Abstract
Algebraic torus-based cryptosystems are public key cryptosystems based on the discrete logarithm problem, and have compact expressions compared with those of finite field-based cryptosystems. In this paper, we propose parameter selection criteria for the algebraic torus-based cryptosystems from the viewpoints of security and efficiency. The criteria include the following conditions: consistent resistance to attacks on algebraic tori and their embedding fields, and a large degree of freedom to select parameters suitable for each implementation. An extension degree and a characteristic size of a finite field on which the algebraic tori are defined are adjustable. We also provide examples of parameters satisfying the criteria.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Smith, P., Skinner, C.: A Public-key Cryptosystem and a Digital Signature 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)
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)
Rubin, K., Silverberg, A.: Torus-based Cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)
Barker, E., Barker, W., Burr, W., Polk, W., Smid, M.: Recommendation for Key Management - Part 1: Genaral (Revised). Special Publication 800/57, NIST (2007)
van Dijk, M., Granger, R., Page, D., Rubin, K., Silverberg, A., Stam, M., Woodruff, D.: Practical Cryptography in High Dimensional Tori. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 234–250. Springer, Heidelberg (2005)
Gower, J.E.: Prime Order Primitive Subgroups in Torus-based Cryptography. Cryptology ePrint Archive, Report 2006/466 (2006)
Granger, R., Vercauteren, F.: On the Discrete Logarithm Problem on Algebraic Tori. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 66–85. Springer, Heidelberg (2005)
Joux, A., Lercier, R.: The Function Field Sieve in the Medium Prime Case. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 254–270. Springer, Heidelberg (2006)
Freeman, D., Scott, M., Teske, E.: A Taxonomy of Pairing-Friendly Elliptic Curves. Journal of Cryptology 23(2), 224–280 (2010)
Miyaji, A., Nakabayashi, M., Takano, S.: New Explicit Conditions of Elliptic Curve Traces for FR-Reduction. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E84-A(5), 1234–1243 (2001)
Granger, R., Page, D., Stam, M.: A Comparison of CEILIDH and XTR. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 235–249. Springer, Heidelberg (2004)
Hitt, L.: On the Minimal Embedding Field. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 294–301. Springer, Heidelberg (2007)
Bosma, W., Hutton, J., Verheul, E.R.: Looking beyond XTR. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 321–332. Springer, Heidelberg (2002)
Galbraith, S.: Disguising Tori and Elliptic Curves. Cryptology ePrint Archive, Report 2006/248 (2006)
Rubin, K., Silverberg, A.: Compression in Finite Fields and Torus-based Cryptography. SIAM Jour. on Computing 37(5), 1401–1428 (2008)
Gordon, D.: Discrete Logarithms in GF (p) Using the Number Field Sieve. SIAM Jour. on Discrete Math. 6, 124–138 (1993)
Adleman, L.M.: The Function Field Sieve. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 108–121. Springer, Heidelberg (1994)
Joux, A., Lercier, R., Smart, N.P., Vercauteren, F.: The Number Field Sieve in the Medium Prime Case. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 326–344. Springer, Heidelberg (2006)
Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yonemura, T., Hanatani, Y., Isogai, T., Ohkuma, K., Muratani, H. (2010). Generating Parameters for Algebraic Torus-Based Cryptosystems. In: Heng, SH., Wright, R.N., Goi, BM. (eds) Cryptology and Network Security. CANS 2010. Lecture Notes in Computer Science, vol 6467. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17619-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-17619-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17618-0
Online ISBN: 978-3-642-17619-7
eBook Packages: Computer ScienceComputer Science (R0)