A Verifiable Secret Sharing Scheme with Statistical zero-knowledge.

In this paper, we first propose a protocol in which the prover can show that a = b holds for two committed integers a and b; also, we present a protocol in which the prover can prove that a 0 holds for committed integer a; then, we construct a protocol to prove that the degree of a polynomial f(x) equals to t 1 exactly, which has been as an open problem(see[21]); finally, we provide a protocol in which the prover proves that a pair (x, y) is generated by a polynomial f(x), i.e., y f(x)(mod m), where m is a prime. Based on above four protocols, we put forward a verifiable (t, n)-secret sharing scheme, which can avoid all known the dealer's cheats. In particular, all above protocols are statistical zero-knowledge.