Abstract
We give a comparison of the performance of the recently proposed torus-based public key cryptosystem CEILIDH, and XTR. Underpinning both systems is the mathematics of the two dimensional algebraic torus \(T_{6}(\mathbb{F}_{p})\). However, while they both attain the same discrete logarithm security and each achieve a compression factor of three for all data transmissions, the arithmetic performed in each is fundamentally different. In its inception, the designers of CEILIDH were reluctant to claim it offers any particular advantages over XTR other than its exact compression and decompression technique. From both an algorithmic and arithmetic perspective, we develop an efficient version of CEILIDH and show that while it seems bound to be inherently slower than XTR, the difference in performance is much smaller than what one might infer from the original description. Also, thanks to CEILIDH’s simple group law, it provides a greater flexibility for applications, and may thus be considered a worthwhile alternative to XTR.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Adleman, L.M., Huang, M.A.: Function Field Sieve Method for Discrete Logarithms over Finite Fields. Info. and Comp. 151(1-2), 5–16 (1999)
Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography. Cambridge University Press, Cambridge (1999)
Bosselaers, A., Govaerts, R., Vandewalle, J.: Comparison of Three Modular Reduction Functions. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 175–186. Springer, Heidelberg (1994)
Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed Coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)
Gaudry, P., Hess, F., Smart, N.P.: Constructive And Destructive Facets Of Weil Descent On Elliptic Curves. J. of Cryptology 15(1), 19–46 (2002)
Gordon, D.M.: Discrete Logarithms in GF(p) Using the Number Field Sieve. SIAM J. of Disc. Math. 6, 124–138 (1993)
Itoh, T., Tsujii, S.: A Fast Algorithm for Computing Multiplicative Inverses in GF(2m) Using Normal Bases. Info. and Comp. 78(3), 171–177 (1988)
Joux, A., Lercier, R.: The Function Field Sieve is Quite Special. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 431–445. Springer, Heidelberg (2002)
Knuth, D.E.: The Art of Computer Programming, vol. 2. Addison-Wesley, Reading (1981)
Koblitz, N.: Elliptic Curve Cryptosystems. Math. Comp. 48, 203–209 (1987)
Koblitz, N.: Hyperelliptic Cryptosystems. J. of Cryptology 1, 139–150 (1989)
Lenstra, A.K.: Using Cyclotomic Polynomials to Construct Efficient Discrete Logarithm Cryptosystems over Finite Fields. In: Mu, Y., Pieprzyk, J.P., Varadharajan, V. (eds.) ACISP 1997. LNCS, vol. 1270, pp. 127–138. Springer, Heidelberg (1997)
Lenstra, A.K., Verheul, E.R.: The XTR Public Key System. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 1–19. Springer, Heidelberg (2000)
Lim, C.H., Lee, P.J.: More flexible exponentiation with precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)
Miller, V.: Uses of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1985)
Montgomery, P.L.: Modular Multiplication Without Trial Division. Math. Comp. 44, 519–521 (1985)
Proos, J.: Joint Sparse Forms and Generating Zero Columns when Combing. University of Waterloo, Technical Report CORR 2003-23 (2003)
Rubin, K., Silverberg, A.: Supersingular abelian varieties in Cryptology. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 336–353. Springer, Heidelberg (2002)
Rubin, K., Silverberg, A.: Torus-Based Cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)
Shoup, V.: Lower Bounds for Discrete Logarithms and Related Problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997)
Solinas, J.A.: Low-Weight Binary Representations for Pairs of Integers. University of Waterloo, Technical Report CORR 2001-41 (2001)
Stam, M., Lenstra, A.: Efficient Subgroup Exponentiation in Quadratic and Sixth Degree Extensions. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 318–332. Springer, Heidelberg (2003)
Stam, M., Lenstra, A.: Speeding Up XTR. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 125–143. Springer, Heidelberg (2001)
Straus, E.G.: Problems and Solutions (5125) Addition Chains of Vectors. American Mathematical Monthly 71, 806–808 (1964)
Smith, P., Skinner, C.: A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 357–364. Springer, Heidelberg (1995)
Teske, E.: Speeding up Pollard’s Rho Method for Computing discrete Logarithms. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 541–554. Springer, Heidelberg (1998)
Thériault, N.: Index Calculus Attack for Hyperelliptic Curves of Small Genus. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 75–92. Springer, Heidelberg (2003)
Voskresenskiĭ, V.E.: Algebraic Groups and Their Birational Invariants. In: Translations of Mathematical Monographs, 179, American Mathematical Society, Providence (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Granger, R., Page, D., Stam, M. (2004). A Comparison of CEILIDH and XTR. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-24847-7_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22156-2
Online ISBN: 978-3-540-24847-7
eBook Packages: Springer Book Archive