This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth ...

If a Banach space E has an equivalent smooth norm, then every bounded sequence in E* has a weak* converging subsequence. Generalizations of this result are obtained.

0.2. Structure of the proof. 1. As in [dJ], we choose a projection X 99K P of relative dimension 1, and apply semistable reduction to obtain a model X ′ → B over a suitable Galois base change B → P , with Galois group G. 2. We apply induction on the dimension to B , therefore we may assume that B is smooth, and that the discriminant locus of X ′ → B is a G-strict divisor of normal crossings. 3....