Abstract
We study different possibilities of implementing the Karatsuba multiplier for polynomials over \({\mathbb F}_{2}\) on FPGAs.
This is a core task for implementing finite fields of characteristic 2. Algorithmic and platform dependent optimizations yield efficient hardware designs. The resulting structure is hybrid in two different aspects. On the one hand, a combination of the classical and the Karatsuba methods decreases the number of bit operations. On the other hand, a mixture of sequential and combinational circuit design techniques includes pipelining and can be adapted flexibly to time-area constraints. The approach—both theory and implementation—can be viewed as a further step towards taming the machinery of fast algorithmics for hardware applications.
Keywords
Download to read the full chapter text
Chapter PDF
References
Bailey, D.V., Paar, C.: Optimal extension fields for fast arithmetic in public-key algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998)
Blahut, R.E.: Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading (1985)
Cantor, D.G.: On arithmetical algorithms over finite fields. Journal of Combinatorial Theory, Series A 50, 285–300 (1989)
von zur Gathen, J., Gerhard, J.: Arithmetic and factorization of polynomials over F2. In: Lakshman, Y.N. (ed.) Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation ISSAC 1996, Zürich, Switzerland. ACM Press, New York (1996); 1–9 Technical report tr-rsfb-96-018, University of Paderborn, Germany, 43 pages (1996). Final version in Mathematics of Computation
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003); First edition 1999
von zur Gathen, J., Nöcker, M.: Polynomial and normal bases for finite fields. Journal of Cryptology (2005) (to appear)
Grabbe, C., Bednara, M., Shokrollahi, J., Teich, J., von zur Gathen, J.: FPGA designs of parallel high performance GF(2233) multipliers. In: Proc. of the IEEE International Symposium on Circuits and Systems (ISCAS 2003), Bangkok, Thailand, vol. II, pp. 268–271 (2003)
Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2003)
Jung, M., Madlener, F., Ernst, M., Huss, S.: A Reconfigurable Coprocessor for Finite Field Multiplication in GF(2n). In: Workshop on Cryptographic Hardware and Embedded Systems, Hamburg. IEEE, Los Alamitos (2002)
Karatsuba, A., Ofman, Y.: Multiplication of multidigit numbers on automata. Soviet Physics–Doklady 7, 595–596 (1963); translated from Doklady Akademii Nauk SSSR 145(2), 293– 294 (July 1962)
Koç, Ç.K., Erdem, S.S.: Improved Karatsuba-Ofman Multiplication in GF(2m). US Patent Application (2002)
Paar, C.: Efficient VLSI Architectures for Bit-Parallel Computation in Galois Fields. PhD thesis, Institute for Experimental Mathematics, University of Essen, Essen, Germany (1994)
U.S. Department of Commerce / National Institute of Standards and Technology: Digital Signature Standard (DSS). Federal Information Processings Standards Publication 186-2 (2000)
Weimerskirch, A., Paar, C.: Generalizations of the karatsuba algorithm for efficient implementations. Technical report, Ruhr-Universität-Bochum, Germany (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
von zur Gathen, J., Shokrollahi, J. (2006). Efficient FPGA-Based Karatsuba Multipliers for Polynomials over \({\mathbb F}_{2}\) . In: Preneel, B., Tavares, S. (eds) Selected Areas in Cryptography. SAC 2005. Lecture Notes in Computer Science, vol 3897. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11693383_25
Download citation
DOI: https://doi.org/10.1007/11693383_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33108-7
Online ISBN: 978-3-540-33109-4
eBook Packages: Computer ScienceComputer Science (R0)