On Probable Prime Testing and the Computation of Square Roots mod n

  • Siguna Müller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1838)

Abstract

We will investigate two well-known square root finding algorithms which return the roots of some quadratic residue modulo a prime p. Instead of running the mechanisms modulo p we will investigate their behaviour when applied modulo any integer n. In most cases the results will not be the square roots, when n is composite. Since the results obtained can easily be verified for correctness we obtain a very rapid probable prime test. Based on the square root finding mechanisms we will introduce two pseudoprimality tests which will be shown to be extremely fast and very efficient. Moreover, the proposed test for n ≡1 mod 4 will be proven to be even more efficient than Grantham’s suggestion in [5].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnault, F.: Rabin-Miller primality test: Composite numbers which pass it. Math. Comp. 64(209), 355–361 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baillie, R., Wagstaff Jr., S.: Lucas pseudoprimes. Math. Comp. 35, 1391–1417 (1980)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bleichenbacher, D.: Efficiency and Security of Cryptosystems based on Number Theory. Dissertation ETH Zürich (1996)Google Scholar
  4. 4.
    Carmichael, R.D.: On Sequences of Integers Defined by Recurrence Relations. Quart. J. Pure Appl. Math. 48, 343–372 (1920)Google Scholar
  5. 5.
    Grantham, J.: A Probable Prime Test with High Confidence. J. Number Theory 72, 32–47 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Grantham, J.: Frobenius Pseudoprimes (1998) (preprint)Google Scholar
  7. 7.
    Jaeschke, G.: On strong pseudoprimes to several bases. Math. Comp. 61, 915–926 (1993)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Koblitz, N.: A Course in Number Theory and Cryptography. Springer, Heidelberg (1994)MATHCrossRefGoogle Scholar
  9. 9.
    Menezes, A., Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)CrossRefGoogle Scholar
  10. 10.
    Montgomery, P.: Evaluating recurrences of form Xm + n = f(Xm, Xn, Xm − n) via Lucas chains (preprint) Google Scholar
  11. 11.
    More, W.: The LD Probable Prime Test. In: Mullin, R.C., Mullen, G. (eds.) Contemporary Mathematics, vol. 225, pp. 185–191 (1999)Google Scholar
  12. 12.
    Müller, S.: On the Combined Fermat/Lucas Probable Prime Test. In: Walker, M. (ed.) Cryptography and Coding 1999. LNCS, vol. 1746, pp. 222–235. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    Müller, W.B., Oswald, A.: In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 512–516. Springer, Heidelberg (1991)Google Scholar
  14. 14.
    Ribenboim, P.: The New Book of Prime Number Records. Springer, Heidelberg (1996)MATHGoogle Scholar
  15. 15.
    Somer, L.: Periodicity Properties of kth Order Linear Recurrences with Irreducible Characteristic Polynomial Over a Finite Field. In: Mullen, G.L., Shiue, P.J.S. (eds.) Finite Fields, Coding Theory and Advances in Communications and Computing, pp. 195–207. Marcel Dekker Inc., New York (1993)Google Scholar
  16. 16.
    Somer, L.: On Lucas d-Pseudoprimes. In: Bergum, G., Philippou, A., Horadam, A. (eds.) Applications of Fibonacci Numbers, vol. 7, pp. 369–375. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  17. 17.
    Joye, M., Quisquater, J.J.: Efficient computation of full Lucas sequences. IEE Electronics Letters 32(6), 537–538 (1996)CrossRefGoogle Scholar
  18. 18.
    Williams, H.C.: Éduard Lucas and primality Testing. John Wiley & Sons, Chichester (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Siguna Müller
    • 1
  1. 1.Dept. of Math.University of KlagenfurtKlagenfurtAustria

Personalised recommendations