How to construct pseudorandom and super pseudorandom permutations from one single pseudorandom function

Abstract

In this paper we will solve two open problems concerning pseudorandom permutations generators. 1. We will see that it is possible to obtain a pseudorandom permutation generator with only three rounds of DES - like permutation and a single pseudorandom function. This will solve an open problem of [6]. 2. We will see that it is possible to obtain a super pseudorandom permutation generator with a single pseudorandom function. This will solve an open problem of [5]. For this we will use only four rounds of DES — like permutation. For example, we will see that if ζ denotes the rotation of one bit, ψ(f, f, f o ζ o f) is a pseudorandom function generator. And ψ(f, f, f, f o ζ o f) is a super pseudorandom function generator. Here the number of rounds used is optimal. It should be noted that here we introduce an important new idea in that we do not use a composition of f, i times, but f o ζ o f for the last round, where ζ is a fixed and public function.