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The **ElGamal signature scheme** is a digital signature scheme which is based on the difficulty of computing discrete logarithms. It was described by Taher ElGamal in 1984.^{[1]}

The ElGamal signature algorithm is rarely used in practice. A variant developed at NSA and known as the Digital Signature Algorithm is much more widely used. There are several other variants.^{[2]} The ElGamal signature scheme must not be confused with ElGamal encryption which was also invented by Taher ElGamal.

The ElGamal signature scheme allows a third-party to confirm the authenticity of a message.

- Let
*H*be a collision-resistant hash function. - Let
*p*be a large prime such that computing discrete logarithms modulo*p*is difficult. - Let
*g*<*p*be a randomly chosen generator of the multiplicative group of integers modulo*p*.

These system parameters may be shared between users.

- Randomly choose a secret key
*x*with 1 <*x*<*p*− 2. - Compute
*y*=*g*^{ x}mod*p*. - The public key is
*y*. - The secret key is
*x*.

These steps are performed once by the signer.

To sign a message *m* the signer performs the following steps.

- Choose a random
*k*such that 1 <*k*<*p*− 1 and gcd(*k*,*p*− 1) = 1. - Compute .
- Compute .
- If start over again.

Then the pair (*r*,*s*) is the digital signature of *m*. The signer repeats these steps for every signature.

A signature (*r*,*s*) of a message *m* is verified as follows.

- and .

The verifier accepts a signature if all conditions are satisfied and rejects it otherwise.

The algorithm is correct in the sense that a signature generated with the signing algorithm will always be accepted by the verifier.

The signature generation implies

Hence Fermat's little theorem implies

A third party can forge signatures either by finding the signer's secret key *x* or by finding collisions in the hash function . Both problems are believed to be difficult. However, as of 2011 no tight reduction to a computational hardness assumption is known.

The signer must be careful to choose a different *k* uniformly at random for each signature and to be certain that *k*, or even partial information about *k*, is not leaked. Otherwise, an attacker may be able to deduce the secret key *x* with reduced difficulty, perhaps enough to allow a practical attack. In particular, if two messages are sent using the same value of *k* and the same key, then an attacker can compute *x* directly.^{[1]}

Another more serious attack: Given attacker can attempt to solve for unknown x,k such that This involves solving linear equation in 2 variables and can discover private key from just 1 signature.^{[3]}

The original paper^{[1]} did not include a hash function as a system parameter. The message *m* was used directly in the algorithm instead of *H(m)*. This enables an attack called existential forgery, as described in section IV of the paper. Pointcheval and Stern generalized that case and described two levels of forgeries:^{[4]}

**The one-parameter forgery.**Let be a random element. If and , the tuple is a valid signature for the message .**The two-parameters forgery.**Let and be random elements and . If and , the tuple is a valid signature for the message .

Improved version (with a hash) is known as Pointcheval–Stern signature algorithm

- Modular arithmetic
- Digital Signature Algorithm
- Elliptic Curve DSA
- ElGamal encryption
- Schnorr signature

- ^
^{a}^{b}^{c}T. ElGamal (1985). "A public key cryptosystem and a signature scheme based on discrete logarithms" (PDF).*IEEE Trans Inf Theory*.**31**(4): 469–472. Archived from the original (PDF) on 2015-05-13. - this article appeared earlier in the proceedings to Crypto '84. **^**K. Nyberg, R. A. Rueppel (1996). "Message recovery for signature schemes based on the discrete logarithm problem".*Designs, Codes and Cryptography*.**7**(1-2): 61–81. doi:10.1007/BF00125076.**^**"Phong NGUYEN -- Publications".*www.di.ens.fr*(in Japanese). Retrieved 2018-05-18.**^**Pointcheval, David; Stern, Jacques (2000). "Security Arguments for Digital Signatures and Blind Signatures" (PDF).*J Cryptology*.**13**(3): 361–396.