Primality Testing with Fewer Random Bits

In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms, to test a number n for primality, the algorithm chooses "candidates" x 1 ; x 2 ; : : : ; x k uniformly and independently at random from Z n , and tests if any are a "witness" to the compositeness of n. For either algorithm, the probability that it errs is at most 2 \Gammak . In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen as x; x+1; : : : ; x+k \Gamma 1, where x is chosen uniformly at random from Z n . We prove that for k = d 1 2 log 2 ne, the error probability of the Miller-Rabin test is no more than n \Gamma1=2+o(1) , which improves on the n \Gamma1=4+o(1) bound 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 of n \Gamma1=2+o(1) if the number of distinct prime factors of n is o(log n= loglog n). 1. Introduction Main..