Faster Multiplication in GF(2)[x]

In this paper, we discuss an implementation of various algorithmsfor multiplying polynomials in GF(2)[x]: variants of the windowmethods, Karatsuba's, Toom-Cook's, Schonhage's and Cantor's algorithms.For most of them, we propose improvements that lead to practicalspeedups.

[1]  Joachim von zur Gathen,et al.  Arithmetic and factorization of polynomials over F_2 , 1996, ISSAC 1996.

[2]  Alfred Menezes,et al.  Guide to Elliptic Curve Cryptography , 2004, Springer Professional Computing.

[3]  David G. Cantor,et al.  On arithmetical algorithms over finite fields , 1989, Journal of combinatorial theory. Series A.

[4]  Joris van der Hoeven.  The truncated fourier transform and applications , 2004, ISSAC '04.

[5]  Susanne Wetzel,et al.  LiDIA—a library for computational number theory , 1997, SIGS.

[6]  S. Cook,et al.  ON THE MINIMUM COMPUTATION TIME OF FUNCTIONS , 1969 .

[7]  晋輝 趙,et al.  H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercauteren (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Math. Appl. (Boca Raton)., Chapman & Hall/CRC, 2006年，xxxiv + 808ページ． , 2009 .

[8]  Pierrick Gaudry,et al.  A gmp-based implementation of schönhage-strassen's large integer multiplication algorithm , 2007, ISSAC '07.

[9]  Arjen K. Lenstra,et al.  A Kilobit Special Number Field Sieve Factorization , 2007, ASIACRYPT.

[10]  Emmanuel Thomé,et al.  Subquadratic Computation of Vector Generating Polynomials and Improvement of the Block Wiedemann Algorithm , 2002, J. Symb. Comput..

[11]  Richard P. Brent,et al.  A Multi-level Blocking Distinct Degree Factorization Algorithm , 2007, ArXiv.

[12]  Kaoru Kurosawa,et al.  Advances in Cryptology - ASIACRYPT 2007, 13th International Conference on the Theory and Application of Cryptology and Information Security, Kuching, Malaysia, December 2-6, 2007, Proceedings , 2007, International Conference on the Theory and Application of Cryptology and Information Security.

[13]  Peter Roelse.  Factoring high-degree polynomials over F2 with Niederreiter's algorithm on the IBM SP2 , 1999, Math. Comput..

[14]  Arnold Schönhage,et al.  Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2 , 1977, Acta Informatica.

[15]  L. G. Lidia,et al.  A library for computational number theory , 1997 .

[16]  Information Security and Privacy , 1996, Lecture Notes in Computer Science.

[17]  André Weimerskirch,et al.  Generic GF(2) Arithmetic in Software and Its Application to ECC , 2003, ACISP.

[18]  Tanja Lange,et al.  Handbook of Elliptic and Hyperelliptic Curve Cryptography , 2005 .

[19]  Joachim von zur Gathen,et al.  Arithmetic and factorization of polynomial over F2 (extended abstract) , 1996, ISSAC '96.

[20]  Joachim von zur Gathen,et al.  Polynomial factorization over F2 , 2002, Math. Comput..

[21]  Peter L. Montgomery,et al.  Five, six, and seven-term Karatsuba-like formulae , 2005, IEEE Transactions on Computers.

[22]  Pierrick Gaudry,et al.  The mpFq library and implementing curve-based key exchanges , 2007 .

[23]  C. Small.  Arithmetic of Finite Fields , 1991 .

[24]  David G. Gantor.  On arithmetical algorithms over finite fields , 1989 .

[25]  Marco Bodrato,et al.  Towards Optimal Toom-Cook Multiplication for Univariate and Multivariate Polynomials in Characteristic 2 and 0 , 2007, WAIFI.