Fast algorithms under the extended riemann hypothesis

Abstract

Several results in theoretical computer science use the following theorem: For a positive integer q, let Z•q denote the multiplicative group of all integers x, 0 that are relatively prime to q. Let G be a proper subgroup of Z•q. Then, assuming the Extended Riemann Hypothesis, there is a constant C such that if q is sufficiently large, Z•q−G contains a positive integer N≤C (logeq)2. We show that for q≥106, one may take C=60. As an application, we discuss a deterministic polynomial-time primality test. Miller proved that such algorithms must exist if the ERH is true, but we are unable to specify one without the concrete information given above. We eliminate this difficulty, and show how to implement a fast primality test.