How to construct pseudorandom permutations from pseudorandom functions
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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,fo 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.