Statistical Zero-Knowledge Proofs from Diophantine Equations

Abstract

A family $(S_t)$ of sets is $p$-bounded Diophantine if $S_t$ has a representing $p$-bounded polynomial $R_{S,t}$, s.t.~$x\\in S_t \\iff (\\exists y)[R_{S}(x;y)=0]$. We say that $(S_t)$ is unbounded Diophantine if additionally, $R_{S,t}$ 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 $S$.