An Extended Quadratic Frobenius Primality Test with Average- and Worst-Case Error Estimate

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and to several other known probabilistic tests. EQFT takes time equivalent to about two or three Miller-Rabin tests, but has a much smaller error probability, namely 256/331776^t for t iterations of the test in the worst case. We also give bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2^-155 for k = 500, t = 2. Compared with earlier similar results for the Miller-Rabin test, the results indicate that our test in the average case has the effect of nine Miller-Rabin tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point.