Statistical Zero-Knowledge Proofs from Diophantine Equations

A family (St) of sets is p-bounded Diophantine if St has a representing p-bounded polynomial RS;t, s.t. x 2 St () (9y)[RS(x; y) = 0℄. We say that (St) is unbounded Diophantine if additionally, RS;t is a fixed t-independent polynomial. We show that p-bounded (resp., unbounded) Diophantine set has a polynomial-size (resp., constant-size) statistical zero-knowledge proof system that a committed tuple x belongs to S. We describe efficient SZK proof systems for several cryptographically interesting sets. Finally, we show how to prove in SZK that an encrypted number belongs to S.