A Dichotomy for Local Small-Bias Generators

We consider pseudorandom generators in which each output bit depends on a constant number of input bits. Such generators have appealingly simple structure: They can be described by a sparse input–output dependency graph G and a small predicate P that is applied at each output. Following the works of Cryan and Miltersen (MFCS’01) and by Mossel et al (STOC’03), we ask: which graphs and predicates yield “small-bias” generators (that fool linear distinguishers)? We identify an explicit class of degenerate predicates and prove the following. For most graphs, all non-degenerate predicates yield small-bias generators, f:{0,1}^n →{0,1}^m , with output length m = n^1 + ϵ for some constant ϵ > 0 . Conversely, we show that for most graphs, degenerate predicates are not secure against linear distinguishers, even when the output length is linear m=n+Ω (n) . Taken together, these results expose a dichotomy: Every predicate is either very hard or very easy, in the sense that it either yields a small-bias generator for almost all graphs or fails to do so for almost all graphs. As a secondary contribution, we attempt to support the view that small-bias is a good measure of pseudorandomness for local functions with large stretch. We do so by demonstrating that resilience to linear distinguishers implies resilience to a larger class of attacks.