On the security of Goldreich's one-way function

Goldreich (ECCC 2000) suggested a simple construction of a candidate one-way function f : {0, 1} (n) -> {0, 1} (m) where each bit of output is a fixed predicate P of a constant number d of (random) input bits. We investigate the security of this construction in the regime m = Dn, where D(d) is a sufficiently large constant. We prove that for any predicate P that correlates with either one or two of its inputs, f can be inverted with high probability. We also prove an amplification claim regarding Goldreich's construction. Suppose we are given an assignment x' is an element of.{0, 1}(n) that has correlation epsilon > 0 with the hidden assignment x is an element of {0, 1} n. Then, given access to x', it is possible to invert f on x with high probability, provided D = D(d, epsilon) is sufficiently large.