On generation of probable primes by incremental search

Abstract

This paper examines the following algorithm for generating a probable prime number: choose a random it bit odd number n, and test the numbers n,n + 2,… for primality using i iterations of Rabin’s test, until a probable prime has been found or some maximum number s of candidates have been tested. We show an explicit upper bound as a function of k, t and s on the probability that this algorithm outputs a composite. From Hardy and Littiewoods prime r- tuple conjecture, an upper bound follows on the probability that the algorithm fails. We propose the entropy of the output distributrion as a natural measure of the quality of the output. Under the prime r-tuple conjecture, we show a lower bound on the entropy of the output distribution over the primes. This bound shows that as k →∞ the entropy becomes almost equal to the largest possible value. Variants allowing repeated choice of starting points or arbitrary search length are also examined. They are guaranteed not to fail, and their error probability and output entropy can be analysed to some extent.