Abstract
We describe improvements to the performance of a key agreement protocol based in the infrastructure of a real quadratic field through investigating fast methods for exponentiating ideals. We present adaptations of non-adjacent form and signed base-3 exponentiation and compare these to the binary method. To adapt these methods, we introduce new algorithms for squaring, cubing, and dividing w-near (f,p) representations of ideals in the infrastructure. Numerical results from an implementation of the key agreement protocol using our new algorithms and all three exponentiation methods are presented, demonstrating that non-adjacent form exponentiation improves the speed of key establishment for most of the currently recommended security levels.
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References
Barker, E., Barker, W., Polk, W., Smid, M.: Recommendation for key management - part 1: General (revised). NIST Special Publication 800-57, National Institute of Standards and Technology (NIST) (March 2007), http://csrc.nist.gov/groups/ST/toolkit/documents/SP800-57Part1_3-8-07.pdf
Biasse, J.-F., Jacobson Jr., M.J., Silvester, A.K.: Security Estimates for Quadratic Field Based Cryptosystems. In: Steinfeld, R., Hawkes, P. (eds.) ACISP 2010. LNCS, vol. 6168, pp. 233–247. Springer, Heidelberg (2010), http://dl.acm.org/citation.cfm?id=1926211.1926229
Buchmann, J., Williams, H.C.: A key-exchange system based on imaginary quadratic fields. Journal of Cryptology 1, 107–118 (1988)
Buchmann, J., Williams, H.C.: A Key Exchange System Based on Real Quadratic Fields. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 335–343. Springer, Heidelberg (1990), http://dl.acm.org/citation.cfm?id=646754.705067
Ciet, M., Joye, M., Lauter, K., Montgomery, P.: Trading inversions for multiplications in elliptic curve cryptography. Designs, Codes and Cryptography 39, 189–206 (2006)
Diffie, W., Hellman, M.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)
Dixon, V.: Fast Exponentiation in the Infrastructure of a Real Quadratic Field. Master’s thesis, University of Calgary, Calgary, Alberta (2011)
Free Software Foundation: The GNU Multiple Precision Arithmetic Library (2011), http://gmplib.org
Guillou, L.C., Quisquater, J.-J.: A Practical Zero-Knowledge Protocol Fitted to Security Microprocessor Minimizing Both Transmission and Memory. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 123–128. Springer, Heidelberg (1988)
Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography, pp. 98–99. Springer Science and Buisness Media, LLC (2004)
Imbert, L., Jacobson Jr., M.J., Schmidt, A.: Fast ideal cubing in imaginary quadratic number and function fields. Advances in Mathematics of Communications 4(2), 237–260 (2010)
Jacobson Jr., M.J., Scheidler, R., Stein, A.: Cryptographic aspects of real hyperelliptic curves. Tatra Mountains Mathematical Publications 45, 1–35 (2010)
Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: The efficiency and security of a real quadratic field based key exchange protocol. In: Public Key Cryptography and Computational Number Theory (Warsaw 2000), pp. 89–112. Walter de Gruyter, Berlin (2001)
Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: An improved real quadratic field based key exchange procedure. Journal of Cryptology 19, 211–239 (2006)
Jacobson Jr., M.J., Williams, H.C.: Solving the Pell Equation. CMS Books in Mathematics. Springer (2009) iSBN 978-0-387-84922-5
Jebelean, T.: A double-digit Lehmer-Euclid algorithm for finding the GCD of long integers. Journal of Symbolic Computation 19, 145–157 (1995)
Lehmer, D.H.: Euclid’s algorithm for large numbers. The American Mathematical Monthly 45(4), 227–233 (1938)
Shanks, D.: The infrastructure of real quadratic fields and its applications. In: Proc. 1972 Number Theory Conf., Boulder, Colorado, pp. 217–224 (1972)
Shanks, D.: On Gauss and composition I, II. In: Proceedings NATO ASI on Number Theory and Applications, pp. 163–204. Kluwer, Dordrecht (1989)
Silvester, A.: Doctoral Dissertation, University of Calgary (in progress, 2012)
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Dixon, V., Jacobson, M.J., Scheidler, R. (2012). Improved Exponentiation and Key Agreement in the Infrastructure of a Real Quadratic Field. In: Hevia, A., Neven, G. (eds) Progress in Cryptology – LATINCRYPT 2012. LATINCRYPT 2012. Lecture Notes in Computer Science, vol 7533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33481-8_12
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DOI: https://doi.org/10.1007/978-3-642-33481-8_12
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