Abstract
In several cryptographic systems, a fixed element g of a group (generally \( \mathbb{Z}/q\mathbb{Z} \) ) is repeatedly raised to many different powers. In this paper we present a practical method of speeding up such systems, using precomputed values to reduce the number of multiplications needed. In practice this provides a substantial improvement over the level of performance that can be obtained using addition chains, and allows the computation of g n for n < N in O(log N/log log N) group multiplications. We also show how these methods can be parallelized, to compute powers in O(log log N) group multiplications with O(log N/log log N) processors.
This research was supported in part by the U.S. Department of Energy under contract number DE-AC04-76DP00789, in part by the Defense Advanced Research Projects Agency under Grant N00014-91-J-1698, and in part by an ONR-NDSEG fellowship.
PATENT CAUTION: This document may reveal patentable subject matter.
Chapter PDF
Similar content being viewed by others
Keywords
- Smart Card
- Modular Multiplication
- Binary Method
- Defense Advance Research Project Agency
- Modular Exponentiation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
A Proposed Federal Information Processing Standard for Digital Signature Standard, Federal Register, Volume 56, No. 169, August 31, 1991, pp. 42980–42982.
J. Bos and M. Coster, Addition Chain Heuristics, in Advances in Cryptology-Proceedings of Crypto’ 89, Lecture Notes in Computer Science, Volume 435, Springer-Verlag, New York, 1990, pp. 400–407.
W. Diffie and M. Hellman, New Directions in Cryptography, IEEE Transactions on Information Theory 22 (1976), 472–492.
E.F. Brickell and K.S. McCurley, An Interactive Identification Scheme Based on Discrete Logarithms and Factoring, to appear in Journal of Cryptology.
Ryo Fuji-Hara, Cipher Algorithms and Computational Complexity, Bit 17 (1985), 954–959 (in Japanese).
D.E. Knuth,The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Second Edition, Addison-Wesley, Massachusetts, 1981.
D.W. Matula, Basic digit sets for radix representation, Journal of the ACM, 29 (1982), pp. 1131–1143.
C.P. Schnorr, Efficient signature generation by smart cards, to appear in Journal of Cryptology.
D.R. Stinson, Some observations on parallel algorithms for fast exponentiation in GF(2n), Siam. J. Comput., 19, (1990), pp. 711–717.
J.S. Vitter and P. Flajolet, Average-case analysis of algorithms and data structures, in Handbook of Theoretical Computer Science, ed. J. van Leeuwen, Elsevier, Amsterdam, 1990, pp. 431–524.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brickell, E.F., Gordon, D.M., McCurley, K.S., Wilson, D.B. (1993). Fast Exponentiation with Precomputation. In: Rueppel, R.A. (eds) Advances in Cryptology — EUROCRYPT’ 92. EUROCRYPT 1992. Lecture Notes in Computer Science, vol 658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47555-9_18
Download citation
DOI: https://doi.org/10.1007/3-540-47555-9_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56413-3
Online ISBN: 978-3-540-47555-2
eBook Packages: Springer Book Archive