Improving the Speed and Accuracy of the Miller-Rabin Primality Test

Currently, even the fastest deterministic primality tests run slowly, with the AgrawalKayal-Saxena (AKS) Primality Test runtime Õ(log(n)), and probabilistic primality tests such as the Fermat and Miller-Rabin Primality Tests are still prone to false results. In this paper, we discuss the accuracy of the Miller-Rabin Primality Test and the number of nonwitnesses for a composite odd integer n. We also extend the MillerRabin Theorem by determining when the number of nonwitnesses N(n) equals φ(n) 4 and by proving that for all n, if N(n) > 5 32 · φ(n) then n must be of one of these 3 forms: n = (2x + 1)(4x + 1), where x is an integer, n = (2x + 1)(6x + 1), where x is an integer, n is a Carmichael number of the form pqr, where p, q, r are distinct primes congruent to 3 (mod 4). We then find witnesses to certain forms of composite numbers with high rates of nonwitnesses and find that Jacobi nonresidues and 2 are both valuable bases for the Miller-Rabin test. Finally, we investigate the frequency of strong pseudoprimes further and analyze common patterns using MATLAB. This work is expected to result in a faster and better primality test for large integers.