The Combinatorics behind Number-Theoretic Sieves

The Combinatorics behind Number-Theoretic Sieves

Ever since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated re nements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more and more areas of combinatorics where sieve methods (or M obius inversion) are applicable. Unfor...

Improved incremental prime number sieves

An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the rst prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: { it employs only (n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for xed n; { the transition from n to n + 1 takes on...

Two Fast Parallel Prime Number Sieves

Applications of Combinatorics to Information-Theoretic Cryptography

3 4 CONTENTS Preface Cryptology, as the scientific study of secure communications, began in World War II. Ever since the foundational work by Shannon [50], and especially after the introduction of public key cryptography in the seventies [21, 46], applications to cryptology have appeared from an increasing number of areas in mathematics. Combinatorics is one of them. For instance, the difficult...

Combinatorics, Automata and Number Theory

numeration systems The motivation for the introduction of abstract numeration systems stemsfrom the celebrated theorem of Cobham dating back to 1969 about the so-called recognisable sets of integers in any integer base numeration system.An abstract numeration system is simply an infinite genealogically ordered(regular) language. In particular, this notion extends the usual integ...