Statistical Zero Knowledge Protocols to Prove Modular Polynomial Relations

This paper proposes a bit commitment scheme, BC(·), and efficient statistical zero knowledge (in short, SZK) protocols in which, for any given multi-variable polynomial f(X 1,..,X t) and any given modulus n, prover P gives (I 1,..,I t) to verifier V and can convince V that V knows (x 1,..,x t) satisfying f(x 1,..,x t) = 0 (mod n) and I i = BC(x i), (i = l,..,t). The proposed protocols are O(n) times more efficient than the corresponding previous ones [Dam93, Dam95, Oka95]. The (knowledge) soundness of our protocols holds under a computational assumption, the intractability of a modified RSA problem (see Def.3), while the (statistical) zero-knowledgeness of the protocols needs no computational assumption. The protocols can be employed to construct various practical cryptographic protocols, such as fair exchange, untraceable electronic cash and verifiable secret snaring protocols.