On constructing parallel pseudorandom generators from one-way functions

We study pseudorandom generator (PRG) constructions G/sup f/ : {0, 1}/sup l/ /spl rarr/ {0, 1}/sup 1+s/ from one-way functions f : {0, 1}/sup n/ /spl rarr/ {0, 1}/sup m/. We consider PRG constructions of the form G/sup f/ (x) = C(f(q/sub 1/) ...f (g/sub poly(n)/)) where C is a polynomial-size constant depth circuit (i.e., AC/sup 0/) 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 /spl les/ 1 /spl middot/ (log/sup 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}/sup n/ /spl rarr/ {0, 1}/sup 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 = /spl Omega/(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}/sup n/ /spl rarr/ {0, 1}/sup 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}/sup l/ /spl rarr/ {0, 1}/sup 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.