A simple unpredictable pseudo random number generator
SIAM Journal on Computing
Fast Multiple-Precision Evaluation of Elementary Functions
Journal of the ACM (JACM)
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Searching Worst Cases of a One-Variable Function Using Lattice Reduction
IEEE Transactions on Computers
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Partial key exposure attacks on RSA up to full size exponents
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Cryptanalysis of RSA with private key d less than N0.292
IEEE Transactions on Information Theory
Speeding-up lattice reduction with random projections
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
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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.