Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Journal of Cryptology
  3. Article
The generation of random numbers that are probably prime
Download PDF
Download PDF
  • Published: January 1988

The generation of random numbers that are probably prime

  • Pierre Beauchemin1,
  • Gilles Brassard1,
  • Claude Crépeau2,
  • Claude Goutier3 &
  • …
  • Carl Pomerance4 

Journal of Cryptology volume 1, pages 53–64 (1988)Cite this article

  • 406 Accesses

  • 21 Citations

  • 3 Altmetric

  • Metrics details

Abstract

In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probability of failing only on composite numbers that can actually be split in expected polynomial time. Therefore, factoring would be easy if Rabin's test systematically failed with a 25% probability on each composite integer (which, of course, it does not). The second observation is more fundamental because it is not restricted to primality testing: it has consequences for the entire field of probabilistic algorithms. The failure probability when using a probabilistic algorithm for the purpose of testing some property is compared with that when using it for the purpose of obtaining a random element hopefully having this property. More specifically, we investigate the question of how reliable Rabin's test is when used to generate a random integer that is probably prime, rather than to test a specific integer for primality.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Adleman, L., and M.-D. Huang, Recognizing primes in random polynomial time, Proceedings of the 19th Annual ACM Symposium on the Theory of Computing, pp. 462–469, 1987.

  2. Adleman, L., C. Pomerance, and R. Rumely, On distinguishing prime numbers from composite numbers, Annals of Mathematics, vol. 117, pp. 173–206, 1983.

    Google Scholar 

  3. Babai, L., Monte Carlo algorithms in graph isomorphism testing, Rapport de Recherches du Département de Mathématiques et de Statistiques, D.M.S. # 79-10, Université de Montréal, 1979.

  4. Baillie, R., and S. S. Wagstaff, Jr., Lucas pseudoprimes, Mathematics of Computation, vol. 35, no. 152, pp. 1392–1417, 1980.

    Google Scholar 

  5. Beauchemin, P., G. Brassard, C. Crépeau, and C. Goutier, Two observations on probabilistic primality testing, Advances in Cryptology—Crypto 86 Proceedings, Springer-Verlag, New York, pp. 443–450, 1987.

    Google Scholar 

  6. Brassard, G., and P. Bratley, Algorithmics: Theory and Practice, Prentice-Hall, Englewood Cliffs, New Jersey, 1988.

    Google Scholar 

  7. Cohen, H., and A. K. Lenstra, Implementation of a new primality test, Mathematics of Computation, vol. 48, no. 177, pp. 103–121, 1987.

    Google Scholar 

  8. Couvreur, C., and J.-J. Quisquater, An introduction to fast generation of large prime numbers, Philips Journal of Research, vol. 37, nos. 5/6, pp. 231–264, 1982.

    Google Scholar 

  9. Erdös, P., and C. Pomerance, On the number of false witnesses for a composite number, Mathematics of Computation, vol. 46, no. 173, pp. 259–279, 1986.

    Google Scholar 

  10. Gill, J., Computational complexity of probabilistic Turing machines, SIAM Journal on Computing, vol. 6, no. 4, pp. 675–695, 1977.

    Google Scholar 

  11. Goldwasser, S., and J. Kilian, Almost all primes can be quickly certified, Proceedings of the 18th Annual ACM Symposium on the Theory of Computing, pp. 316–329,1986.

  12. Hardy, G. H., and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Oxford Science Publications, 1979.

  13. Knuth, D. E., The Art of Computer Programming, Volume 2, Second edition, Addison-Wesley, Reading, Massachusetts, 1981.

    Google Scholar 

  14. Kranakis, E., Primality and Cryptography, Wiley-Teubner Series in Computer Science, 1986.

  15. Miller, G. L., Riemann's hypothesis and tests for primality, Journal of Computer and System Sciences, vol. 13, pp. 300–317, 1976.

    Google Scholar 

  16. Monier, L., Evaluation and comparison of two efficient probabilistic primality testing algorithms, Theoretical Computer Science, vol. 11, pp. 97–108, 1980.

    Google Scholar 

  17. Pomerance, C., The search for prime numbers, Scientific American, vol. 247, no. 6, pp. 136–147, 1982.

    Google Scholar 

  18. Pomerance, C., J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25.109, Mathematics of Computation, vol. 35, no. 151, pp. 1003–1026, 1980.

    Google Scholar 

  19. Pratt, V., Every prime has a succinct certificate, SIAM Journal on Computing, vol. 4, no. 3, pp. 214–220, 1975.

    Google Scholar 

  20. Rabin, M. O., Probabilistic algorithm for testing primality, Journal of Number Theory, vol. 12, pp. 128–138, 1980.

    Google Scholar 

  21. Rivest, R. L., A. Shamir, and L. Adleman, A method for obtaining digital signatures and publickey cryptosystems, Communications of the Association for Computing Machinery, vol. 21, no. 2, pp. 120–126, 1978.

    Google Scholar 

  22. Solovay, R., and V. Strassen, A fast Monte Carlo test for primality, SIAM Journal on Computing, vol. 6, pp. 84–85, 1977; erratum in vol. 7, p. 118,1978.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département d'informatique et de recherche opérationnelle, Université de Montréal, C.P. 6128, Succ. “A”, H3C3J7, Montréal, Québec, Canada

    Pierre Beauchemin & Gilles Brassard

  2. Department of Computer Science, Massachusetts Institute of Technology, 545 Technology Square, 02139, Cambridge, MA, USA

    Claude Crépeau

  3. Centre de Calcul, Université de Montréal, C.P. 6128, Succ. “A”, H3C3J7, Montréal, Québec, Canada

    Claude Goutier

  4. Department of Mathematics, University of Georgia, 30602, Athens, GA, USA

    Carl Pomerance

Authors
  1. Pierre Beauchemin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles Brassard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Claude Crépeau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Claude Goutier
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Carl Pomerance
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by NSERC grant A4107. Part of the research was performed while this author was at the CWI, Amsterdam.

Supported in part by an NSERC Posgraduate Scholarship. Part of the research was performed while this author was at the Université de Montréal.

Supported in part by an NSF grant.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Beauchemin, P., Brassard, G., Crépeau, C. et al. The generation of random numbers that are probably prime. J. Cryptology 1, 53–64 (1988). https://doi.org/10.1007/BF00206325

Download citation

  • Issue Date: January 1988

  • DOI: https://doi.org/10.1007/BF00206325

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Key words

  • Factorization
  • False witnesses
  • Primality testing
  • Probabilistic algorithms
  • Rabin's test
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature