Abstract not found

A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudo-prime of the same size that passes the Miller-Rabin test for only one base. Therefore our algorithm is even faster than presently-used algorithms for generating only pseudo-primes because several Miller-Rabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval. Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA public-key cryptosystem is presented. The prime generation algorithm can easily be modified to generate nearly random primes or RSA-moduli that satisfy t...

. For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several special cases. In particular, we prove that C 1 is finite and that C k is infinite, whenever 1 \Gamma k ? 1 is square-free. We also discuss generalized Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers n which satisfy the equation a n j a mod n only for a = 2, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares. 1 Introduction: Historical Background On October 18th, 1640, Pierre de Fermat wrote in a letter to Bernard Frenicle de Bessy that if p is a prime number, then p divides a p\Gamma1 \Gamma 1 for all in...

We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.

: Goldach's conjecture states that every even integer greater or equal to 6 is the sum of two prime numbers. This result is still unproved. This conjecture has been numerically checked up to 4:10 11 on an IBM 3083 mainframe. We describe here an implementation on a less powerful machine which raises the bound to 10 12 . Key-words: Prime numbers; Goldbach's problem (R'esum'e : tsvp) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientifique Institut National de Recherche en Informatique (URA 227) Universit e de Rennes 1 -- Insa de Rennes et en Automatique -- unit e de recherche de Rennes Am'eliorations de bornes au sujet de la conjecture de Goldbach (premi`ere version) R'esum'e : La conjecture de Goldbach stipule que tout nombre pair sup'erieur ou 'egal `a 6 est somme de deux nombres premiers. Ce r'esultat est `a ce jour non d'emontr'e. Il a 'et'e v'erifi'e num'eriquement jusqu'`a 4:10 11 sur un IBM 3083. Nous d'ecrivons ici une impl'ementation...

Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, non-primes: if for a certain a coprime to n, an−1 is not congruent to 1 mod n, then, by the theorem, n is not

Carmichael numbers with three prime factors

Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, non-primes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is