Expander-Based Cryptography Meets Natural Proofs

Abstract

We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich’s pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander’s neighbor function and/or of the local predicate. Our two key conceptual contributions are: 1. We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbor function has low circuit complexity might compromise the security of Goldreich’s PRG and OWF in certain settings. 2. We show that the security of Goldreich’s PRG and OWF over arbitrary expanders is closely related to two other long-standing problems: The existence of unbalanced lossless expanders with low-complexity neighbor function, and limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits. We prove two types of technical results. First, in the regime of quasipolynomial stretch (in which the output length of the PRG and the running time of the distinguisher are quasipolynomial in the seed length) we unconditionally break Goldreich’s PRG, when instantiated with a specific expander whose existence we prove, and for a class of predicates that match the parameters of the currently-best “hard” candidates. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich’s PRG in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich’s PRG is insecure in the regime of a large polynomial stretch for some expander graphs, regardless of the predicate used.