Modelling the yield of number field sieve polynomials

  • Brian Murphy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1423)

Abstract

Understanding the yield of number field sieve polynomials is crucial to improving the performance of the algorithm, and to assessing its potential impact on the practical security of cryptosystems relying on integer factorisation. In this paper we examine the yield of these polynomials, concentrating on those produced by Montgomery's selection algorithm. Given such a polynomial f, we consider the influence of two factors; the size of values taken by f and the effect of the knowing the primes p for which f has roots mod p. Experiments show the influence of the first property, particularly whilst sieving close to real roots. Estimates of the effect of the second property show that it may effect yield by as much as a factor of two. We present sieving experiments demonstrating the effect to that extent. Finally, we suggest a preliminary model to approximate the behaviour of these polynomials across the sieving region.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E Bach and R Peralta, “Asymptotic Semismoothness Probabilities” Math. Comp. 65 (1996), pp 1717–1735.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    H Boender, “Factoring Integers with the Quadratic Sieve”, PhD Thesis, University of Leiden, 1997.Google Scholar
  3. 3.
    J P Buhler, H W Lenstra Jr, C Pomerance, “Factoring Integers with the Number Field Sieve”, The Development of the Number Field Sieve, LNM 1554 (1993) pp 50–94.MATHMathSciNetGoogle Scholar
  4. 4.
    K Dickman, “On the Frequency of Numbers Containing Prime Factors of a Certain Relative Magnitude”, Ark. Mat. Astronomi och Fysik 22A 10 (1930), pp 1–14.Google Scholar
  5. 5.
    M Elkenbracht-Huizing, “An Implementation of the Number Field Sieve”, Experimental Mathematics 5(3) (1996) pp 375–389.MathSciNetGoogle Scholar
  6. 6.
    M Elkenbracht-Huizing, “A Multiple Polynomial General Number Field Sieve”, Algorithmic Number Theory, LNCS 1122 (1996) pp 99–114.MATHMathSciNetGoogle Scholar
  7. 7.
    R A Golliver, A K Lenstra and K S McCurley, “Lattice Sieving and Trial Division”, Algorithmic Number Theory, LNCS 877 (1994) pp 18–27.MATHMathSciNetGoogle Scholar
  8. 8.
    D E Knuth and L T Pardo, “Analysis of a Simple Factorization Algorithm”, Theor. Comp. Sci. 3 (1976) pp 321–348.CrossRefGoogle Scholar
  9. 9.
    R Lambert, “Computational Aspects of Discrete Logarithms”, PhD Thesis, Univeristy of Waterloo, 1996.Google Scholar
  10. 10.
    B Murphy and R P Brent, “On Quadratic Polynomials for the Number Field Sieve”, Computing Theory 98, ACSC 20(3) (1998), Springer, pp 199–215.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Brian Murphy
    • 1
  1. 1.Computer Sciences Laboratory Research School of Information Sciences and EngineeringAustralian National UniversityCanberra

Personalised recommendations