Pseudorandom Number Generation and Space Complexity

Recently, Blum and Micali (1982) described a pseudorandum number generator that transforms m-bit seeds to ink-bit pseudorandom numbers, for any integer k. Under the assumption that the discrete logarithm problem cannot be solved by polynomial-size combinational logic circuits, they show that the pseudorandom numbers generated are good in the sense that no polynomial-size circuit can determine the tth bit given the 1st through ( t 1 ) t h bits, with better than 50% accuracy. Yao (1982) has shown under the same assumption about the nonpolynomial complexity of the discrete logarithm problem, that these pseudorandom numbers can be used in place of truly random numbers by any polynomial-time probabilistic Turing machine. Thus, given a time n ~ probabilistic Turing machine M and given any e > 0 , a deterministic Turing machine can simulate M by cycling through all seeds of length n ~, giving a deterministic simulation in time 2 n~, an improvement over the time 2 "k taken by the obvious simulation. Yao also shows that other problems, for example, integer factorization, can be used instead of the discrete logarithm in the intractability assumption.