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.
- L.Adleman and M.-D.Huang: Recognizing Primes in Random Polynomial Time, Proc. of STOC 1987, 462–469.
- E.Bach and J.Shallit: Factoring with Cyclotomic Polynomials, Math. Comp. (1989), 52: 201–219.
- P.Beauchemin, G.Brassard,C.Crépeau, C.Goutier and C.Pomerance: The Generation of Numbers that are Probably Prime, J.Cryptology (1988) 1:53–64. CrossRef
- J.Brillhart, D.H.Lehmer and J.L.Selfridge: New Primality Criteria and Factorizations of 2m±1, Math. Comp. (1975), 29: 620–647.
- Damgård and Landrock: Improved Bounds for the Rabin Primality Test, to appear.
- P.Erdös and C.Pomerance: On the Number of False Witnesses for a Composite Number, Math. Comp. (1986), 46: 259–279.
- J.Gordon: Strong Primes are Easy to find, Proc. of Crypto 84.
- R.K. Guy: How to Factor a Number, Proc. of the 5'th Manitoba Conference on Numerical Mathematics, 1975, University of Manitoba, Winnipeg.
- S.H. Kim and C. Pomerance: The Probability that a Randomly Probable Prime is Composite, Math. Comp. (1989), 53: 721–741.
- U.Maurer: The Generation of Secure RSA Products With Almost Maximal Diversity, Proc. of EuroCrypt 89 (to appear).
- C.Pomerance: On the Distribution of Pseudoprimes, Math. Comp. (1981), 37: 587–593.
- M.O. Rabin: Probabilistic Algorithm for Testing Primality, J.Number Theory (1980), 12: 128–138. CrossRef
- Speeding up prime number generation
- Book Title
- Advances in Cryptology — ASIACRYPT '91
- Book Subtitle
- International Conference on the Theory and Application of Cryptology Fujiyosida, Japan, November 1991 Proceedings
- pp 440-449
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Additional Links
- Industry Sectors
- eBook Packages
To view the rest of this content please follow the download PDF link above.