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 variables, f can be inverted with high probability. We also prove an amplification claim regarding Goldreich’s construction. Suppose we are given an assignment x ∈ {0, 1}n that has correlation > 0 with the hidden assignment x ∈ {0, 1}n . Then, given access to x , it is possible to invert f on x with high probability, provided D = D(d, ε) is sufficiently large.

• Type: Conference
• Year: 2009
• Where: APPROX
• Authors: Andrej Bogdanov, Youming Qiao
• Keywords: Algorithms, APPROX 2009, Candidate One-way Function, Simple Construction, fixed Predicate