Abstract
This paper studies τ-adic expansions of scalars, which are important in the design of scalar multiplication algorithms on Koblitz Curves, and are less understood than their binary counterparts.
At Crypto ’97 Solinas introduced the width-w τ-adic non-adjacent form for use with Koblitz curves. It is an expansion of integers \(z=\sum_{i=0}^\ell z_i\tau^i\), where τ is a quadratic integer depending on the curve, such that \(z_i\ne 0\) implies z w + i − 1 = ... = z i + 1= 0, like the sliding window binary recodings of integers. We show that the digit sets described by Solinas, formed by elements of minimal norm in their residue classes, are uniquely determined. However, unlike for binary representations, syntactic constraints do not necessarily imply minimality of weight.
Digit sets that permit recoding of all inputs are characterized, thus extending the line of research begun by Muir and Stinson at SAC 2003 to Koblitz Curves.
Two new useful digit sets are introduced: one set makes precomputations easier, the second set is suitable for low-memory applications, generalising an approach started by Avanzi, Ciet, and Sica at PKC 2004 and continued by several authors since. Results by Solinas, and by Blake, Murty, and Xu are generalized.
Termination, optimality, and cryptographic applications are considered. We show how to perform a “windowed” scalar multiplication on Koblitz curves without doing precomputations first, thus reducing memory storage dependent on the base point to just one point.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Avanzi, R.M.: A Note on the Signed Sliding Window Integer Recoding and its Left-to-Right Analogue. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 130–143. Springer, Heidelberg (2004)
Avanzi, R.M.: Delaying and Merging Operations in Scalar Multiplication: Applications to Curve-Based Cryptosystems. In: Bihamel, E. (ed.) SAC 2006. LNCS, vol. 4356, pp. 203–219. Springer, Heidelberg (to appear)
Avanzi, R.M., Ciet, M., Sica, F.: Faster Scalar Multiplication on Koblitz Curves combining Point Halving with the Frobenius Endomorphism. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 28–40. Springer, Heidelberg (2004)
Avanzi, R.M., Dimitrov, V., Doche, C., Sica, F.: Extending Scalar Multiplication using Double Bases. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 130–144. Springer, Heidelberg (2006)
Avanzi, R.M., Heuberger, C., Prodinger, H.: Minimality of the Hamming Weight of the τ-NAF for Koblitz Curves and Improved Combination with Point Halving. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 332–344. Springer, Heidelberg (2006)
Avanzi, R.M., Heuberger, C., Prodinger, H.: Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and its Combination with Point Halving: Extensions and Mathematical Analysis. Algorithmica 46, 249–270 (2006)
Avanzi, R.M., Sica, F.: Scalar Multiplication on Koblitz Curves Using Double Bases. Cryptology ePrint Archive, Report 2006/067
Blake, I.F., Murty, V.K., Xu, G.: A note on window τ-NAF algorithm. Information Processing Letters 95, 496–502 (2005)
Cohen, H., Frey, G. (eds.): The Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, Boca Raton (2005)
Coron, J.-S., M’Raïhi, D., Tymen, C.: Fast generation of pairs (k,[k]p) for Koblitz elliptic curves. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 151–164. Springer, Heidelberg (2001)
Heuberger, C., Prodinger, H.: Analysis of Alternative Digit Sets for Nonadjacent Representations. Monatshefte für Mathematik, pp. 219–248 (2006)
Kátai, I., Kovács, B.: Canonical number systems in imaginary quadratic fields. Acta Math. Hungar. 37, 159–164 (1981)
Kátai, I., Szabó, J.: Canonical Number Systems for Complex Integers. Acta Scientiarum Mathematicarum 1975, 255–260
Knudsen, E.W.: Elliptic Scalar Multiplication Using Point Halving. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 135–149. Springer, Heidelberg (1999)
Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987)
Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992)
Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)
Muir, J.A., Stinson, D.R.: Alternative digit sets for nonadjacent representations. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 306–319. Springer, Heidelberg (2004)
Muir, J.A., Stinson, D.R.: Minimality and other properties of the width-w nonadjacent form. Math. Comp. 75, 369–384 (2006)
Okeya, K., Takagi, T., Vuillaume, C.: Short Memory Scalar Multiplication on Koblitz Curves. In: Rao, J.R., Sunar, B. (eds.) CHES 2005. LNCS, vol. 3659, pp. 91–105. Springer, Heidelberg (2005)
Park, D.J., Sim, S.G., Lee, P.J.: Fast scalar multiplication method using change-of-basis matrix to prevent power analysis attacks on Koblitz curves. In: Chae, K.-J., Yung, M. (eds.) Information Security Applications. LNCS, vol. 2908, pp. 474–488. Springer, Heidelberg (2004)
Schroeppel, R.: Elliptic curve point ambiguity resolution apparatus and method. International Application Number PCT/US00/31014 (filed 9 November, 2000)
Solinas, J.A.: An improved algorithm for arithmetic on a family of elliptic curves. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 357–371. Springer, Heidelberg (1997)
Solinas, J.A.: Efficient Arithmetic on Koblitz Curves. Designs, Codes and Cryptography 19(2/3), 125–179 (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Avanzi, R.M., Heuberger, C., Prodinger, H. (2007). On Redundant τ-Adic Expansions and Non-adjacent Digit Sets. In: Biham, E., Youssef, A.M. (eds) Selected Areas in Cryptography. SAC 2006. Lecture Notes in Computer Science, vol 4356. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74462-7_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-74462-7_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74461-0
Online ISBN: 978-3-540-74462-7
eBook Packages: Computer ScienceComputer Science (R0)