In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm. Its security is based on the intractability of certain discrete logarithm problems. The Schnorr signature is considered the simplest digital signature scheme to be provably secure in a random oracle model. It is efficient and generates short signatures. It was covered by U.S. Patent 4,995,082 which expired in February 2008.
In the following,
To sign a message,
The signature is the pair,
Proof of correctness
It is relatively easy to see that
This shows only that a correctly signed message will verify correctly; many other properties are required for a secure signature algorithm.
The signature scheme was constructed by applying the Fiat–Shamir transform to Schnorr's identification protocol. Therefore, (per Fiat and Shamir's arguments), it is secure if
Its security can also be argued in the generic group model, under the assumption that
In 2012, Seurin provided an exact proof of the Schnorr signature scheme. In particular, Seurin shows that the security proof using the Forking lemma is the best possible result for any signature schemes based on one-way group homomorphisms including Schnorr-Type signatures and the Guillou-Quisquater signature schemes. Namely, under the ROMDL assumption, any algebraic reduction must lose a factor