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 (FOCS '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, , 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 give evidence in support of 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 for such functions.