On the randomness of bits generated by sufficiently smooth functions

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 axe 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. © Springer-Verlag Berlin Heidelberg 2006.