Abstract
Using a method based on Chinese Remainder Theorem for polynomial multiplication and suitable reductions, we obtain an efficient multiplication method for finite fields of characteristic 3. Large finite fields of characteristic 3 are important for pairing based cryptography [3]. For 5 ≤ ℓ ≤ 18, we show that our method gives canonical multiplication formulae over \(\mathbb{F}_{3^{\ell m}}\) for any m ≥ 1 with the best multiplicative complexity improving the bounds in [6]. We give explicit formula in the case \(\mathbb{F}_{3^{6 \cdot 97}}\).
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
Gorla, E., Puttmann, C., Shokrollahi, J.: Explicit formulas for efficient multiplication in
. In: Adams, C., Miri, A., Wiener, M. (eds.) Selected Areas in Cryptography (SAC 2007). LNCS, vol. 4876, pp. 173–183. Springer, Heidelberg (2007),
http://www.arxiv.org/PS_cache/arxiv/pdf/0708/0708.3014v1.pdf
Shokrollahi, J., Gorla, E., Puttmann, C.: Efficient FPGA-Based Multipliers for
and 
. In: Field Programmable Logic and Applications (FPL 2007),
http://www.arxiv.org/PS_cache/arxiv/pdf/0708/0708.3022v1.pdf
Kerins, T., Marnane, W.P., Popovici, E.M., Barreto, P.S.L.M.: Efficient hardware for the tate pairing calculation in characteristic three. In: Rao, J.R., Sunar, B. (eds.) CHES 2005. LNCS, vol. 3659, pp. 412–426. Springer, Heidelberg (2005)
Karatsuba, A., Ofman, Y.: Multiplication of multidigit numbers by automata. Soviet Physics-Doklady 7, 595–596 (1963)
Winograd, S.: Arithmetic Complexity of Computations. SIAM, Philadelphia (1980)
Montgomery, P.L.: Five, six, and seven-term Karatsuba-like formulae. IEEE Transactions on Computers 54(3), 362–369 (2005)
Fan, H., Anwar Hasan, M.: Comments on Five, Six, and Seven-Term Karatsuba-Like Formulae. IEEE Transactions on Computers 56(5), 716–717 (2007)
Cenk, M., Özbudak, F.: Improved Polynomial Multiplication Formulae over
Using Chinese Remainder Theorem. IEEE Transactions on Computers (submitted)Weimerskirch, A., Paar, C.: Generalizations of the Karatsuba Algorithm for Polynomial Multiplication. Technical Report, Ruhr-Universität Bochum, Germany (2003), http://www.crypto.ruhr-uni-bochum.de/imperia/md/content/texte/publications/tecreports/kaweb.pdf
Wagh, M.D., Morgera, S.D.: A new structured design method for convolutions over finite fields. Part I”, IEEE Transactions on Information Theory 29(4), 583–594 (1983)
. In: Adams, C., Miri, A., Wiener, M. (eds.) Selected Areas in Cryptography (SAC 2007). LNCS, vol. 4876, pp. 173–183. Springer, Heidelberg (2007), 
and 
. In: Field Programmable Logic and Applications (FPL 2007), 
Using Chinese Remainder Theorem. IEEE Transactions on Computers (submitted)Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cenk, M., Özbudak, F. (2008). Efficient Multiplication in \(\mathbb{F}_{3^{\ell m}}\), m ≥ 1 and 5 ≤ ℓ ≤ 18. In: Vaudenay, S. (eds) Progress in Cryptology – AFRICACRYPT 2008. AFRICACRYPT 2008. Lecture Notes in Computer Science, vol 5023. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68164-9_27
Download citation
DOI: https://doi.org/10.1007/978-3-540-68164-9_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68159-5
Online ISBN: 978-3-540-68164-9
eBook Packages: Computer ScienceComputer Science (R0)