This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Arithmetic Operations in Finite Fields of Medium Prime Characteristic Using the Lagrange Representation
September 2006 (vol. 55 no. 9)
pp. 1167-1177
In this paper, we propose a complete set of algorithms for the arithmetic operations in finite fields of prime medium characteristic. The elements of the fields {\hbox{\rlap{I}\kern 2.0 pt{\hbox{F}}}}_{p^k} are represented using the newly defined Lagrange representation, where polynomials are expressed using their values at sufficiently many points. Our multiplication algorithm, which uses a Montgomery approach, can be implemented in O(k) multiplications and O(k^2 \log k) additions in the base field {\hbox{\rlap{I}\kern 2.0 pt{\hbox{F}}}}_p. For the inversion, we propose a variant of the extended Euclidean GCD algorithm, where the inputs are given in the Lagrange representation. The Lagrange representation scheme and the arithmetic algorithms presented in the present work represent an interesting alternative for elliptic curve cryptography.

[1] N. Koblitz, “Elliptic Curve Cryptosystems,” Math. Computation, vol. 48, no. 177, pp. 203-209, Jan. 1987.
[2] V.S. Miller, “Uses of Elliptic Curves in Cryptography,” Advances in Cryptology, Proc. CRYPTO '85, H.C. Williams, ed., pp. 417-428, 1986.
[3] A. Menezes, P.C. Van Oorschot, and S.A. Vanstone, Handbook of Applied Cryptography. CRC Press, 1997.
[4] IEEE, IEEE 1363-2000 Standard Specifications for Public-Key Cryptography, 2000.
[5] Nat'l Inst. of Standards and Technology, FIPS PUB 186-2: Digital Signature Standard (DSS), Jan. 2000.
[6] D. Bailey and C. Paar, “Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms,” Advances in Cryptography, Proc. CRYPTO '98, H. Krawczyk, ed., pp. 472-485, 1998.
[7] D. Bailey and C. Paar, “Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography,” J. Cryptology, vol. 14, no. 3, pp. 153-176, 2001.
[8] D. Hankerson, A. Menezes, and S. Vanstone, Guide to Elliptic Curve Cryptography. Springer-Verlag, 2004.
[9] N.P. Smart and E.J. Westwood, “Point Muliplication on Ordinary Elliptic Curves over Fields of Characteristic Three,” Applicable Algebra in Eng., Comm., and Computing, vol. 13, no. 6, pp. 485-497, Apr. 2003.
[10] J.A. Solinas, “Improved Algorithms for Arithmetic on Anomalous Binary Curves,” Research Report CORR-99-46, Center for Applied Cryptographic Research, Univ. of Waterloo, Canada, updated version of the paper appearing in the Proc. CRYPTO '97, 1999.
[11] N. Koblitz, “CM Curves with Good Cryptographic Properties,” Advances in Cryptography, Proc. CRYPTO '91, pp. 279-287, 1992.
[12] D.H. Lehmer, “Euclid's Algorithm for Large Numbers,” Am. Math. Monthly, vol. 45, no. 4, pp. 227-233, 1938.
[13] J. Von Zur Gathen and J. Gerhard, Modern Computer Algebra. Cambridge Univ. Press, 1999.
[14] P.L. Montgomery, “Modular Multiplication without Trial Division,” Math. Computation, vol. 44, no. 170, pp. 519-521, Apr. 1985.
[15] Ç.K. Koç and T. Acar, “Montgomery Multiplication in ${\rm GF}(2^k)$ ,” Designs, Codes, and Cryptography, vol. 14, no. 1, pp. 57-69, Apr. 1998.
[16] V. Lefèvre, “Multiplication by an Integer Constant,” Research Report 4192, INRIA, May 2001.
[17] N. Boullis and A. Tisserand, “Some Optimizations of Hardware Multiplication by Constant Matrices,” IEEE Trans. Computers, vol. 54, no. 10, pp. 1271-1282, Oct. 2005.
[18] J.-C. Bajard, L. Imbert, C. Nègre, and T. Plantard, “Multiplication in ${\rm GF}(p^k)$ for Elliptic Curve Cryptography,” Proc. 16th IEEE Symp. Computer Arithmetic, pp. 181-187, 2003.
[19] D.E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, third ed. Reading, Mass.: Addison-Wesley, 1997.
[20] R. Crandall and C. Pomerance, Prime Numbers. A Computational Perspective. Springer-Verlag, 2001.
[21] J. Sorenson, “Two Fast GCD Algorithms,” J. Algorithms, vol. 16, no. 1, pp. 110-144, 1994.
[22] J. Sorenson, “An Analysis of Lehmer's Euclidean Algorithm,” Proc. 1995 Int'l Symp. Symbolic and Algebraic Computation (ISSAC '95), pp. 254-258, 1995.
[23] J.M. Pollard, “Monte Carlo Methods for Index Computation mod $p$ ,” Math. Computation, vol. 32, no. 143, pp. 918-924, July 1978.
[24] P. Gaudry, “Index Calculus for Abelian Varieties and the Elliptic Curve Discrete Logarithm Problem,” preprint, Oct. 2004.
[25] R.M. Avanzi, H. Cohen, C. Doche, G. Frey, T. Lange, K. Nguyen, and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, 2005.

Index Terms:
Finite field arithmetic, optimal extension fields, Newton interpolation, Euclidean algorithm, elliptic curve cryptography.
Citation:
Jean-Claude Bajard, Laurent Imbert, Christophe N?gre, "Arithmetic Operations in Finite Fields of Medium Prime Characteristic Using the Lagrange Representation," IEEE Transactions on Computers, vol. 55, no. 9, pp. 1167-1177, Sept. 2006, doi:10.1109/TC.2006.136
Usage of this product signifies your acceptance of the Terms of Use.