ON NONLINEAR POLYNOMIAL SELECTION AND GEOMETRIC PROGRESSION (MOD N) FOR NUMBER FIELD SIEVE

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 53, Issue 1, 2016, pp.1-20
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2016.53.1.001

Title & Authors

ON NONLINEAR POLYNOMIAL SELECTION AND GEOMETRIC PROGRESSION (MOD N) FOR NUMBER FIELD SIEVE

Cho, Gook Hwa; Koo, Namhun; Kwon, Soonhak;

Cho, Gook Hwa; Koo, Namhun; Kwon, Soonhak;

Abstract

The general number field sieve (GNFS) is asymptotically the fastest known factoring algorithm. One of the most important steps of GNFS is to select a good polynomial pair. A standard way of polynomial selection (being used in factoring RSA challenge numbers) is to select a nonlinear polynomial for algebraic sieving and a linear polynomial for rational sieving. There is another method called a nonlinear method which selects two polynomials of the same degree greater than one. In this paper, we generalize Montgomery`s method [12] using geometric progression (GP) (mod N) to construct a pair of nonlinear polynomials. We also introduce GP of length d + k with and show that we can construct polynomials of degree d having common root (mod N), where the number of such polynomials and the size of the coefficients can be precisely determined.

Keywords

polynomial selection;number field sieve;geometric progression;LLL algorithm;

Language

English

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