**An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates**

*Ivan Damgård and Gudmund Frandsen*

**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.

**Category / Keywords: **public-key cryptography / Primality test, number theory

**Date: **received 22 Nov 2001

**Contact author: **ivan at daimi au dk

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20011125:205221 (All versions of this report)

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