Abstract
Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security [6]. XTR is a new method to represent elements of a subgroup of a multiplicative group of a finite field GF(p 6) and it can be generalized to the field GF(p 6m) [6],[9]. This paper proposes optimal extension fields for XTR among Galois fields GF(p 6m) which can be applied to XTR. In order to select such fields, we introduce a new notion of Generalized Optimal Extension Fields(GOEFs) and suggest a condition of prime p, a defining polynomial of GF(p 2m) and a fast method of multiplication in GF(p 2m) to achieve fast finite field arithmetic in GF(p 2m). From our implementation results, GF(p 36) → GF(p 12) is the most efficient extension fields for XTR and computing Tr(g n) given Tr(g) in GF(p 12) is on average more than twice faster than that of the XTR system [6],[10] on Pentium III/700MHz which has 32-bit architecture.
This work was supported by both Ministry of Information and Communication and Korea Information Security Agency, Korea, under project 2002-130
Chapter PDF
Similar content being viewed by others
References
Aho, A., Hopcroft, J., Ullman, J., The Design and Analysis of Computer Algorithms., Addison-Wesley, Reading Mass,1974.
Bach, E, Shallit, J., Algorithmic Number Theory., Vol 1, The MIT Press, Mass, 1996.
Bailey. D.V. and Paar C, Optimal extension fields for fast arithmetic in public-key algorithms., Crypto’ 98, Springer-Verlag pp.472–485, 1998.
H. Cohen, A.K. Lenstra, Implementation of a new primality test., Math.Comp.48 (1987) 103–121.
D.E. Knuth, The art of computer programming., Volume 2, Seminumerical Algorithms, second edition, Addison-Wesley, 1981.
A.K. Lenstra, E.R. Verheul, The XTR public key system., Proceedings of Crypto 2000, LNCS 1880,Springer-Verlag, 2000,1–19; available from http://www.ecstr.com.
A.K. Lenstra, Using Cyclotomic Polynomials to Construct Efficient Discrete Logarithm Cryptosystems over Finite Fields., Proceedings of ACISP 1997, LNCS 1270,Springer-Verlag, 1997,127–138.
A.K. Lenstra, Lip 1.1, available at http://www.ecstr.com.
Seongan Lim, Seungjoo Kim, Ikkwon Yie, Jaemoon Kim, Hongsub Lee, XTR Extended to GF(p 6m). Procee dings of SAC 2001,317–328, LNCS 2259, Springer-Verlag, 2001,125-143.
Martijn Stam, A.K. Lenstra, Speeding Up XTR. Proceedings of Asiacrypt 2001, LNCS 2248, Springer-Verlag, 2001,125–143; available from http://www.ecstr.com.
A.J Menezes, Applications of Finite Fields., Waterloo, 1993.
S.B. Mohan and B.S. Adiga, Fast Algorithms for Implementating RSA Public Key Cryptosystem., Electronics Letters, 21917):761,1985.
S. Oh, S. Hong, D. Cheon, C. Kim, J. Lim and M. Sung, An Extension Field of Characteristic Greater than Two and its Applicatins. Technical Report 99-2, CIST,1999. Available from http://cist.korea.ac.kr/.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Han, DG., Yoon, K.S., Park, YH., Kim, C.H., Lim, J. (2003). Optimal Extension Fields for XTR. In: Nyberg, K., Heys, H. (eds) Selected Areas in Cryptography. SAC 2002. Lecture Notes in Computer Science, vol 2595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36492-7_24
Download citation
DOI: https://doi.org/10.1007/3-540-36492-7_24
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00622-0
Online ISBN: 978-3-540-36492-4
eBook Packages: Springer Book Archive