4-move perfect ZKIP for some promise problems

In this paper, we consider ZKIPs for promise problems. A promise problem is a pair of predicates (Q;R). A Turning machine T solves the promise problem (Q;R) if, for every x satisfying Q(x), machine T halts and it answers \yes" i R(x). When :Q(x), we do not care what T does. First, we de ne \promised BPP" which is a promise problem version of BPP. Then, we prove that a promise problem (Q;R) has a 3-move interactive proof system which is black-box simulation zero knowledge if and only if (Q;R) 2 promised BPP. Next, we show a \4-move" perfect ZKIPs (black-box simulation) for a promise problem of Quadratic Residuosity and that of Blum Numbers under no cryptographic assumption.