Abstract
The problem of exponentiation over a finite field is to compute A e for a field element A and a positive integer e. This problem has many useful applications in cryptography and information security. In this paper, we present an efficient exponentiation algorithm in optimal extension field (OEF) GF(p m), which uses the fact that the Frobenius map, i.e., the p-th powering operation is very efficient in OEFs. Our analysis shows that the new algorithm is twice as fast as the conventional square-and-multiply exponentiation. One of the important applications of our new algorithm is random generation of a base point for elliptic curve cryptography, which is an attractive public-key mechanism for resource-constrained devices. We present a further optimized exponentiation algorithm for this application. Our experimental results show that the new technique accelerates the generation process by factors of 1.62–6.55 over various practical elliptic curves.
This work was supported by INHA UNIVERSITY Research Grant.
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
Gordon, D.M.: A survey of fast exponentiation methods. Journal of Algorithms 27, 129–146 (1998)
Agnew, G.B., Mullin, R.C., Vanstone, S.A.: Fast exponentiation in GF(2n). In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 251–255. Springer, Heidelberg (1988)
von zur Gathen, J.: Processor-efficient exponentiation in finite fields. Information Processing Letters 41, 81–86 (1992)
Lee, M.K., Kim, Y., Park, K., Cho, Y.: Efficient parallel exponentiation in GF(q n) using normal basis representations. Journal of Algorithms 54, 205–221 (2005)
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)
Bailey, D.V., Paar, C.: Efficient arithmetic in finite field extensions with application in elliptic curve cryptography. Journal of Cryptology 14, 153–176 (2001)
TTAS.KO-12.0015: Digital Signature Mechanism with Appendix– Part 3: Korean Certificate-based Digital Signature Algorithm using Elliptic Curves (2001)
Kobayashi, T.: Base-φ method for elliptic curves of OEF. IEICE Trans. Fundamentals E83-A, 679–686 (2000)
IEEE P1363-2000: IEEE Standard Specifications for Public-Key Cryptography (2000)
Barreto, P.S., Kim, H.Y., Lynn, B., Scott, M.: Efficient algorithms for pairing-based cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–369. Springer, Heidelberg (2002)
Feng, W., Nogami, Y., Morikawa, Y.: A fast square root computation using the Frobenius mapping. In: Qing, S., Gollmann, D., Zhou, J. (eds.) ICICS 2003. LNCS, vol. 2836, pp. 1–10. Springer, Heidelberg (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
Lee, MK., Kim, H., Hong, D., Chung, K. (2006). Efficient Exponentiation in GF(p m) Using the Frobenius Map. In: Gavrilova, M.L., et al. Computational Science and Its Applications - ICCSA 2006. ICCSA 2006. Lecture Notes in Computer Science, vol 3983. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11751632_64
Download citation
DOI: https://doi.org/10.1007/11751632_64
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34077-5
Online ISBN: 978-3-540-34078-2
eBook Packages: Computer ScienceComputer Science (R0)