Speeding up prime number generation

  • Jorgen Brandt
  • Ivan Damgård
  • Peter Landrock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 739)

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AdHu]
    L.Adleman and M.-D.Huang: Recognizing Primes in Random Polynomial Time, Proc. of STOC 1987, 462–469.Google Scholar
  2. [BaSh]
    E.Bach and J.Shallit: Factoring with Cyclotomic Polynomials, Math. Comp. (1989), 52: 201–219.Google Scholar
  3. [BBCGP]
    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.CrossRefGoogle Scholar
  4. [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
  5. [DaLa]
    Damgård and Landrock: Improved Bounds for the Rabin Primality Test, to appear.Google Scholar
  6. [EdPo]
    P.Erdös and C.Pomerance: On the Number of False Witnesses for a Composite Number, Math. Comp. (1986), 46: 259–279.Google Scholar
  7. [Go]
    J.Gordon: Strong Primes are Easy to find, Proc. of Crypto 84.Google Scholar
  8. [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
  9. [KiPo]
    S.H. Kim and C. Pomerance: The Probability that a Randomly Probable Prime is Composite, Math. Comp. (1989), 53: 721–741.Google Scholar
  10. [Ma]
    U.Maurer: The Generation of Secure RSA Products With Almost Maximal Diversity, Proc. of EuroCrypt 89 (to appear).Google Scholar
  11. [Po]
    C.Pomerance: On the Distribution of Pseudoprimes, Math. Comp. (1981), 37: 587–593.Google Scholar
  12. [Ra]
    M.O. Rabin: Probabilistic Algorithm for Testing Primality, J.Number Theory (1980), 12: 128–138.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Jorgen Brandt
    • 1
  • Ivan Damgård
    • 1
  • Peter Landrock
    • 1
  1. 1.Mathematical InstituteAarhus UniversityAarhus CDenmark

Personalised recommendations