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Secure Hash Algorithms  

Concepts  
hash functions · SHA · DSA  
Main standards  
SHA0 · SHA1 · SHA2 · SHA3


The Digital Signature Algorithm (DSA) is a Federal Information Processing Standard for digital signatures. In August 1991 the National Institute of Standards and Technology (NIST) proposed DSA for use in their Digital Signature Standard (DSS) and adopted it as FIPS 186 in 1993.^{[1]}^{[not in citation given]} Four revisions to the initial specification have been released: FIPS 1861 in 1996,^{[2]} FIPS 1862 in 2000,^{[3]} FIPS 1863 in 2009,^{[4]} and FIPS 1864 in 2013.^{[5]}
DSA is covered by U.S. Patent 5,231,668, filed July 26, 1991 and attributed to David W. Kravitz,^{[6]} 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 NIST has made this patent available worldwide royaltyfree.^{[7]} Claus P. Schnorr claims that his U.S. Patent 4,995,082 (expired) covered DSA; this claim is disputed.^{[8]} DSA is a variant of the ElGamal signature scheme.
Key generation has two phases. The first phase is a choice of algorithm parameters which may be shared between different users of the system, while the second phase computes public and private keys for a single user.
The algorithm parameters (p, q, g) may be shared between different users of the system.
Given a set of parameters, the second phase computes private and public keys for a single user:
There exist efficient algorithms for computing the modular exponentiations h^{(p − 1)/q} mod p and g^{x} mod p, such as exponentiation by squaring.
Let be the hashing function and the message:
The first two steps amount to creating a new permessage key. The modular exponentiation here is the most computationally expensive part of the signing operation, and it may be computed before the message hash is known. The modular inverse is the second most expensive part, and it may also be computed before the message hash is known. It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as .
DSA is similar to the ElGamal signature scheme.
The signature scheme is correct in the sense that the verifier will always accept genuine signatures. This can be shown as follows:
First, if , it follows that by Fermat's little theorem. Since and is prime, must have order .
The signer computes
Thus
Since has order we have
Finally, the correctness of DSA follows from
With DSA, the entropy, secrecy, and uniqueness of the random signature value k are critical. It is so critical that violating any one of those three requirements can reveal the entire private key to an attacker.^{[11]} Using the same value twice (even while keeping k secret), using a predictable value, or leaking even a few bits of k in each of several signatures, is enough to reveal the private key x.^{[12]}
This issue affects both DSA and ECDSA – in December 2010, a group calling itself fail0verflow announced recovery of the ECDSA private key used by Sony to sign software for the PlayStation 3 game console. The attack was made possible because Sony failed to generate a new random k for each signature.^{[13]}
This issue can be prevented by deriving k deterministically from the private key and the message hash, as described by RFC 6979. This ensures that k is different for each H(m) and unpredictable for attackers who do not know the private key x.
In addition, malicious implementations of DSA and ECDSA can be created where k is chosen in order to subliminally leak information via signatures. For example, an offline private key could be leaked from a perfect offline device that only released innocentlooking signatures.^{[14]}