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[1] digital signature scheme to be provably secure in a random oracle model.[2] It is efficient and generates short signatures. It is 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, .Note that ; if , then the signature representation can fit into 40 bytes.Ifthen the signature is verified.It is relatively easy to see thatif the signed message equals the verified message:, and hence .Public elements: , , , , , , . Private elements: , .The signature scheme was constructed by applying the Fiat–Shamir transform[3] to Schnorr's identification protocol.[4] Therefore (per Fiat and Shamir's arguments), it is secure ifis modeled as a random oracle.Its security can also be argued in the generic group model, under the assumption thatis "random-prefix preimage resistant" and "random-prefix second-preimage resistant".[5] In particular,does not need to be collision resistant.In 2012, Seurin[2] 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 factorin its time-to-success ratio, whereis a function that remains close to 1 as long as " is noticeably smaller than 1", whereis the probability of forging an error making at mostqueries to the random oracle.
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