# Factoring with cubic integers

Conference paper

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## 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:- i.
the integer

*a*+*bx*is smooth, - 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 (2*F*_{7}=*x*^{3}+2, where *x*=2^{43}).

## 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.MathSciNetCrossRefzbMATHGoogle Scholar - 2.I.N. Stewart, D.O. Tall,
*Algebraic number theory*, second edition, Chapman and Hall, London, 1987.zbMATHGoogle Scholar - 3.M. Morrison, J. Brillhart,
*A method of factoring and the factorization of F*_{7}, Math. Comp.**29**(1975), 183–205.MathSciNetzbMATHGoogle Scholar - 4.J.L. Gerver,
*Factoring large numbers with a quadratic sieve*, Math. Comp.**41**(1983), 287–294.MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

© Springer-Verlag 1993