The least witness of a composite number

  • R. Balasubramanian
  • S. V. Nagaraj
Public-Key Cryptography
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1396)


We consider the problem of finding the least witness of a composite number. If n is a composite number then a number w for which n is not a strong pseudo-prime to the base w is called a witness for n. Let w(n) be the least witness for a composite n. Bach [7] assuming the Generalized Riemann Hypothesis (GRH) showed that w(n) < 2log2 n. In this paper we are interested in obtaining upper bounds for w(n) without assuming the GRH.

Burthe [15) showed that w(n) = O(n1/(8√e)+ε) for all composite numbers n which are not a product of three distinct prime factors. For the three prime factor case he was able to show that w(n) = O(n1/(6√e)+). We improve his result to show w(n) = O(n1/(8√e)+) for all composite numbers n except Carmichael numbers n = pqr for which v2(p − 1) = v2 (q − 1) = v2 (r − 1). For the special Carmichaels we use an argument due to Heath-Brown to get w(n) = O(n1/(6.568√e)+).

We conjecture w(n) = O(n1/(8√e)+) for every composite number n and look at open problems. It appears to be very difficult to settle our conjecture.


Deterministic Algorithm Average Running Time Composite Number Generalize Riemann Hypothesis Deterministic Polynomial Time 
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  1. 1.
    L. Adleman and M. Huang, Primality testing and two dimensional Abelian varieties over finite fields, Lec. Notes in Math, 1512, Springer-Verlag (1994).Google Scholar
  2. 2.
    L. Adleman and F. T. Leighton, An O(n 1/10.89) primality testing algorithm, Math. Comp. 36 (1981) 261–266.Google Scholar
  3. 3.
    L. Adleman and K. S. McCurley, Open problems in number-theoretic complexity-II, in: L. M. Adleman and M. D. Huang (eds.), Algorithmic Number Theory, LNCS 877, Springer-Verlag, Berlin (1994), 291–322.Google Scholar
  4. 4.
    L. Adleman, C. Pomerance and R. Rumely, On distinguishing prime numbers from composite numbers, Ann. of Math. 117 (1983) 173–206.Google Scholar
  5. 5.
    W. R. Alford, A. Granville and C. Poinerance, There are infinitely many Carmichael numbers, Ann. of Math. 140 (1994) 1–20.Google Scholar
  6. 6.
    W. R. Alford, A. Granville and C. Pomerance, On the difficulty of finding reliable witnesses, in: L. M. Adleman and M. D. Huang (eds.), Algorithmic Number Theory, LNCS 877, Springer-Verlag, Berlin (1994), 1–16.Google Scholar
  7. 7.
    E. Bach, Analytic methods in the analysis and design of number-theoretic algorithms, MIT Press, Cambridge, Mass. (1985).Google Scholar
  8. 8.
    E. Bach and L. Huelsbergen, Statistical evidence for small generating sets, Math. Comp 61 (1993), 69–82.Google Scholar
  9. 9.
    R. Balasubramanian and S. V. Nagaraj, Density of Carmichael numbers with three prime factors, Math. Comp. 66 (1997), 1705–1708.Google Scholar
  10. 10.
    P. Beauchemin, G. Brassard, C. Crepeau, C. Goutier and C. Pomerance, The generation of random numbers that are probably prime, J. Cryptology 1 (1988) 53–64.Google Scholar
  11. 11.
    D. Bleichenbacher, Efficiency and security of crypto-systems based on number theory, Ph.D Thesis, Swiss Federal Institute of Technology, Diss. ETH No. 11404, Zurich 1996.Google Scholar
  12. 12.
    W. Bosma and M. P. van der Hulst, Primality testing with cyclotomy, Ph.D Thesis, Faculteit Wiskunde en Informatica, Univ. of Amsterdam (1990).Google Scholar
  13. 13.
    D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc. 12 (1962) 179–192.Google Scholar
  14. 14.
    D. A. Burgess, On character sums and L-series II, Proc. London Math. Soc. 13 (1963) 524–536.Google Scholar
  15. 15.
    R. J. Burthe, The average witness is 2, Ph.D Thesis, University of Georgia (1995).Google Scholar
  16. 16.
    R. J. Burthe Jr., Uper bounds for least witnesses and generating sets, Acta Arith. 80 (1997) 311–326.Google Scholar
  17. 17.
    R. J. Burthe Jr., The average witness is 2, Acta Arith. 80 (1997) 327–341.Google Scholar
  18. 18.
    H. Davenport, Multiplicative Number Theory, 2nd Ed., (Springer Verlag, New York, 1980).Google Scholar
  19. 19.
    I. Damgaard, P. Landrock and C. Pomerance, Average case error estimates for the strong probable prime test, Math. Comp 61 (1993) 177–194.Google Scholar
  20. 20.
    A. Granville, Some conjectures related to Fermat's last theorem, in: Proc. of the First Conference of the CNTA, Alberta, April 1988, pp. 177–192, (Walter de Gruyter, Berlin 1990).Google Scholar
  21. 21.
    A. Granville, On pairs of co-prime integers with no large prime factors, Expo. Math. 9 (1991), 335–350.Google Scholar
  22. 22.
    D. R. Heath-Brown, Personal Communication, April 1997.Google Scholar
  23. 23.
    N. Koblitz, A Course in Number Theory and Cryptography Graduate texts in Mathematics, (Springer Verlag, New York 1987).Google Scholar
  24. 24.
    S. Konyagin and C. Pomerance, On primes recognisable in deterministic polynomial time, in: R. L. Graham and J. Nesetril (eds.), Mathematics of Paul Erdos, Springer-Verlag, Berlin (1997).Google Scholar
  25. 25.
    H. W. Lenstra, Jr., Miller's primality test, Info. Proc. Lett. 8 (1979) 86–88.Google Scholar
  26. 26.
    G. L. Miller, Riemann hypothesis and tests for primality, J. Comput. System Sci. 13 (1976), 300–317.Google Scholar
  27. 27.
    R. Peralta and V. Shoup, Primality testing with fewer random bits, Comp. Compl. 3 (1993), 355–367.Google Scholar
  28. 28.
    R. Pinch, Personal Communication, April 1997.Google Scholar
  29. 29.
    M. O. Rabin, Probabilistic algorithm for testing primality, J. Number Theory 12 (1980), 128–138.Google Scholar
  30. 30.
    G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics No. 46, (Cambridge University Press, 1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • R. Balasubramanian
    • 1
  • S. V. Nagaraj
    • 1
    • 2
  1. 1.The Institute of Mathematical SciencesMadrasIndia
  2. 2.IBM Tokyo Research LaboratoryKanagawakenJapan

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