A probable prime test with very high confidence for n = 3 mod 4

Abstract

The workhorse of most compositeness tests is Miller-Rabin, which works very fast in practice, but may fail for one-quarter of all bases. We present an alternative method to decide quickly whether a large number n is composite or probably prime. Our algorithm is both based on the ideas of Pomerance, Baillie, Selfridge, and Wagstaff, and on a suitable combination of square, third, and fourth root testing conditions. A composite number ≡ 3 mod 4 will pass our test with probability less than 1/331,000, in the worst case. For most numbers, the failure rate is even smaller. Depending on the the respective residue classes n modulo 3 and 8, we prove a worst-case failure rate of less than 1/5,300,000, 1/480,000, and 1/331,000, respectively, for any iteration of our test. Along with some fixed precomputation, our test has running time about three times the time as for the Miller-Rabin test. Implementation can be achieved very efficiently by naive arithmetic only. © 2003 International Association for Cryptologlc Research.