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

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Abstract

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to an extends the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability, namely 256/331776 t for t iterations of the test in the worst case. We 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− 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.