On nonlinear polynomial selection for the number field sieve

Details On nonlinear polynomial selection for the number field sieve On nonlinear polynomial selection for the number field sieve

2011-09-29

arxiv.org/abs/1109.6398v1

Mathematics

Number Theory

Scientific paper

The number field sieve is asymptotically the fastest known integer factorisation algorithm. The algorithm begins with the selection of a pair of low-degree integer polynomials. The coefficient size of the chosen polynomials then plays a key role in determining the running time of the algorithm. Nonlinear polynomial selection algorithms approach the problem of constructing polynomials with small coefficients by employing a reduction to the well-studied problem of finding short vectors in lattices. The reduction rests upon the construction of modular geometric progressions with small terms. In this paper, tools are developed to aid in the analysis of nonlinear algorithms. Precise criteria for the selection of geometric progressions are given. Existing nonlinear algorithms are extended and analysed.

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