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Factoring with cubic integers

  • J. M. Pollard
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1554)

Abstract

We describe an experimental factoring method for numbers of form x3+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 z3=−k. This is the same as asking that its norm, the integer a3 - kb3 shall be smooth (at least, it is when k=2).

     

We used the method to repeat the factorisation of F7 on an 8-bit computer (2F7=x3+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. 1.
    D. Coppersmith, A. M. Odlyzko, R. Schroeppel, Discrete logarithms in GF(p), Algorithmica 1 (1986), 1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I.N. Stewart, D.O. Tall, Algebraic number theory, second edition, Chapman and Hall, London, 1987.zbMATHGoogle Scholar
  3. 3.
    M. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975), 183–205.MathSciNetzbMATHGoogle Scholar
  4. 4.
    J.L. Gerver, Factoring large numbers with a quadratic sieve, Math. Comp. 41 (1983), 287–294.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • J. M. Pollard
    • 1
  1. 1.Tidmarsh CottageReadingEngland

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