Constant round perfect ZKIP of computational ability

In this paper, we show that without any unproven assumption, there exists a 'four' move blackbox simulation perfect zero-knowledge interactive proof system of computational ability for any random self-reducible relation R whose domain is in B P P, and that without any unproven assumption, there exists a 'four' move blackbox simulation perfect zero-knowledge interactive proof system of knowledge on the prime factorization. These results are optimal in the light of the round complexity, because it is shown that if a relation R has a three move blackbox simulation (perfect) zero-knowledge interactive proof system of computational ability (or of knowledge), then there exists a probabilistic polynomial time algorithm that on input x ε {0, 1}, outputs y such that (x,y) ε R with overwhelming probability if x ε dom R, and outputs '⊥' with probability 1 if x ε dom R.