A low communication competitive interactive proof system for promised quadratic residuosity

Abstract

A notion of “competitive” interactive proof systems is defined by Bellare and Goldwasser as a natural extension of a problem whether computing a witness w of x ∈ L is harder than deciding x ∈ L for a language L ∈ NP. It is widely believed that quadratic residuosity (QR) does not have a competitive interactive proof system. Bellare and Goldwasser however introduced a notion of “representative” of ZN* and showed that there exists a competitive interactive proof system for promised QR, i.e., the moduli N is guaranteed to be the product of k = O(log log |iV|) distinct odd primes. In this paper, we consider how to reduce the communication complexity of a competitive interactive proof system for promised QR and how to relax the constraint on k from O(loglog |N|) to O(log|N|). To do this, we introduce a notion of “dominant” of ZN*, and show that promised QR with the constraint that k = O(log |N|) has a competitive interactive proof system with considerably low communication complexity.