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Factoring with cubic integers

Part of the Lecture Notes in Mathematics book series (LNM,volume 1554)

Abstract

We describe an experimental factoring method for numbers of form x 3+k; at present we have used only k=2. The method is the cubic version of the idea given by Coppersmith, Odlyzko and Schroeppel (Algorithmica 1 (1986), 1–15), in their section ‘Gaussian integers’. We look for pairs of small coprime integers a and b such that:

  1. i.

    the integer a+bx is smooth,

  2. ii.

    the algebraic integer a+bz is smooth, where z 3=−k. This is the same as asking that its norm, the integer a 3 - kb 3 shall be smooth (at least, it is when k=2).

We used the method to repeat the factorisation of F 7 on an 8-bit computer (2F 7=x 3+2, where x=243).

Keywords

  • Algebraic Integer
  • Factoring Algorithm
  • Rational Integer
  • Algebraic Number Theory
  • Algebraic Number Field

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. D. Coppersmith, A. M. Odlyzko, R. Schroeppel, Discrete logarithms in GF(p), Algorithmica 1 (1986), 1–15.

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  2. I.N. Stewart, D.O. Tall, Algebraic number theory, second edition, Chapman and Hall, London, 1987.

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  3. M. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975), 183–205.

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  4. J.L. Gerver, Factoring large numbers with a quadratic sieve, Math. Comp. 41 (1983), 287–294.

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© 1993 Springer-Verlag

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Pollard, J.M. (1993). Factoring with cubic integers. In: Lenstra, A.K., Lenstra, H.W. (eds) The development of the number field sieve. Lecture Notes in Mathematics, vol 1554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091536

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  • DOI: https://doi.org/10.1007/BFb0091536

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57013-4

  • Online ISBN: 978-3-540-47892-8

  • eBook Packages: Springer Book Archive