On constructing parallel pseudorandom generators from one-way functions

We study pseudorandom generator (PRG) constructions G^f : {0, 1}^l → {0, 1}^1+s from one-way functions f : {0, 1}ⁿ → {0, 1}^m. We consider PRG constructions of the form G^f (x) = C(f(q₁) ...f (g_poly(n))) where C is a polynomial-size constant depth circuit (i.e., AC⁰) and C and the q´s are generated from x arbitrarily. We show that every black-box PRG construction of this form must have stretch s bounded as s ≤ 1 · (log^O(1) n)/m + O(1) = o(l). This holds even if the PRG construction starts from a one-to-one function f : {0,1}ⁿ → {0, 1}^m where m > 5n. This shows that either adaptive queries or sequential computation are necessary for black-box PRG constructions with constant factor stretch (i.e. s = Ω(l)) from one-way functions, even if the functions are one-to-one. On the positive side we show that if there is a one-way function f : {0, 1}ⁿ → {0, 1}^m that is regular (i.e. the number of preimages of f(x) depends on |x| but not on x) and computable by polynomial-size constant depth circuits then there is a PRG : {0, 1}^l → {0, 1}^{l + 1} computable by polynomial-size constant depth circuits. This complements our negative result above because one-to-one functions are regular. We also study constructions of average-case hard functions starting from worst-case hard ones, i.e. hardness amplifications. We show that if there is an oracle procedure Ampf in the polynomial time hierarchy (PH) such that Ampf is average-case hard for every worst-case hard f, then there is an average-case hard function in PH unconditionally. Bogdanov and Trevisan (FOGS ´03) and Viola (CCC´03) show related. but incomparable negative results.