On the Existence of Pseudorandom Generators

Pseudorandom generators (suggested and developed by Blum and Micali and Yao) are efficient deterministic programs that expand a randomly selected k -bit seed into a much longer pseudorandom bit sequence which is indistinguishable in polynomial time from an (equally long) sequence of unbiased coin tosses. Pseudorandom generators are known to exist assuming the existence of functions that cannot be efficiently inverted on the distributions induced by applying the function iteratively polynomially many times. This sufficient condition is also a necessary one, but it seems difficult to check whether particular functions, assumed to be one-way, are also one-way on their iterates. This raises the fundamental question whether the mere existence of one-way functions suffices for the construction of pseudorandom generators.In this paper we present progress towards resolving this question. We consider regular functions, in which every image of a k -bit string has the same number of preimages of length k. We show that if a regular function is one-way then pseudorandom generators do exist. In particular, assuming the intractability of general factoring, we can now prove that pseudorandom generators do exist. Other applications are the construction of pseudorandom generators based on the conjectured intractability of decoding random linear codes, and on the assumed average case difficulty of combinatorial problems as subset-sum.