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( Appeared in the Federal Register dated August 30, 1991 )
DEPARTMENT OF COMMERCE
National Institute of Standards and Technology
A PROPOSED FEDERAL INFORMATION PROCESSING STANDARD
FOR
DIGITAL SIGNATURE STANDARD (DSS)
AGENCY: National Institute of Standards and Technology (NIST),
Commerce.
ACTION: Notice; Request for Comments.
SUMMARY: A Federal Information Processing Standard (FIPS) for
Digital Signature Standard (DSS) is being proposed. This
proposed standard specifies a public-key based digital signature
algorithm (DSA) appropriate for Federal digital signature
applications. The proposed DSS uses a public key to verify to a
recipient the integrity of data and the identity of the sender of
the data. The DSS can also be used by a third party to ascertain
the authenticity of a signature and the data associated with it.
This proposed standard adopts a public-key signature system that
uses a pair of transformations to generate and verify a digital
value called a signature. The government has applied to the U.S.
Patent Office for a patent on this technique. The government
will also seek foreign patents as appropriate. NIST intends to
make this DSS technique available world-wide on a royalty-free
basis in the public interest. We believe this technique is
patentable and that no other patents would apply to the DSS, but
we cannot give firm assurances to such effect in advance of
issuance of the patent.
The purpose of this notice is to solicit views from the public,
manufacturers, and Federal, state, and local government users so
that their needs can be considered prior to submission of this
proposed standard to the Secretary of Commerce for review and
approval.
The proposed standard contains two sections: (1) an announcement
section, which provides information concerning the applicability,
implementation, and maintenance of the standard; and (2) a
specifications section which deals with the technical aspects of
the standard. Only the announcement section of the standard is
provided in this notice. Interested parties may obtain copies of
the specifications section from the Standards Processing
Coordinator (ADP), National Institute of Standards and
Technology, Technology Building, Room B-64, Gaithersburg, MD
20899, telephone (301) 975-2816.
DATE: Comments on this proposed standard must be received on or
before (please insert date which is ninety (90) days from the
date of publication of this notice in the Federal Register).
ADDRESS: Written comments concerning the proposed standard
should be sent to: Director, Computer Systems Laboratory, ATTN:
Proposed FIPS for DSS, Technology Building, Room B-154, National
Institute of Standards and Technology, Gaithersburg, MD 20899.
Written comments received in response to this notice will be made
part of the public record and will be made available for
inspection and copying in the Central Reference and Records
Inspection Facility, Room 6020, Herbert C. Hoover Building, 14th
Street between Pennsylvania and Constitution Avenues, NW,
Washington, DC 20230.
FOR FURTHER INFORMATION CONTACT: Mr. Miles Smid, National
Institute of Standards and Technology, Gaithersburg, MD 20899,
telephone (301) 975-2938.
SUPPLEMENTARY INFORMATION: This proposed FIPS is the result of
evaluating a number of alternative digital signature techniques.
In making the selection, the NIST has followed the mandate
contained in section 2 of the Computer Security Act of 1987 that
NIST develop standards and guidelines to "... assure the cost-
effective security and privacy of sensitive information in
Federal systems". In meeting this statutory responsibility, NIST
has placed primary emphasis on selecting the technology that best
assures the appropriate security of Federal information and,
among technologies offering comparable protection, on selecting
the option with the most desirable operating and use
characteristics.
Among the factors that were considered during this process were
the level of security provided, the ease of implementation in
both hardware and software, the ease of export from the U.S., the
applicability of patents, impact on national security and law
enforcement and the level of efficiency in both the signing and
verification functions. A number of techniques were deemed to
provide appropriate protection for Federal systems. The
technique selected has the following desirable characteristics:
-- NIST expects it to be available for public use on a
royalty-free basis. Broader use of this technique
resulting from public availability should be an
economic benefit to the government and the public.
-- The technique selected provides for efficient
implementation of the signature operations in smart
card applications. In these applications the signing
operations are performed in the computationally modest
environment of the smart card while the verification
process is implemented in a more computationally rich
environment such as a personal computer, a hardware
cryptographic module, or a mainframe computer.
NIST has received agreement from Department of Defense
authorities that this digital signature technique may be used to
sign unclassified data processed by "Warner Amendment" systems
(10 U.S.C. 2315 and 44 U.S.C. 3502(2)) as well as classified data
in selected applications.
A hashing function has not been specified by NIST at this time
for use with the DSS. NIST has been reviewing various candidate
hashing functions; however, we are not satisfied with any of the
functions we have studied thus far. NIST does intend to publish
a hashing function that is complementary to the DSS.
(NOTE: Original signed by Dr. John Lyons, NIST Director)
***********************************************************************
( The following is the Proposed Digital Signature Standard)
A PROPOSED
DIGITAL SIGNATURE STANDARD (DSS)
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 user who generates the
signature. 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 signer of the data. This is known
as nonrepudiation since the signer of data 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 XX
ANNOUNCING A
DIGITAL SIGNATURE STANDARD
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.
Category of Standard: ADP Operations, Computer Security.
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 it can be used to verify
the identity of the originator and integrity of the data. The DSA
includes signature generation and verification. Generation makes
use of a private key to generate a digital signature. Verification
of the signature makes use of a public key which corresponds to,
but is not the same as, the private key used to generate the
signature. Each user possesses a private and public key pair.
Public keys are assumed to be known to all members of a group of
users or to the public in general. Private keys must be known
only by their creators. 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. The
message digest is then signed. The digital signature is sent to
the intended recipient along with the signed data (often called
the message). The recipient 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
will be specified in a separate standard. Similar procedures may
be used to generate and verify signatures for stored as well as
transmitted data.
Approving Authority: Secretary of Commerce.
Maintenance Agency: Computer Systems Laboratory, National
Institute of Standards and Technology.
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.
Private and commercial organizations are encouraged to adopt and
use this standard.
Applications: The DSA authenticates the integrity of the signed
data and the identity of the signer. 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
or hardware. Only implementations of the DSA that are validated
by NIST will be considered as complying with this standard.
Information about the requirements for validating implementations
of this standard 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: Implementations of the DSA in this standard may be
covered by U.S. and foreign patents.
Implementation Schedule: This standard becomes effective six
months after the publication date of this FIPS PUB.
Specifications: Federal Information Processing Standard (FIPS XX)
Digital Signature Standard (affixed).
Cross Index:
a. FIPS PUB 46-1, Data Encryption Standard.
b. FIPS PUB 73, Guidelines for Security of Computer
Applications.
c. FIPS PUB 140-1, Security Requirements for Cryptographic
Modules.
Qualifications: The security of a digital signature system is
dependent on maintaining the secrecy of users' private keys.
Users must therefore guard against the unauthorized acquisition 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 a particular
implementation is secure. The responsible authority in each agency
or department shall assure that an overall implementation provides
an acceptable level of security. This standard will be reviewed
every five years in order to assess its adequacy.
Waiver Procedure: Under certain exceptional circumstances, the
heads of Federal departments and agencies may approve waivers to
Federal Information Processing Standards (FIPS). The head of
such agency may redelegate such authority only to a senior official
designated pursuant to section 3506(b) of Title 44, United States
Code. Waiver shall be granted only when:
a. Compliance with a standard would adversely affect the
accomplishment of the mission of an operator of a Federal
computer system; or
b. Compliance with a standard would cause a major adverse
financial impact on the operator which is not offset by
Government-wide savings.
Agency heads may act upon a written waiver request containing the
information detailed above. Agency heads may also act without a
written waiver request when they determine that conditions for
meeting the standard cannot be met. Agency heads may approve
waivers only by a written decision which explains the basis on
which the agency head made with required finding(s). A copy of
each decision, with procurement sensitive or classified portions
clearly identified, shall be sent to: National Institute of
Standards and Technology; ATTN: FIPS Waiver Decisions, Technology
Building, Room B-154, Gaithersburg, MD 20899.
In addition, notice of each waiver granted and each delegation of
authority to approve waivers shall be sent promptly to the
Committee on Government Operations of the House of
Representatives and the Committee on Government Affairs of the Senate and
shall be published promptly in the Federal Register.
When the determination on a waiver applies to the procurement of
equipment and/or services, a notice of the waiver determination
must be published in the Commerce Business Daily as a part of the
notice of solicitation for offers of an acquisition or, if the
waiver determination is made after that notice is published, by
amendment to such notice.
A copy of the waiver, any supporting documents, the document
approving the waiver and any accompanying documents, with such
deletions as the agency is authorized and decides to make under 5
United States Code Section 552(b), shall be part of the
procurement documentation and retained by the agency.
Where to Obtain Copies of the Standard: Copies of this
publication are for sale by the National Technical Information Service, U.S.
Department of Commerce, Springfield, VA 22161. When ordering,
refer to Federal Information Processing Standards Publication XX
(FIPS PUB XX), and identify the title. When microfiche is
desired, this should be specified. Prices are published by NTIS in
current catalogs and other issuances. Payment may be made by check,
money order, deposit account or charged to a credit card accepted by
NTIS.
Federal Information
Processing Standards Publication XX
Specifications for a
DIGITAL SIGNATURE STANDARD (DSS)
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 transmitted from one party to another, the
recipient may desire to know that the message has not been
altered in transit. Furthermore, the recipient may wish to be certain of
the origin of the message. 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 a third party that the message was, in fact, signed by
the originator. Unlike their written counterparts, digital
signatures also verify the integrity of messages. It is also
desirable to be able to generate digital signatures 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
A private and public key pair is used to generate and verify a
digital signature. These keys are employed in conjunction with a
hash function H, which is not specified in this standard. The
holder of a private key can generate a signature for data,
referred to as the message, m. A holder of the corresponding public key
can verify the signature. Both signature generation and verification
use H. An adversary, who does not know the private key of a
user, cannot generate that user's signature for a message. In other
words, signatures cannot be forged: an adversary cannot generate
a correct signature for another person for any message. However,
by using the appropriate public key, anyone can check that a
given signature is valid.
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
Let
A. p = a prime modulus where 2^511 < p < 2^512.
B. q = a prime divisor of p - 1 where 2^159 < q < 2^160.
C. g = h^((p-1)/q) mod p, where h is any integer with 0 < h < p
such that h^((p-1)/q) mod p > 1.
D. x = an integer with 0 < x < q.
E. y = g^x mod p.
F. m = the message to be signed and transmitted.
G. k = a random integer with 0 < k < q.
H. H = a one-way hash function.
The integers p, q, and g can be public and can be common to a
group of users. A user's private and public keys are x and y,
respectively. x and k must be secret. k must be changed for each
signature. H is not specified in this standard. However, H must
be chosen so that it is computationally infeasible to create a
message which results in a given hash value and it must also be
computationally infeasible to find any two different messages
which result in the same hash value.
5. SIGNATURE GENERATION
To send a signed message m the user chooses a random k and
computes
r = (g^k mod p) mod q
s = (k^-1 (H(m) + xr)) mod q.
where k^-1 is the multiplicative inverse of k, mod q; i.e., (k^-1 k)
mod q = 1 and 0 < k^-1 < q.
The values r and s constitute the signature of the message.
These are transmitted along with the message m to the recipient.
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 recipient in an authenticated manner.
Let m', r', and s' be the received versions of m, r, and s,
respectively, and let y be the public key of the sender. To
verify the signature, the recipient first checks to see that 0 < r' < q
and 0 < s' < q; if either condition is violated the signature is
rejected. If these two conditions are satisfied, the recipient
computes:
w = (s')^-1 mod q
u1 = ((H(m'))w) mod q
u2 = ((r')w) mod q
v = (((g)^(u1) (y)^(u2)) mod p) mod q.
If v = r', then the signature is verified and the receiver 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 = r' when m' = m, r' = r, and s' = s, see Appendix 1.
If v does not equal r', then the message may have been modified,
the message may have been incorrectly signed by the sender, or
the message may have been signed by an impostor. The message should
be considered invalid.
Appendix 1
A Proof that v = r'
This appendix is for informational purposes only and is not
required to meet the standard.
The purpose of this appendix is to provide a rigorous proof that
in the signature verification (Section 6 of the DSS) we have v =
r' when m' = m, r' = r, and s' = s. The proof is given by the Theorem
below; it is preceded by four lemmas.
LEMMA 1. For any nonnegative integer t, if g = h^((p-1)/q) mod
p, then
g^t mod p = g^(t mod q) mod p.
Proof: By the Euler/Fermat theorem, since h is relatively prime
to p, we have h^(p-1) mod p = 1. Hence for any nonnegative
integer n,
g^(nq) mod p = (h^((p-1)/q) mod p)^(nq) mod p
= h^(((p-1)/q)nq) mod p
= h^((p-1)n) mod p
= ((h^(p-1) mod p)^n) mod p
= 1^n mod p
= 1.
Thus, for any nonnegative integers n and z we have
g^(nq+z) mod p = (g^(nq) g^z) mod p
= ((g^(nq) mod p) (g^z mod p)) mod p
= g^z mod p.
Any nonnegative integer t can be represented uniquely as t = nq
+ z where n and z are nonnegative integers and 0 s z < q. Then
g^t mod p = g^z mod p.
Also, z = t mod q. The result follows. QED.
LEMMA 2. For any nonnegative integers a and b,
g^(a mod q + b mod q) mod p = g^((a + b) mod q) mod p.
Proof: By Lemma 1 we have
g^(a mod q + b mod q) mod p = g^((a mod q + b mod q) mod q) mod p
= g^((a + b) mod q) mod p.
QED.
LEMMA 3.
y^((rw) mod q) mod p = g^((xrw) mod q) mod p.
Proof: Since y = g^x mod p, using Lemma 1 we have
y^((rw) mod q) mod p = (g^x mod p)^((rw) mod q) mod p
= g^(x((rw) mod q)) mod p
= g^((x((rw) mod q)) mod q) mod p (by Lemma 1)
= g^((xrw) mod q) mod p.
QED.
LEMMA 4.
((H(m) + xr)w) mod q = k.
Proof: We have
s = (k^-1 (H(m) + xr)) mod q.
Since (k k^-1) mod q = 1,
(ks) mod q = (k((k^-1 (H(m) + xr)) mod q)) mod q
= ((k(k^-1 (H(m) + xr)))) mod q
= (((k k^-1) mod q) ((H(m) + xr) mod q)) mod q
= (H(m) + xr) mod q.
Since w = s^-1 mod q we have (ws) mod q = 1, and thus
((H(m) + xr)w) mod q = (((H(m) + xr) mod q) (w mod q)) mod q
= (((ks) mod q) (w mod q)) mod q
= (kws) mod q
= ((k mod q) ((ws) mod q)) mod q
= k mod q.
Since 0 < k < q we have k mod q = k. QED.
THEOREM. If m' = m, r' = r, and s' = s, then v = r'.
Proof: Using Lemmas 2, 3 and 4 we find
v = ((g^(u1) y^(u2)) mod p) mod q
= ((g^((H(m)w) mod q) y^((rw) mod q)) mod p) mod q
= ((g^((H(m)w) mod q) g^((xrw) mod q)) mod p) mod q (by Lemma 3)
= ((g^((H(m)w) mod q + (xrw) mod q)) mod p) mod q
= ((g^(((H(m)w) mod q + (xrw) mod q) mod q)) mod p) mod q
= ((g^((H(m)w + xrw) mod q)) mod p) mod q (by Lemma 2)
= ((g^(((H(m) + xr)w) mod q)) mod p) mod q
= (g^k mod p) mod q (by Lemma 4)
= r
= r'.
QED.
Appendix 2
Generation of Parameters for the DSA
This appendix is for informational purposes only and is not
required to meet the standard.
This appendix includes suggestions for generating the parameters
and performing the functions needed to implement the DSA. These
algorithms require a random number generator (see Appendix 3),
and an efficient modular exponentiation algorithm (see Appendix 4).
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 1/4^n.
Therefore, n=50 should give an acceptable probability of error.
To test whether an integer is prime:
1. Set i = 1 and n = 50.
2. Set w = the integer to be tested, w = 1 + 2^a m,
where m is odd and 2^a is the largest power of 2 dividing w-1.
3. Generate a random integer b in the range 1<b<w.
4. Set j = 0 and z = b^m mod w.
5. If j = 0 and z = 1, or if z = w-1, go to step 9.
6. If j> 0 and z = 1, go to step 8.
7. j = j + 1. If j < a, set z = z^2 mod w and go to step 6.
8. w is not prime. Stop.
9. If i< n, set i = i + 1 and go to step 3. Otherwise, w
is probably prime.
To generate a prime q where 2^159 < q < 2^160:
1. Set n = a random number between 2^158 and 2^159 - 1, inclusive.
2. Set t = 2n + 1.
3. If t < 2^160, test whether t is prime. Otherwise, go to step
1.
4. If t is prime, set q = t. Otherwise, set t = t + 2 and go to
step 3.
To generate a prime p where 2^511 < p < 2^512 and q divides p - 1:
1. Generate q as specified above.
2. Generate a random integer n, where n is between (2^511-1)/(2q)
and (2^512-1)/(2q).
3. Set t = 2nq + 1.
4. Test whether t is prime.
5. If t is prime, set p = t. Otherwise, go to step 2.
To generate an element g of order q mod p:
1. Generate p and q as specified above.
2. Set h = a random number, where 1 < h < p-1.
3. Set t = h^((p-1)/q) mod p.
4. If t = 1 then go to step 2. Otherwise, set g = t.
To generate integers x and k:
Set x and k equal to random integers between 0 and q.
To generate y:
Set y = g^x mod p.
To compute the signature (r,s) of a message m:
1. Set t = g^k mod p.
2. Set r = t mod q.
3. Set h = H(m) where H is a hash function.
4. Calculate k^-1 mod q as described below.
5. Set s = ((k^-1) (h + xr)) mod q.
To verify the signature (r,s) of a message m:
1. m, r, and s are received as m', r', and s', respectively.
2. If r' <= 0 or r' >= q then reject the signature as invalid.
3. If s' <= 0 or s' >= q then reject the signature as invalid.
4. Set h = H(m').
5. Calculate w = s'^-1 mod q as described below.
6. Set u1 = (hw) mod q.
7. Set u2 = (r'w) mod q.
8. Set a = g^(u1) mod p.
9. Set b = y^(u2) mod p.
10. Set t = (ab) mod p.
11. Set v = t mod q.
12. If v = r', signature is verified. Otherwise, reject
signature as invalid.
To compute the multiplicative inverse n^-1 mod q for n with 0 < n
< q:
1. Set i = q, h = n, v = 0 and d = 1.
2. While h > 0 repeat steps 3 thru 5.
3. t = i DIV h where DIV is defined as integer division.
4. Set x = h, h = i - tx and i = x.
5. Set x = d, d = v - tx and v = x.
6. n^-1 = v mod q.
Appendix 3
Random Number Generation for the DSA
This appendix is for informational purposes only and is not
required to meet the standard.
Any implementation of the DSA requires the ability to generate
random numbers. Random numbers are used to derive a user's private
key and a user's per message random secret number. These random
numbers are selected to be between 0 and the 160-bit prime q (as
specified in the standard). The numbers can be generated by either
a true noise hardware randomizer or via a pseudorandom function.
Such a function would employ a user generated and secret "seed"
key to initialize the number generator. The generator then would
produce a stream of bits or numbers which could be converted into
the integers mod q discussed above. One such pseudorandom number
generator is the key generation methodology found in Appendix C of
ANSI X9.17, "Financial Institution Key Management (Wholesale)".
Appendix 4
Modular Arithmetic for the DSA
This appendix is for informational purposes only and is not
required to meet the standard.
One key to efficient implementation of the Digital Signature
Standard is the development of a modular arithmetic package suited
to the processor one intends to use. For the purposes of this
discussion, we will consider two types of processors: hardware and
software and suggest some techniques which may allow for
efficient implementation of the DSA in those systems.
With respect to hardware implementations an excellent reference
is "VLSI Implementation of Public Key Encryption Algorithms" by
Orton, Roy, Scott, Peppard and Tavares from the Proceedings of
CRYPTO-86. That paper and the references found at its end will
provide a good basis for anyone looking into hardware
implementations of the DSA. It is also worth noting that several
commercial firms have custom processors to do modular arithmetic
on the market.
With respect to software implementations, there are many
different ways to proceed and a rich literature containing
different techniques for different computing environments. For
instance, the algorithms one chooses for 32-bit processors may be
substantially different than those which would be appropriate for
8-bit processors. Likewise, one can make trade offs on
computation versus memory in various algorithms. Good starting places for
modular arithmetic algorithms in software are found in "A
Cryptographic Library for the Motorola DSP56000" by Dusse and
Kaliski from the Proceedings of EUROCRYPT-90 and "Implementing the
Rivest Shamir and Adleman Public Key Encryption Algorithm on a
Standard Digital Signal Processor" by Barrett from the Proceedings
of CRYPTO-86. These papers contain pseudo-code for some of the
more popular techniques used in modular arithmetic.
Appendix 5
Example of the DSA
This appendix is for informational purposes only and is not
required to meet the standard.
p = Prime Modulus =
d0451ffe 2c64c4ed 6b0ae636 5b7fef9c 15425e40 a37ca5f8 39865e2c
fb4169a0 d825c913 0f8864ff fcf3bfbe b0273660 67aa27e2 7bfcaf40
00000000 00000001
q = Prime Divisor =
d9525756 704a663e 7323caf2 6fb8fc25 77e4fbeb
g = Element of Order q Mod p =
acf958c4 0d301efc 5153e7dc d5ef75fe c9e8fb0f ae6a80ee 5c3b84b9
c0e51305 1b2b7542 e66b8d3a 25e93891 1ad6be5c 24395099 c6ddaa86
e18942f2 984275a
x = Private Key =
12345678 9abcdef0 12345678 9abcdef
y = g^x mod p = Public Key =
9d168087 c60c5cb3 aeb1e8ac c622f167 f1e97151 0b34876c 080d81b5
20329817 e3e279fa 86eb6a9d 5e9e5897 5clf3d0d 3786ce04 abb0cab4
dfd9fa13 50bb3aa3
k = Random Secret Integer =
bf27aa41 6c006dd4 b4f2806c 71171cc4 ce28db
r = (g^k mod p) mod q =
1c3d5143 a7beb085 9cbd08a2 039d7148 27ceddf9
h = H(m) = Hash(message) =
2a2a2a2a 2a2a2a2a 2a2a2a2a 2a2a2a2a 2a2a2a2a
(h + xr) mod q =
20215b52 aa605b7f 967991de b947a07f 13413a8a 8753d457 d1897afe
786c8f82 5ecaal
k^-1 mod q =
5dfb26c8 208eda68 d8bb5fce 89732bae 595392e6
s = (k^-1 (h + xr)) mod q =
6f0be90c 72350564 77c69e89 ab6416b2 f365d95c
w = s^-1 mod q =
27fd6332 9e1cddf1 883b1d62 a345d9c5 f115d821
u1 = (hw) mod q =
6521b9e7 e0824239 0754396d c5327f98 c74fc641
u2 = (rw) mod q =
52d754b2 153d3c14 483e65de 9895abe9 6bbb7751
g^(u1) mod p =
c50370ba af79d463 7bf6ea24 579495de d0fd1190 2e5f8c5f 8524dd53
ee13c1e8 fb4ad43c db2e86f7 b892dc81 5e7676ab 11b48803 916e453b
f95bdfeb 93003009
y^(u2) mod p =
53b07663 b7078870 b640044f 592d1076 8b82fb49 1cfd10fe 4310a315
f3cf4de9 5ccff3df 926f837d 69afe707 640dd5a2 6fd41b11 1ff5cc6d
51aa7453 d79ca533
v = ((g^(u1) mod p)(y^(u2) mod p) mod p) mod q =
1c3d5143 a7beb085 9cbd08a2 039d7148 27ceddf9
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