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

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 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. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order dividing 24. 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^{-143} for k=500, t = 2. Compared to earlier similar results for the Miller-Rabin test, the results indicate 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.

[1]  Ivan Damgård,et al.  On Generation of Probable Primes By Incremental Search , 1992, CRYPTO.

[2]  Siguna Müller,et al.  A Probable Prime Test with Very High Confidence for n equiv 1 mod 4 , 2001, ASIACRYPT.

[3]  Claus Brabrand,et al.  The metafront System: Extensible Parsing and Transformation , 2003, LDTA@ETAPS.

[4]  Ivan Damgård,et al.  Speeding up Prime Number Generation , 1991, ASIACRYPT.

[5]  I. Damgård,et al.  Average case error estimates for the strong probable prime test , 1993 .

[6]  Siguna Müller,et al.  A Probable Prime Test with Very High Confidence for n ≡ 3 mod 4 , 2003, Journal of Cryptology.

[7]  Vladimiro Sassone,et al.  Deriving Bisimulation Congruences: 2-Categories Vs Precategories , 2003, FoSSaCS.

[8]  Ivan Damgård,et al.  Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers , 2003 .

[9]  An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates , 2003 .

[10]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

[11]  Vladimiro Sassone,et al.  Jeeg: temporal constraints for the synchronization of concurrent objects , 2005, Concurr. Pract. Exp..

[12]  C. Crépeau,et al.  On the Computational Collapse of Quantum Information , 2003 .

[13]  Ronald Joseph Burthe Further investigations with the strong probable prime test , 1996, Math. Comput..

[14]  Renate Scheidler,et al.  A public-key cryptosystem utilizing cyclotomic fields , 1995, Des. Codes Cryptogr..

[15]  Ivan Damgård,et al.  Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers , 2003, J. Symb. Comput..

[16]  Olivier Danvy,et al.  Tagging, Encoding, and Jones Optimality , 2003, ESOP.

[17]  Jon Grantham,et al.  A Probable Prime Test with High Confidence , 1998, 1903.06823.