An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates

Abstract

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT is well-suited for generating large, random prime numbers since on a random input number, it takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order 3 and 4. We obtain a simple closed expression that upper bounds the probability of acceptance for any input number. This in turn allows us to give strong 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^{-143} for k=500, t=2 . Compared to earlier similar results for the Miller-Rabin test, the results indicates that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point. While EQFT is slower than the average case on a small set of inputs, we present a variant that is always fast, i.e. takes time about 2 Miller-Rabin tests. The variant has slightly larger worst case error probability than EQFT, but still improves on previous proposed tests.