A Dichotomy for Local Small-Bias Generators
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AbstractWe 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= n1 + ϵfor 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.
All Author(s) ListApplebaum B., Bogdanov A., Rosen A.
Journal nameJournal of Cryptology
Detailed descriptionTo ORKTS: Supported in part by Hong Kong RGC GRF Grant CUHK 410309.
Volume Number29
Issue Number3
PublisherSpringer Verlag
Place of PublicationGermany
Pages577 - 596
LanguagesEnglish-United Kingdom
KeywordsDichotomy, Local functions, NC0, Small-bias generator

Last updated on 2021-18-01 at 01:06