Speeding up prime number generation
We present various ways of speeding up the standard methods for generating provable, resp. probable primes. For probable primes, the effect of using test division and 2 as a fixed base for the Rabin test is analysed, showing that a speedup of almost 50% can be achieved with the same confidence level, compared to the standard method. For Maurer's algorithm generating provable primes p, we show that a small extension of the algorithm will mean that only one prime factor of p−1 has to be generated, implying a gain in efficiency. Further savings can be obtained by combining with the Rabin test. Finally, we show how to combine the algorithms of Maurer and Gordon to make ”strong provable primes” that satisfy additional security constraints.
KeywordsPrime Factor Error Probability Cyclotomic Polynomial Large Prime Factor Incremental Search
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- [AdHu]L.Adleman and M.-D.Huang: Recognizing Primes in Random Polynomial Time, Proc. of STOC 1987, 462–469.Google Scholar
- [BaSh]E.Bach and J.Shallit: Factoring with Cyclotomic Polynomials, Math. Comp. (1989), 52: 201–219.Google Scholar
- [BLS]J.Brillhart, D.H.Lehmer and J.L.Selfridge: New Primality Criteria and Factorizations of 2m±1, Math. Comp. (1975), 29: 620–647.Google Scholar
- [DaLa]Damgård and Landrock: Improved Bounds for the Rabin Primality Test, to appear.Google Scholar
- [EdPo]P.Erdös and C.Pomerance: On the Number of False Witnesses for a Composite Number, Math. Comp. (1986), 46: 259–279.Google Scholar
- [Go]J.Gordon: Strong Primes are Easy to find, Proc. of Crypto 84.Google Scholar
- [Gu]R.K. Guy: How to Factor a Number, Proc. of the 5'th Manitoba Conference on Numerical Mathematics, 1975, University of Manitoba, Winnipeg.Google Scholar
- [KiPo]S.H. Kim and C. Pomerance: The Probability that a Randomly Probable Prime is Composite, Math. Comp. (1989), 53: 721–741.Google Scholar
- [Ma]U.Maurer: The Generation of Secure RSA Products With Almost Maximal Diversity, Proc. of EuroCrypt 89 (to appear).Google Scholar
- [Po]C.Pomerance: On the Distribution of Pseudoprimes, Math. Comp. (1981), 37: 587–593.Google Scholar