Optimal Security Proofs for Signatures from Identification Schemes

We perform a concrete security treatment of digital signature schemes obtained from canonical identification schemes via the Fiat-Shamir transform. If the identification scheme is random selfreducible and satisfies the weakest possible security notion (key-recoverability), then the signature scheme obtained via Fiat-Shamir is unforgeable against chosen-message attacks in the multi-user setting. Our security reduction is in the random oracle model and loses a factor of roughly Qh, the number of hash queries. Previous reductions incorporated an additional multiplicative loss of N , the number of users in the system. Our analysis is done in small steps via intermediate security notions, and all our implications have relatively simple proofs. Furthermore, for each step, we show the optimality of the given reduction in terms of model assumptions and tightness. As an important application of our framework, we obtain a concrete security treatment for Schnorr signatures.