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Modelling the yield of number field sieve polynomials

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Algorithmic Number Theory (ANTS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

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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.

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References

  1. E Bach and R Peralta, “Asymptotic Semismoothness Probabilities” Math. Comp. 65 (1996), pp 1717–1735.

    Article  MATH  MathSciNet  Google Scholar 

  2. H Boender, “Factoring Integers with the Quadratic Sieve”, PhD Thesis, University of Leiden, 1997.

    Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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. M Elkenbracht-Huizing, “An Implementation of the Number Field Sieve”, Experimental Mathematics 5(3) (1996) pp 375–389.

    MathSciNet  Google Scholar 

  6. M Elkenbracht-Huizing, “A Multiple Polynomial General Number Field Sieve”, Algorithmic Number Theory, LNCS 1122 (1996) pp 99–114.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  8. D E Knuth and L T Pardo, “Analysis of a Simple Factorization Algorithm”, Theor. Comp. Sci. 3 (1976) pp 321–348.

    Article  Google Scholar 

  9. R Lambert, “Computational Aspects of Discrete Logarithms”, PhD Thesis, Univeristy of Waterloo, 1996.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Computer Sciences Laboratory Research School of Information Sciences and Engineering, Australian National University, 0200, Canberra, ACT

    Brian Murphy

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  1. Brian Murphy
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Joe P. Buhler

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© 1998 Springer-Verlag Berlin Heidelberg

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Murphy, B. (1998). Modelling the yield of number field sieve polynomials. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054858

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  • DOI: https://doi.org/10.1007/BFb0054858

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64657-0

  • Online ISBN: 978-3-540-69113-6

  • eBook Packages: Springer Book Archive

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