Abstract
The efficiency of cryptographic protocols rely on the speed of the underlying arithmetic and finite field computation. In the literature, several methods on how to improve the multiplication over extensions fields \(\mathbb{F}_{q^{m}}\), for prime q were developped. These optimisations are often related to the Karatsuba and Toom Cook methods. However, the speeding-up is only interesting when m is a product of powers of 2 and 3. In general cases, a fast multiplication over \(\mathbb{F}_{q^{m}}\) is implemented through the use of the naive school-book method. In this paper, we propose a new efficient multiplication over \(\mathbb{F}_{q^{m}}\) for any power m. The multiplication relies on the notion of Adapted Modular Number System (AMNS), introduced in 2004 by [3]. We improve the construction of an AMNS basis and we provide a fast implementation of the multiplication over \(\mathbb{F}_{q^{m}}\), which is faster than GMP and NTL.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
NIST Key Length Recommendations, http://www.keylength.com/
Recommendations for Key Management. Special Publication 800-57 Part 1 (2007)
Bajard, J.-C., Imbert, L., Plantard, T.: Modular Number Systems: Beyond the Mersenne Family. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 159–169. Springer, Heidelberg (2004)
Bodrato, M.: Towards Optimal Toom-Cook Multiplication for Univariate and Multivariate Polynomials in Characteristic 2 and 0. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 116–133. Springer, Heidelberg (2007)
Boneh, D., Franklin, M.: Identity-Based Encryption from the Weil Pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)
Boneh, D., Lynn, B., Shacham, H.: Short Signatures from the Weil Pairing. J. Cryptology 17(4), 297–319 (2004)
Brier, E., Joye, M.: Fast Point Multiplication on Elliptic Curves Through Isogenies. In: Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 43–50. Springer, Heidelberg (2003)
Chung, J., Hasan, A.: More Generalized Mersenne Numbers (Extended Abstract). In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 335–347. Springer, Heidelberg (2004)
Devegili, A.J., Ó hÉigeartaigh, C., Scott, M., Dahab, R.: Multiplication and squaring on pairing-friendly fields. Cryptology ePrint Archive, Report 2006/471 (2006), http://eprint.iacr.org/
El Mrabet, N., Negre, C.: Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography. In: Boyd, C., González Nieto, J. (eds.) ACISP 2009. LNCS, vol. 5594, pp. 422–436. Springer, Heidelberg (2009)
Gama, N., Nguyen, P.Q.: Predicting Lattice Reduction. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008)
Gama, N., Nguyen, P.Q., Regev, O.: Lattice Enumeration Using Extreme Pruning. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 257–278. Springer, Heidelberg (2010)
Gentry, C., Halevi, S.: Implementing Gentry’s Fully-Homomorphic Encryption Scheme. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 129–148. Springer, Heidelberg (2011)
Katti, R., Brennan, J.: Low Complexity Multiplication in a Finite Field Using Ring Representation. IEEE Transactions on Computers 52(4), 418–427 (2003)
Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Ann. 261, 513–534 (1982)
Miller, V.S.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)
Minkowski, H.: Geometrie der Zahlen. Druck und Verlag von B.G. Teubner, Leipzig und Berlin (1910)
Montgomery, P.L.: Five, six, and seven-term Karatsuba-like formulae. IEEE Transactions on Computers 54(3), 362–369 (2005)
El Mrabet, N., Guillevic, A., Ionica, S.: Efficient Multiplication in Finite Field Extensions of Degree 5. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 188–205. Springer, Heidelberg (2011)
Negre, C., Plantard, T.: Efficient Modular Arithmetic in Adapted Modular Number System Using Lagrange Representation. In: Mu, Y., Susilo, W., Seberry, J. (eds.) ACISP 2008. LNCS, vol. 5107, pp. 463–477. Springer, Heidelberg (2008)
Rubin, K., Silverberg, A.: Torus-Based Cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)
Schnorr, C.-P.: A hierarchy of polynomial lattice basis reduction algorithms. Theoretical Computer Science 53, 201–224 (1987)
Schnorr, C.-P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Programming 66, 181–199 (1994)
Shoup, V.: Number Theory Library (1996), http://www.shoup.net/ntl
Silverman, J.H.: Rings of low multiplicative complexity. In: Finite Fields and Their Applications, vol. 6, pp. 175–191. Academic Press (2000)
van Dijk, M., Granger, R., Page, D., Rubin, K., Silverberg, A., Stam, M., Woodruff, D.: Practical Cryptography in High Dimensional Tori. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 234–250. Springer, Heidelberg (2005)
Von ZurGathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, New York (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
El Mrabet, N., Gama, N. (2012). Efficient Multiplication over Extension Fields. In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-31662-3_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31661-6
Online ISBN: 978-3-642-31662-3
eBook Packages: Computer ScienceComputer Science (R0)