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. 1

Abstract. Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm. 1