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:
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i.
the integer a+bx is smooth,
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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).
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References
D. Coppersmith, A. M. Odlyzko, R. Schroeppel, Discrete logarithms in GF(p), Algorithmica 1 (1986), 1–15.
I.N. Stewart, D.O. Tall, Algebraic number theory, second edition, Chapman and Hall, London, 1987.
M. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975), 183–205.
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
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