Some Number-theoretic Conjectures and Their Relation to the Generation of Cryptographic Primes 1

The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) of a number n on the order of N is derived, given that n satisses one of the conditions 2n+1 is prime, 2an+1 is prime for a given a, or the d integers u 1 ; : : : ; u d , where u 1 = 2a 1 n + 1 and u t = 2a t u t?1 + 1 for 2 t d, are all primes for a given list of integers a 1 ; : : : ; a d. In particular, the conditional probabilities that n is itself a prime, or is of the form \k times a prime" for k = 2; 3; : : : ; is treated for the above conditions. It is shown that although for all k these probabilities strongly depend on the condition placed on n, the probability distribution of the relative size 1 (n) = log N p 1 (n) of the largest prime factor of n is virtually independent of any of these conditions. Some number-theoretic conjectures related to this analysis are stated. Furthermore, the probability distribution of the size of the smaller prime factor of an RSA-modulus of a certain size is investigated.