only one user’s transactions are affected, rather than a domain of users. Furthermore, 

if the p or q value generated for some user is composite, the problem will not be undiscovered 

for long, since it is almost certain that signatures generated by that user will not be verifiable.

Therefore, when generating the RSA primes p and q, it is sufficient to use the number of rounds

derived from (1) and (2) in Appendix F.1 as the minimum number of M-R tests to be performed. 

However, if the definition of pk, t is not considered to be sufficiently conservative when testing p

and q, it is recommended that the t rounds of Miller-Rabin tests be followed by a single Lucas 

test.

The lengths for p and q that are recommended for use in RSA signature algorithms are 512, 1024 

and 1536 bits; recall that n = pq, so the corresponding lengths for n are 1024, 2048 and 3072 bits,

respectively. As currently specified in SP 800-57, Part 1, these lengths correspond to security 

strengths of 80, 112 and 128 bits, respectively. Hence, it makes sense to match the number of 

rounds of Miller-Rabin testing to the target error probability values of 2–80, 2–112, and 2–128. A 

probability of 2–100 is included for all prime lengths, since this probability has often been used in 

the past and may be acceptable for many applications. 

When generating the RSA primes p and q with conditions, it is sufficient to use the value t

derived from (1) and (2) as the minimum number of M-R tests to be performed when generating 
120

the auxiliary primes p1, p2, q1 and q2. It is not necessary to use an additional Lucas test on these 

numbers. In the extremely unlikely event that one of the numbers p1, p2, q1 or q2 is composite, 

there is still a high probability that the corresponding RSA prime (p or q) will satisfy the 

requisite conditions.

The sizes of 1 p , p2 , q1 , and q2 were chosen to ensure that, for an adversary with significant but 

not overwhelming resources, Lenstra’s elliptic curve factoring method [2] (against which there is 

no protection beyond choosing large p and q) is a more effective factoring algorithm than either 

the Pollard P–1 method [2], the Williams P+1 method [3] or various cycling methods [2]. For an 

adversary with overwhelming resources, the best all-purpose factoring algorithm is assumed to 

be the General Number Field Sieve [2].

Tables C.2 and C.3 in Appendix C.3 specify the minimum number of rounds of M-R testing 

when generating primes to be used in the construction of RSA signature key pairs. 
121

Appendix G: References

[1] I. Damgard, P. Landrock, and C. Pomerance, C. “Average Case Error Estimates for the 

Strong Probable Prime Test,” Mathematics of Computation, v. 61, No, 203, pp. 177-194, 

1993.

[2] A.J Menezes, P.C. Oorschot, and S.A. Vanstone. Handbook of Applied Cryptography. CRC 

Press, 1996.

[3] H.C. Williams. “A p+1 Method of factoring”. Math. Comp. 39, 225-234, 1982.

[4] D.E. Knuth, The Art of Computer Programming, Vol. 2, 3rd Ed., Addison-Wesley, 1998, 

Algorithm P, page 395.

[5] R. Baillie and S.S. Wagstaff Jr., Mathematics of Computation, V. 35 (1980), pages 1391 –

1417.