Digital Signature Algorithm
The Digital Signature Algorithm (DSA) is a United States Federal Government standard or FIPS for digital signatures. It was proposed by the National Institute of Standards and Technology (NIST) in August 1991 for use in their Digital Signature Standard (DSS), specified in FIPS 186 [1], adopted in 1993. A minor revision was issued in 1996 as FIPS 186-1 [2]. The standard was expanded further in 2000 as FIPS 186-2 and again in 2009 as FIPS 186-3 [3].
DSA is covered by Template:US patent, filed July 26, 1991, and attributed to David W. Kravitz, a former NSA employee. This patent was given to "The United States of America as represented by the Secretary of Commerce, Washington, D.C." and the NIST has made this patent available worldwide royalty-free. [4] Dr. Claus P. Schnorr claims that his Template:US patent covers DSA; this claim is disputed (see minutes of the Sept. 94 meeting of the Computer System Security and Privacy Advisory Board).
Key generation
Key generation has two phases. The first phase is a choice of algorithm parameters which may be shared between different users of the system:
- Choose an approved cryptographic hash function H. In the original DSS, H was always SHA-1, but the stronger SHA-2 hash functions are approved for use in the current DSS. The hash output may be truncated to the size of a key pair.
- Decide on a key length L and N. This is the primary measure of the cryptographic strength of the key. The original DSS constrained L to be a multiple of 64 between 512 and 1024 (inclusive). NIST 800-57 recommends lengths of 2048 (or 3072) for keys with security lifetimes extending beyond 2010 (or 2030), using correspondingly longer N. FIPS 186-3 specifies L and N length pairs of (1024,160), (2048,224), (2048,256), and (3072,256).
- Choose an N-bit prime q. N must be less than or equal to the hash output length.
- Choose an L-bit prime modulus p such that p–1 is a multiple of q.
- Choose g, a number whose multiplicative order modulo p is q. This may be done by setting g = h(p–1)/q mod p for some arbitrary h (1 < h < p-1), and trying again with a different h if the result comes out as 1. Most choices of h will lead to a usable g; commonly h=2 is used.
The algorithm parameters (p, q, g) may be shared between different users of the system. The second phase computes private and public keys for a single user:
- Choose x by some random method, where 0 < x < q.
- Calculate y = gx mod p.
- Public key is (p, q, g, y). Private key is x.
There exist efficient algorithms for computing the modular exponentiations ha mod p and gx mod p, such as exponentiation by squaring.
Signing
Let H be the hashing function and m the message:
- Generate a random per-message value k where 0 < k < q
- Calculate r = (gk mod p) mod q
- Calculate s = (k−1(H(m) + x*r)) mod q
- Recalculate the signature in the unlikely case that r = 0 or s = 0
- The signature is (r, s)
The extended Euclidean algorithm can be used to compute the modular inverse k−1 mod q.
Verifying
- Reject the signature if either 0 < r <q or 0 < s < q is not satisfied.
- Calculate w = (s)−1 mod q
- Calculate u1 = (H(m)*w) mod q
- Calculate u2 = (r*w) mod q
- Calculate v = ((gu1*yu2) mod p) mod q
- The signature is valid if v = r
DSA is similar to the ElGamal signature scheme.
Correctness of the algorithm
The signature scheme is correct in the sense that the verifier will always accept genuine signatures. This can be shown as follows:
First, if g = h(p − 1)/q mod p it follows that gq ≡ hp − 1 ≡ 1 (mod p) by Fermat's little theorem. Since g > 1 and q is prime, g must have order q.
The signer computes
- <math>s=k^{-1}(H(m)+xr) \bmod q. \, </math>
Thus
- <math>
\begin{align} k & \equiv H(m)s^{-1}+xrs^{-1}\\
& \equiv H(m)w + xrw \pmod{q}.
\end{align} </math>
Since g has order q we have
- <math>
\begin{align} g^k & \equiv g^{H(m)w}g^{xrw}\\
& \equiv g^{H(m)w}y^{rw}\\
& \equiv g^{u1}y^{u2} \pmod{p}.
\end{align} </math>
Finally, the correctness of DSA follows from
- <math>r=(g^k \bmod p) \bmod q = (g^{u1}y^{u2} \bmod p) \bmod q = v.\,</math>
See also
External links
- FIPS-186, the first version of the official DSA specification.
- FIPS-186, change notice No.1, the first change notice to the first version of the specification.
- FIPS-186-1, the first revision to the official DSA specification.
- FIPS-186-3, the third and current revision to the official DSA specification.
- FIPS-186-3 Approval, Approval announcement of the third revision to the official DSA specification.
- Recommendation for Key Management -- Part 1: general, NIST Special Publication 800-57, p. 62–63