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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Stehlé, D. (2006). On the Randomness of Bits Generated by Sufficiently Smooth Functions. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_19
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DOI: https://doi.org/10.1007/11792086_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36075-9
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