On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms

We introduce a version of the cavity method for diluted mean-field spin models that allows the computation of thermodynamic quantities similar to the Franz-Parisi quenched potential in sparse random graph models. This method is developed in the particular case of partially decimated random constraint satisfaction problems. This allows to develop a theoretical understanding of a class of algorithms for solving constraint satisfaction problems, in which elementary degrees of freedom are sequentially assigned according to the results of a message passing procedure (belief-propagation). We confront this theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure

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 ɛ> 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.