On the Randomness of Bits Generated by Sufficiently Smooth Functions

Damien Stehlé

Abstract: Elementary functions such as sin or exp may naively be considered as good generators of random bits: the bit-runs output by these functions are believed to be statistically random most of the time. Here we investigate their computational hardness: given a part of the binary expansion of exp(x), can one recover x? We describe a heuristic technique to address this type of questions. It relies upon Coppersmith's heuristic technique --- itself based on lattice reduction --- for finding the small roots of multivariate polynomials modulo an integer. For our needs, we improve the lattice construction step of Coppersmith's method: we describe a way to find a subset of a set of vectors that decreases the Minkowski theorem bound, in a rather general setup including Coppersmith-type lattices.

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