Abstract
We propose a scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems, which uses affine arithmetic and is resistant against simple power attacks. Also, using a modification of known techniques the algorithm can be made immune against differential power attacks. The algorithm uses Montgomery’s trick and a precomputed table consisting of multiples of the base point. Consequently, the algorithm is useful in a scenario where the base point is fixed, like Elgamal encryption or signature generation. Under such circumstances, for hyperelliptic curves, the algorithm compares favourably with other known algorithms over all fields. For elliptic curves, under similar circumstances, the algorithm performs better than other algorithms over prime fields. The increase in speed is due to a proper application of Montgomery’s trick to efficiently perform the simultaneous inversion of several field elements.
Chapter PDF
Similar content being viewed by others
Keywords
References
Avanzi, R.M.: Countermeasures Against Differential Power Analysis for Hyperelliptic Curve Cryptosystems. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 366–381. Springer, Heidelberg (2003) (to appear)
Brior, E., Joye, M.: Weierstrass Elliptic Curves and Side-Channel Attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 335–345. Springer, Heidelberg (2002)
Cantor, D.G.: Computing in the Jacobian of a Hyperelliptic curve. Mathematics of Computation 48, 95–101 (1987)
Coron, J.-S.: Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)
Fong, K., Hankerson, D., López, J., Menezes, A.: Field inversion and point halving revisited. Technical Report, CORR 2003-18, Department of Combinatorics and Optimization, University of Waterloo, Canada (2003)
Harley, R.: Fast Arithmetic on Genus 2 Curves (2000) Avaiable at, http://cristal.inria.fr/harley/hyper
Izu, T., Takagi, T.: A Fast Parallel Elliptic Curve Multiplication Resistant against Side-Channel Attacks Technical Report CORR 2002-03, University of Waterloo (2002), Available at, http://www.cacr.math.uwaterloo.ca
Izu, T., Moller, B., Takagi, T.: Improved Elliptic Curve Multiplication Methods Resistant Against Side Channel Attacks. In: Menezes, A., Sarkar, P. (eds.) INDOCRYPT 2002. LNCS, vol. 2551, pp. 296–313. Springer, Heidelberg (2002)
Joye, M., Tymen, C.: Protection against differential attacks for elliptic curve cryptography. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402–410. Springer, Heidelberg (2001)
Koblitz, N.: Hyperelliptic Cryptosystems. Journal of Cryptology 1, 139–150 (1989)
Koblitz, N.: Algebraic Aspects of Cryptology, Algorithms and Computation in Mathematics. Springer, Heidelberg (1998)
Lange, T.: Efficient Arithmetic on Genus 2 Curves over Finite Fields via Explicit Formulae. Cryptology ePrint Archive, Report 2002/121 (2002), http://eprint.iacr.org/
Lange, T.: Inversion-free Arithmetic on Genus 2 Hyperelliptic Curves. Cryptology ePrint Archive, Report 2002/147 (2002), http://eprint.iacr.org/
Menezes, J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)
Menezes, A., Wu, Y., Zuccherato, R.: An Elementary Introduction to Hyperelliptic Curve. Technical Report CORR 96-19. University of Waterloo, Canada (1996) Available at, http://www.cacr.math.uwaterloo.ca
Miyamoto, Y., Doi, H., Matsuo, K., Chao, J., Tsujii, S.: A fast addition algorithm for genus 2 hyperelliptic curves. In: Proc of SCIS 2002, IEICE, Japan, pp. 497–502 (2002) (in Japanese)
Montgomery, P.: Speeding the Pollard and Elliptic Curve Methods for Factorisation. Math. Comp. 48, 243–264 (1987)
Nagao, K.: Improving Group Law Algorithms for Jacobians of Hyperelliptic Curves. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 439–447. Springer, Heidelberg (2000)
Pelzl, J., Wollinger, T., Guajardo, J., Paar, C.: Hyperelliptic Curve Cryptosystems: Closing the Performance Gap to Elliptic Curves. Cryptology ePrint Archive, Report 2003/26 (2003), http://eprint.iacr.org/
Pelzl, J., Wollinger, T., Guajardo, J., Paar, C.: Low Cost Security: Explicit Formulae for Genus 4 Hyperelliptic Curves. Cryptology ePrint Archive, Report 2003/97 (2003), http://eprint.iacr.org/
Okeya, K., Sakurai, K.: Efficient Elliptic Curve Cryptosystems from a Scalar Multiplication Algorithm with Recovery of the y-coordinate on a Montgomery form Elliptic Curve. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 126–141. Springer, Heidelberg (2001)
Shacham, H., Boneh, D.: Improving SSL Handshake Performance via Batching. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 28–43. Springer, Heidelberg (2001)
Spallek, M.: Kurven vom Geschletch 2 und irhe Anwendung in Public-Key- Kryptosystemen. Ph D Thesis, Universitat Gesamthochschule, Essen (1994)
Takahashi, M.: Improving Harley Algorithms for Jacobians of Genus 2 Hyperelliptic Curves. In: Proc of SCIS 2002, ICICE, Japan (2002) (in Japanese)
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
Mishra, P.K., Sarkar, P. (2004). Application of Montgomery’s Trick to Scalar Multiplication for Elliptic and Hyperelliptic Curves Using a Fixed Base Point. In: Bao, F., Deng, R., Zhou, J. (eds) Public Key Cryptography – PKC 2004. PKC 2004. Lecture Notes in Computer Science, vol 2947. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24632-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-24632-9_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21018-4
Online ISBN: 978-3-540-24632-9
eBook Packages: Springer Book Archive