Limits on the Stretch of Non-adaptive Constructions of Pseudo-Random Generators

The standard approach for constructing a large-stretch pseudorandom generator given a one-way permutation or given a smaller-stretch pseudo-random generator involves repeatedly composing the given primitive with itself. In this paper, we consider whether this approach is necessary, that is, whether there are constructions that do not involve composition. More formally, we consider black-box constructions of pseudorandom generators from pseudo-random generators of smaller stretch or from one-way permutations, where the constructions make only nonadaptive queries to the given object. We consider three classes of such constructions, and for each class, we give a black-box impossibility result that demonstrates a contrast between the stretch that can be achieved by adaptive and non-adaptive black-box constructions. We first consider constructions that make constantly-many non-adaptive queries to a given pseudo-random generator, where the seed length of the construction is at most O(logn) bits longer than the length n of each oracle query. We show that such constructions cannot achieve stretch that is even a single bit greater than the stretch of the given pseudo-