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http://dx.doi.org/10.7840/KICS.2011.36C.10.614

Two Cubic Polynomials Selection for the Number Field Sieve  

Jo, Gooc-Hwa (성균관대학교 수학과)
Koo, Nam-Hun (성균관대학교 수학과)
Kwon, Soon-Hak (성균관대학교 수학과)
Publication Information
The Journal of Korean Institute of Communications and Information Sciences / v.36, no.10C, 2011 , pp. 614-620 More about this Journal
Abstract
RSA, the most commonly used public-key cryptosystem, is based on the difficulty of factoring very large integers. The fastest known factoring algorithm is the Number Field Sieve(NFS). NFS first chooses two polynomials having common root modulo N and consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root, of which the most time consuming step is the Sieving step. However, in recent years, the importance of the Polynomial Selection step has been studied widely, because one can save a lot of time and memory in sieving and matrix step if one chooses optimal polynomial for NFS. One of the ideal ways of choosing sieving polynomial is to choose two polynomials with same degree. Montgomery proposed the method of selecting two (nonlinear) quadratic sieving polynomials. We proposed two cubic polynomials using 5-term geometric progression.
Keywords
NFS; LLL algorithm; RSA; Extended Euclidean Algorithm; Geometric Progression;
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1 T. Kleinjung, K. Aoki, J. Franke, A. Lenstra, E. Thome, J. Bos, P. Gaudry, A. Kruppa, P. Montgomery, D. Osvik, H. te Riele, A. Timofeev, P. Zimmermann, "Factorization of a 768-bit RSA modulus", Proceeding of Crypto 2010, LNCS 6223, pp.333-350, 2010
2 P. Montgomery, "Small geometric progressions modulo n". Unpublished note of 2 pages. December 1993, revised 1995 and 2005
3 P. Montgomery, Searching for Higher-Degree Polynomials for the General Number Field Sieve. PowerPoint Presentation. October, 2006
4 B. Murphy, "Polynomial Selection for the Number Field Sieve Integer Factorization Algorithm", Ph.D thesis, Australian National Universit, July 1999
5 T. Prest, P. Zimmermann, "Non-linear polynomial selection for the number field sieve", available at http://hal.archives-ouvertes.fr/docs/00/54/04/83/PDF/polyselect.pdf
6 J.P. Buhler, H.W. Lenstra, C. Pomerance, Factoring Integers with the Number Field Sieve. Reprinted in The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554. A.K. Lenstra, HW. Lenstra, Jr., Eds. (1993)
7 T. Kleinjung, "On polynomial selection for the general number field sieve". Mathematics of Computation 75 (2006), 2037-2047   DOI   ScienceOn
8 A. K. Lenstra, H. W. Lenstra, Jr, and L. Lov´asz, "Factoring polynomials with rational coefficients", Mathematische Ann., 261, 513-534, 1982