Primality testing with fewer random bits

Abstract

In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a numbern, the algorithm chooses “candidates”x ₁,x ₂, ...,x _k uniformly and independently at random from ℤ _n , and tests if any is a “witness” to the compositeness ofn. For either algorithm, the probabilty that it errs is at most 2^−k.

In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen asx, x+1, ..., x+k−1, wherex is chosen uniformly at random from ℤ _n . We prove that fork=[1/2log₂ n], the error probability of the Miller-Rabin test is no more thann ^−1/2+o(1), which improves on the boundn ^−1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound ofn ^−1/2+o(1) if the number of distinct prime factors ofn iso(logn/loglogn).