Linear Programming Relaxations for Goldreich's Generators over Non-Binary Alphabets

Goldreich suggested candidates of one-way functions and pseudorandom generators included in NC. It is known that randomly generated Goldreich’s generator using (r−1)-wise independent predicates with n input variables and m = Cn output variables is not pseudorandom generator with high probability for sufficiently large constant C. Most of the previous works assume that the alphabet is binary and use techniques available only for the binary alphabet. In this paper, we deal with non-binary generalization of Goldreich’s generator and derives the tight threshold for linear programming relaxation attack using local marginal polytope for randomly generated Goldreich’s generators. We assume that u(n) ∈ ω(1) ∩ o(n) input variables are known. In that case, we show that when r ≥ 3, there is an exact threshold μc(k, r) := ( k r ) −1 (r−2) r(r−1)r−1 such that for m = μ n r−1 u(n)r−2 , the LP relaxation can determine linearly many input variables of Goldreich’s generator if μ > μc(k, r), and that the LP relaxation cannot determine 1 r−2 u(n) input variables of Goldreich’s generator if μ < μc(k, r). This paper uses characterization of LP solutions by combinatorial structures called stopping sets on a bipartite graph, which is related to a simple algorithm called peeling algorithm.