On Tight Security Proofs for Schnorr Signatures

The Schnorr signature scheme is the most efficient signature scheme based on the discrete logarithm problem and a long line of research investigates the existence of a

tight

security reduction for this scheme in the random oracle. Almost all recent works present lower tightness bounds and most recently Seurin (Eurocrypt 2012) showed that under certain assumptions the

non

-tight security proof for Schnorr signatures in the random oracle by Pointcheval and Stern (Eurocrypt 1996) is essentially optimal. All previous works in this direction rule out tight reductions from the (one-more) discrete logarithm problem. In this paper we introduce a new meta-reduction technique, which shows lower bounds for the large and very natural class of

generic

reductions. A generic reduction is independent of a particular representation of group elements and most reductions in state-of-the-art security proofs have this desirable property. Our approach shows

unconditionally

that there is no tight generic reduction from any

natural

computational problem Π defined over algebraic groups (including even interactive problems) to breaking Schnorr signatures, unless solving Π is easy.