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 256/331776t 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.
Partially supported by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT).
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References
Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Preprint 2002. Department of Computer Science & Engineering, Indian Institute of Technology, Kanpur Kanpur-208016, INDIA (2002)
Bach, E., Shallit, J.: Algorithmic number theory. Foundations of Computing Series, vol. 1. MIT Press, Cambridge (1996)
Brandt, J., Damgård, I.: On generation of probable primes by incremental search. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 358–370. Springer, Heidelberg (1993)
Brandt, J., Damgård, I., Landrock, P.: Speeding up prime number generation. In: Matsumoto, T., Imai, H., Rivest, R.L. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 440–449. Springer, Heidelberg (1993)
Damgård, I., Landrock, P., Pomerance, C.: Average case error estimates for the strong probable prime test. Math. Comp. 61(203), 177–194 (1993)
Grantham, J.: A probable prime test with high confidence. J. Number Theory 72(1), 32–47 (1998)
Müller, S.: A probable prime test with very high confidence for n ≡ 1 mod 4. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 87–106. Springer, Heidelberg (2001)
Müller, S.: A probable prime test with very high confidence for n ≡ 3 mod 4. J. Cryptology 16(2), 117–139 (2003)
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Damgård, I.B., Frandsen, G.S. (2003). An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates. In: Lingas, A., Nilsson, B.J. (eds) Fundamentals of Computation Theory. FCT 2003. Lecture Notes in Computer Science, vol 2751. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45077-1_12
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DOI: https://doi.org/10.1007/978-3-540-45077-1_12
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