Saving Private Randomness in One-Way Functions and Pseudorandom Generators

Can a one-way function f on n input bits be used with fewer than n bits while retaining comparable hardness of inversion? We show that the answer to this fundamental question is negative, if one is limited black-box reductions. Instead, we ask whether one can save on secret random bits at the expense of more public random bits. Using a shorter secret input is highly desirable, not only because it saves resources, but also because it can yield tighter reductions from higher-level primitives to one-way functions. Our first main result shows that if the number of output elements of f is at most 2k, then a simple construction using pairwise-independent hash functions results in a new one-way function that uses only k secret bits. We also demonstrate that it is not the knowledge of security of f, but rather of its structure, that enables the savings: a black-box reduction cannot, for a general f, reduce the secret-input length, even given the knowledge that security of f is only 2-k; nor can a black-box reduction use fewer than k secret input bits when f has 2k distinct outputs. Our second main result is an application of the public-randomness approach: we show a construction of a pseudorandom generator based on any regular one-way function with output range of known size 2k. The construction requires a seed of only 2n + O(k log k) bits (as opposed to O(n log n) in previous constructions); the savings come from the reusability of public randomness. The secret part of the seed is of length only k (as opposed to n in previous constructions), less than the length of the one-way function input.