FIPS PUB 186 - DIGITAL SIGNATURE STANDARD (DSS)

FIPS PUB 186

FEDERAL INFORMATION PROCESSING STANDARDS PUBLICATION

1994 May 19

U.S. DEPARTMENT OF COMMERCE/National Institute of Standards and Technology

CATEGORY: COMPUTER SECURITY SUBCATEGORY: CRYPTOGRAPHY

U.S. DEPARTMENT OF COMMERCE, Ronald Brown, Secretary NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, 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 Section 111(d) of the Federal Property and Administrative Services Act of 1949 as amended by the Computer Security Act of 1987, Public Law 100-235. These mandates have given the Secretary of Commerce and NIST important responsibilities for improving the utilization and management of computer and related telecommunications systems in the Federal Government. The NIST, through the Computer Systems Laboratory, provides leadership, technical guidance, and coordination of Government efforts in the development of standards and guidelines in these areas.

Comments concerning Federal Information Processing Standards Publications are welcomed and should be addressed to the Director, Computer Systems Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899.

James H. Burrows, Director Computer Systems Laboratory

Abstract

This standard specifies a Digital Signature Algorithm (DSA) which 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 in proving to a third party that the signature was in fact generated by the signatory. This is known as nonrepudiation since the signatory cannot, at a later time, repudiate the signature.

Key words: ADP security, computer security, digital signatures, public-key cryptography, Federal Information Processing Standard.

Federal Information Processing Standards Publication 186

1994 May 19

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 111(d) of the Federal Property and Administrative Services Act of 1949 as amended by the Computer Security Act of 1987, Public Law 100-235.

Name of Standard: Digital Signature Standard (DSS).

Category of Standard: Computer Security; Cryptography.

Explanation: This Standard specifies a Digital Signature Algorithm (DSA) appropriate for applications requiring a digital rather than written signature. The DSA digital signature is a pair of large numbers represented in a computer as strings of binary digits. The digital signature is computed using a set of rules (i.e., the DSA) and a set of parameters such that the identity of the signatory and integrity of the data can be verified. The DSA provides the capability to generate and verify signatures. Signature generation makes use of a private key to generate a digital signature. Signature verification makes use of a public key which corresponds to, but is not the same as, the private key. Each user possesses a private and public key pair. Public keys are assumed to be known to the public in general. Private keys are never shared. Anyone can verify the signature of a user by employing that user's public key. Signature generation can be performed only by the possessor of the user's private key.

A hash function is used in the signature generation process to obtain a condensed version of data, called a message digest (see Figure 1). The message digest is then input to the DSA to generate the digital signature. The digital signature is sent to the intended verifier along with the signed data (often called the message). The verifier of the message and signature verifies the signature by using the sender's public key. The same hash function must also be used in the verification process. The hash function is specified in a separate standard, the Secure Hash Standard (SHS), FIPS 180. Similar procedures may be used to generate and verify signatures for stored as well as transmitted data.

Approving Authority: Secretary of Commerce.

Maintenance Agency: U.S. Department of Commerce, National Institute of Standards and Technology (NIST), Computer Systems Laboratory (CSL).

Applicability: This standard is applicable to all Federal departments and agencies for the protection of 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 which Federal departments and agencies operate or which are operated for them under contract. Adoption and use of this standard is available to private and commercial organizations.

Applications: The DSA authenticates the integrity of the signed data and the identity of the signatory. The DSA may also be used in proving to a third party that data was actually signed by the generator of the signature. The DSA is intended for use in electronic mail, electronic funds transfer, electronic data interchange, software distribution, data storage, and other applications which require data integrity assurance and data origin authentication.

Implementations: The DSA may be implemented in software, firmware, hardware, or any combination thereof. NIST is developing a validation program to test implementations for conformance to this standard. Information about the planned validation program can be obtained from the National Institute of Standards and Technology, Computer Systems Laboratory, Attn: DSS Validation, Gaithersburg, MD 20899.

Export Control: Implementations of this standard are subject to Federal Government export controls as specified in Title 15, Code of Federal Regulations, Parts 768 through 799. Exporters are advised to contact the Department of Commerce, Bureau of Export Administration for more information.

Patents: The Department of Commerce is not aware of any patents that would be infringed by this standard.

Implementation Schedule: This standard becomes effective December 1, 1994.

Specifications: Federal Information Processing Standard (FIPS186) Digital Signature Standard (DSS), (affixed).

1. INTRODUCTION

This publication prescribes the Digital Signature Algorithm (DSA) for digital signature generation and verification. Additional information is provided in Appendices 1 through 5.

2. GENERAL

When a message is received, the recipient may desire to verify that the message has not been altered in transit. Furthermore, the recipient may wish to be certain of the originator's identity. Both of these services can be provided by the DSA. A digital signature is an electronic analogue of a written signature in that the digital signature can be used in proving to the recipient or a third party that the message was, in fact, signed by the originator. Digital signatures may also be generated for stored data and programs so that the integrity of the data and programs may be verified at any later time.

This publication prescribes the DSA for digital signature generation and verification. In addition, the criteria for the public and private keys required by the algorithm are provided.

3. USE OF THE DSA ALGORITHM

The DSA is used by a signatory to generate a digital signature on data and by a verifier to verify the authenticity of the signature. Each signatory has a public and private key. The private key is used in the signature generation process and the public key is used in the signature verification process. For both signature generation and verification, the data which is referred to as a message, M, is reduced by means of the Secure Hash Algorithm (SHA) specified in FIPS YY. An adversary, who does not know the private key of the signatory, cannot generate the correct signature of the signatory. In other words, signatures cannot be forged. However, by using the signatory's public key, anyone can verify a correctly signed message. A means of associating public and private key pairs to the corresponding users is required. That is, there must be a binding of a user's identity and the user's public key. This binding may be certified by a mutually trusted party. For example, a certifying authority could sign credentials containing a user's public key and identity to form a certificate. Systems for certifying credentials and distributing certificates are beyond the scope of this standard. NIST intends to publish separate document(s) on certifying credentials and distributing certificates.

4. DSA PARAMETERS

The signature of a message M is the pair of numbers r and s computed according to the equations below:

r = (gk mod p) mod q and

s = (k-1(SHA(M) + xr)) mod q.

In the above, k-1 is the multiplicative inverse of k, mod q; i.e., (k- 1 k) mod q = 1 and 0 < k-1 < q. The value of SHA(M) is a 160-bit string output by the Secure Hash Algorithm specified in FIPS 180. For use in computing s, this string must be converted to an integer. The conversion rule is given in Appendix 2.2.

As an option, one may wish to check if r = 0 or s = 0. If either r = 0 or s = 0, a new value of k should be generated and the signature should be recalculated (it is extremely unlikely that r = 0 or s = 0 if signatures are generated properly).

The signature is transmitted along with the message to the verifier.

6. SIGNATURE VERIFICATION

Prior to verifying the signature in a signed message, p, q and g plus the sender's public key and identity are made available to the verifier in an authenticated manner.

Let Mþ, rþ, and sþ be the received versions of M, r, and s, respectively, and let y be the public key of the signatory. To verify the signature, the verifier first checks to see that 0 < rþ < q and 0 < sþ < q; if either condition is violated the signature shall be rejected. If these two conditions are satisfied, the verifier computes

w = (sþ)-1 mod q

u1 = ((SHA(Mþ))w) mod q

u2 = ((rþ)w) mod q v = (((g)u1 (y)u2) mod p) mod q.

If v = rþ, then the signature is verified and the verifier can have high confidence that the received message was sent by the party holding the secret key x corresponding to y. For a proof that v = rþ when Mþ = M, rþ = r, and sþ = s, see Appendix 1.

If v does not equal rþ, then the message may have been modified, the message may have been incorrectly signed by the signatory, or the message may have been signed by an impostor. The message should be considered invalid. APPENDIX 1. A PROOF THAT v = rþ

This appendix is for informational purposes only and is not required to meet the standard.

The purpose of this appendix is to show that if Mþ = M, rþ = r and sþ = s in the signature verification then v = rþ. We need the following easy result.

LEMMA. Let p and q be primes so that q divides p - 1, h a positive integer less than p, and g = h(p-1)/q mod p. Then gq mod p = 1, and if m mod q = n mod q, then gm mod p = gn mod p.

Proof: We have

gq mod 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. þ

We are now ready to prove the main result.

THEOREM. If Mþ = M, rþ = r, and sþ = s in the signature verification, then v = rþ.

Proof: We have

w = (sþ)-1 mod q = s-1 mod q

u1 = ((SHA(Mþ))w) mod q = ((SHA(M))w) mod q

u2 = ((rþ)w) mod q = (rw) mod q.

Now y = gx mod p, so that by the lemma,

v = ((gu1 yu2) mod p) mod q

= ((gSHA(M)w yrw) mod p) mod q

= ((gSHA(M)w gxrw) mod p) mod q = ((g(SHA(M)+xr)w) mod p) mod q.

Also

s = (k-1(SHA(M) + xr)) mod q.

Hence

w = (k(SHA(M) + xr)-1) mod q

(SHA(M) + xr)w mod q = k mod q.

Thus by the lemma,

v = (gk mod p) mod q

= r

= rþ. þ

APPENDIX 2. GENERATION OF PRIMES FOR THE DSA

This appendix includes algorithms for generating the primes p and q used in the DSA. These algorithms require a random number generator (see Appendix 3), and an efficient modular exponentiation algorithm. Generation of p and q shall be performed as specified in this appendix, or using other FIPS approved security methods.

2.1. A PROBABILISTIC PRIMALITY TEST

In order to generate the primes p and q, a primality test is required.

There are several fast probabilistic algorithms available. The following algorithm is a simplified version of a procedure due to M.O. Rabin, based in part on ideas of Gary L. Miller. [See Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1981, Algorithm P, page 379.] If this algorithm is iterated n times, it will produce a false prime with probability no greater than 1/4n. Therefore, n ò 50 will give an acceptable probability of error. To test whether an integer is prime:

Step 1. Set i = 1 and n ò 50.

Step 2. Set w = the integer to be tested, w = 1 + 2am, where m is odd and 2a is the largest power of 2 dividing w - 1.

Step 3. Generate a random integer b in the range 1 < b < w.

Step 4. Set j = 0 and z = bm mod w.

Step 5. If j = 0 and z = 1, or if z = w - 1, go to step 9.

Step 6. If j > 0 and z = 1, go to step 8.

Step 7. j = j + 1. If j < a, set z = z2 mod w and go to step 5.

Step 8. w is not prime. Stop.

Step 9. If i < n, set i = i + 1 and go to step 3. Otherwise, w is probably prime.

2.2. GENERATION OF PRIMES

Any implementation of the DSA requires the ability to generate random or pseudorandom integers. Such numbers are used to derive a user's private key, x, and a user's per message secret number, k. These randomly or pseudorandomly generated integers are selected to be between 0 and the 160-bit prime q (as specified in the standard). They shall be generated by the techniques given in this appendix, or using other FIPS approved security methods. One FIPS approved pseudorandom integer generator is supplied in Appendix C of ANSI X9.17, "Financial Institution Key Management (Wholesale)."

Other pseudorandom integer generators are given in this appendix. These permit generation of pseudorandom values of x and k for use in the DSA. The algorithm in section 3.1 may be used to generate values for x. An algorithm for k and r is given in section 3.2. The latter algorithm allows most of the signature computation to be precomputed without knowledge of the message to be signed.

The algorithms employ a one-way function G(t,c), where t is 160 bits, c is b bits (160 ó b ó 512) and G(t,c) is 160 bits. One way to construct G is via the Secure Hash Algorithm (SHA), as defined in the Secure Hash Standard (SHS). The 160-bit message digest output of the SHA algorithm when message M is input is denoted by SHA(M). A second method for constructing G is to use the Data Encryption Standard (DES). The construction of G by these techniques is discussed in sections 3.3 and 3.4 of this appendix.

In the algorithms in sections 3.1 and 3.2, a secret b-bit seed-key is used. The algorithm in section 3.1 optionally allows the use of a user provided input. If G is constructed via the SHA as defined in section 3.3, then b is between 160 and 512. If DES is used to construct G as defined in section 3.4, then b is equal to 160.

3.1. ALGORITHM FOR COMPUTING m VALUES OF x

Let x be the signer's private key. The following may be used to generate m values of x:

Step 1. Choose a new, secret value for the seed-key, XKEY.

Step 2. In hexadecimal notation let

t = 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0.

This is the initial value for H0 || H1 || H2 || H3 || H4 in the SHS.

This algorithm can be used to precompute k, k-1, and r for m messages at a time. Algorithm:

Step 1. Choose a secret initial value for the seed-key, KKEY.

Step 2. In hexadecimal notation let

t = EFCDAB89 98BADCFE 10325476 C3D2E1F0 67452301.

This is a cyclic shift of the initial value for H0 || H1 || H2 || H3 || H4 in the SHS.

G(t,c) may be constructed using steps (a) - (e) in section 7 of the Specifications for the Secure Hash Standard. Before executing these steps, {Hj} and M1 must be initialized as follows:

i. Initialize the {Hj} by dividing the 160 bit value t into five 32-bit segments as follows:

t = t0 || t1 || t2 || t3 || t4

Then Hj = tj for j = 0 through 4.

ii. There will be only one message block, M1, which is initialized as follows:

M1 = c || 0512-b

(The first b bits of M1 contain c, and the remaining (512-b) bits are set to zero).

Then steps (a) through (e) of section 7 are executed, and G(t,c) is the 160 bit string represented by the five words:

H0 || H1 || H2 || H3 || H4 at the end of step (e).

3.4. CONSTRUCTING THE FUNCTION G FROM THE DES

Let a XOR b denote the bitwise exclusive-or of bit strings a and b. Suppose a1, a2, b1, b2 are 32-bit strings. Let b1' be the 24 least significant bits of b1. Let K = b1' || b2 and A = a1 || a2. Define

DESb1,b2(a1,a2) = DESK(A)

In the above, DESK(A) represents ordinary DES encryption of the 64- bit block A using the 56-bit key K. Now suppose t and c are each 160 bits. To compute G(t,c):

Step 1. Write

t = t1 || t2 || t3 || t4 || t5

c = c1 || c2 || c3 || c4 || c5

In the above, each ti and ci is 32 bits.

Step 2. For i = 1 to 5 do

xi = ti XOR ci

Step 3. For i = 1 to 5 do

b1 = c((i+3) mod 5) + 1

b2 = c((i+2) mod 5) + 1

a1 = xi

a2 = x(i mod 5) + 1 XOR x((i+3) mod 5) + 1

yi,1 || yi,2 = DESb1,b2(a1,a2) (yi,1, yi,2 = 32 bits) Step 4. For i = 1 to 5 do

zi = yi,1 XOR y((i+1) mod 5)+1,2 XOR y((i+2) mod 5)+1,1

Step 5. Let

G(t,c) = z1 || z2 || z3 || z4 || z5

APPENDIX 4. GENERATION OF OTHER QUANTITIES

This appendix is for informational purposes only and is not required to meet the standard.

The algorithms given in this appendix may be used to generate the quantities g, k-1, and s-1 used in the DSS. To generate g:

Step 1. Generate p and q as specified in Appendix 2.

Step 2. Let e = (p - 1)/q.

Step 3. Set h = any integer, where 1 < h < p - 1 and h differs from any value previously tried.

Step 4. Set g = he mod p.

Step 5. If g = 1, go to step 3. To compute the multiplicative inverse n-1 mod q for n with 0 < n < q, where 0 < n-1 < q:

Step 1. Set i = q, h = n, v = 0, and d = 1.

Step 2. Let t = i DIV h, where DIV is defined as integer division.

Step 3. Set x = h.

Step 4. Set h = i - tx.

Step 5. Set i = x.

Step 6. Set x = d.

Step 7. Set d = v - tx.

Step 8. Set v = x.

Step 9. If h > 0, go to step 2.

Step 10. Let n-1 = v mod q.

Note that in step 10, v may be negative. The v mod q operation should yield a value between 1 and q - 1 inclusive.

APPENDIX 5. EXAMPLE OF THE DSA

This appendix is for informational purposes only and is not required to meet the standard.

Let L = 512 (size of p). The values in this example are expressed in hexadecimal notation. The p and q given here were generated by the prime generation standard described in appendix 2 using the 160- bit SEED:

d5014e4b 60ef2ba8 b6211b40 62ba3224 e0427dbd

With this SEED, the algorithm found p and q when the counter was at 38.

x was generated by the algorithm described in appendix 3, section 3.1, using the SHA to construct G (as in appendix 3, section 3.3) and a 160-bit XSEED:

XSEED =

bd029bbe 7f51960b cf9edb2b 61f06f0f eb5a38b6

t =

67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0

x = G(t,XSEED) mod q

k was generated by the algorithm described in appendix 3, section 3.2, using the SHA to construct G (as in appendix 3, section 3.3) and a 160-bit KSEED:

KSEED =

687a66d9 0648f993 867e121f 4ddf9ddb 01205584

t =

EFCDAB89 98BADCFE 10325476 C3D2E1F0 67452301

k = G(t,KSEED) mod q

Finally:

h = 2

p =

d411a4a0 e393f6aa b0f08b14 d1845866 5b3e4dbd ce254454 3fe365cf 71c86224 12db6e7d d02bbe13 d88c58d7 263e9023 6af17ac8 a9fe5f24 9cc81f42 7fc543f7

q =

b20db0b1 01df0c66 24fc1392 ba55f77d 577481e5

g =

b3085510 021f9990 49a9e7cd 3872ce99 58186b50 07e7adaf 25248b58 a3dc4f71 781d21f2 df89b717 47bd54b3 23bbecc4 43ec1d3e 020dadab bf782257 8255c104

x =

6b2cd935 d0192d54 e2c942b5 74c80102 c8f8ef67

k =

79577ddc aafddc03 8b865b19 f8eb1ada 8a2838c6

kinv =

2784e3d6 72d972a7 4e22c67f 4f4f726e cc751efa

M = ASCII form of "abc" (See FIPS PUB YY, Appendix A)

SHA(M) =

0164b8a9 14cd2a5e 74c4f7ff 082c4d97 fledf880

y =

b32fbec0 3175791d f08c3f86 1c81df7d e7e0cba7 f1c4f726 9bb12d6c 628784fb 742e66ed 315754df e38b5984 e94d3725 37f655cb 3ea4767c 878cbd2d 783ee662

r =

9b77f705 4c81531c 4e46a469 2fbfe0f7 7f7ebff2

s =

95b4f608 1f8f890e 4b5a199e f10ffe21 f52b2d68

w =

0ceb5f6b 875f6b67 7e093134 df70b0d4 3226680c

u1 =

347089a2 9897273b fc7a774f a70e0e0e 153bcc95

u2 =

793d9312 a41b88af aa2c1bd9 49ec3bee 2e75d2f5

gu1 mod p =

57a198ab 2c8ea0b6 4810767a ff732fb2 da5fcafb 278889f1 96b60b9c 1285b848 1d08505e 201a5c68 523a15ee 2fb62a56 d141dc4d 71925ef0 6acde0a5 b89c5671

yu2 mod p =

5d983d20 be604e23 fb19bec8 7860490a 41b865dc 0f5623f4 0724a795 021bcd8c 93a39ddf 51cae380 fb6d682a 676608f7 65227ff0 5e44ccf4 9767e4a6 0832d33f

v =

9b77f705 4c81531c 4e46a469 2fbfe0f7 7f7ebff2