Abstract
We introduce the concept of torus-based cryptography, give a new public key system called CEILIDH, and compare it to other discrete log based systems including Lucas-based systems and XTR. Like those systems, we obtain small key sizes. While Lucas-based systems and XTR are essentially restricted to exponentiation, we are able to perform multiplication as well. We also disprove the open conjectures from [2], and give a new algebro-geometric interpretation of the approach in that paper and of LUC and XTR.
Chapter PDF
Similar content being viewed by others
References
Bleichenbacher, D., Bosma, W., Lenstra, A.K.: Some remarks on Lucas-based cryptosystems. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 386–396. Springer, Heidelberg (1995)
Bosma, W., Hutton, J., Verheul, E.R.: Looking beyond XTR. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 46–63. Springer, Heidelberg (2002)
Brouwer, A.E., Pellikaan, R., Verheul, E.R.: Doing more with fewer bits. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 321–332. Springer, Heidelberg (1999)
de Bruijn, N.G.: On the factorization of cyclic groups. Nederl. Akad. Wetensch. Proc. Ser. A 56 (= Indagationes Math. 15), 370–377 (1953)
Gong, G., Harn, L.: Public-key cryptosystems based on cubic finite field extensions. IEEE Trans. Inform. Theory 45, 2601–2605 (1999)
Klyachko, A.A.: On the rationality of tori with cyclic splitting field. In: Arithmetic and geometry of varieties, pp. 73–78. Kuybyshev Univ. Press, Kuybyshev (1988) (Russian)
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)
Lenstra, A.K., Verheul, E.R.: An overview of the XTR public key system. In: Publickey cryptography and computational number theory (Warsaw, 2000), pp. 151–180. de Gruyter, Berlin (2001)
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. CRC Press, Boca Raton (1997)
Müller, W.B., Nöbauer, W.: Some remarks on public-key cryptosystems. Studia Sci. Math. Hungar 16, 71–76 (1981)
Ono, T.: Arithmetic of algebraic tori. Ann. of Math. 74, 101–139 (1961)
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.: Algebraic tori in cryptography. In: To appear in High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Fields Institute Communications Series. American Mathematical Society, Providence
Schoenberg, I.J.: A note on the cyclotomic polynomial. Mathematika 11, 131–136 (1964)
Smith, P.J., Lennon, M.J.J.: LUC: A New Public Key System. In: Proceedings of the IFIP TC11 Ninth International Conference on Information Security IFIP/Sec 1993, pp. 103–117. North-Holland, Amsterdam (1993)
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)
Voskresenskii, V.E.: Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179. American Mathematical Society, Providence (1998)
Voskresenskii, V.E.: Stably rational algebraic tori, Les XXèmes Journées Arithmétiques (Limoges, 1997). J. Théor. Nombres Bordeaux 11, 263–268 (1999)
Weil, A.: Adeles and algebraic groups. Progress in Math. 23, Birkhäuser, Boston (1982)
Williams, H.C.: A p + 1 method of factoring. Math. Comp. 39, 225–234 (1982)
Williams, H.C.: Some public-key crypto-functions as intractable as factorization. Cryptologia 9, 223–237 (1985)
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
Rubin, K., Silverberg, A. (2003). Torus-Based Cryptography. In: Boneh, D. (eds) Advances in Cryptology - CRYPTO 2003. CRYPTO 2003. Lecture Notes in Computer Science, vol 2729. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45146-4_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-45146-4_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40674-7
Online ISBN: 978-3-540-45146-4
eBook Packages: Springer Book Archive