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 $$G$$G and a small predicate $$P$$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, $$f:\{0,1\}^n \rightarrow \{0,1\}^m$$f:{0,1}n→{0,1}m, with output length $$m = n^{1 + \epsilon }$$m=n1+ϵ for some constant $$\epsilon > 0$$ϵ>0. Conversely, we show that for most graphs, degenerate predicates are not secure against linear distinguishers, even when the output length is linear $$m=n+\Omega (n)$$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.