On the Construction of One-Way Functions from Average Case Hardness

In this paper we study the possibility of proving the existence of one-way functions based on average case hardness. It is well-known that if there exists a polynomial-time sampler that outputs instancesolution pairs such that the distribution on the instances is hard on average, then one-way functions exist. We study the possibility of constructing such a sampler based on the assumption that there exists a sampler that outputs only instances, where the distribution on the instances is hard on the average. Namely, we study the possibility of "modifying" an ordinary sampler S that outputs (only) hard instances of some search problem R, to a sampler that outputs instance-solution pairs of the same search problem R. We show that under some restriction, not every sampler can be so modified. That is, we show that if some hard problem with certain properties exists (which, in particular is implied by the existence of one-way permutations), then for every polynomial λ, there exists a search problem R and a polynomial-time sampler S such that (1) R is hard under the distribution induced by S, and (2) there is no sampler S∗ with randomness complexity bounded by λ, that outputs instance-solution pairs of R, where the distribution on the instances of S∗ is closely related to that of S (i.e., dominates it). A possible interpretation of our result is that a generic approach for transforming samplers of instances to samplers of instance-solution pairs cannot succeed.