Abstract
For an integer w ≥ 2, a radix 2 representation is called a width-w nonadjacent form (w-NAF, for short) if each nonzero digit is an odd integer with absolute value less than 2w − − 1, and, of any w consecutive digits, at most one is nonzero. In elliptic curve cryptography, the w-NAF window method is used to efficiently compute nP where n is an integer and P is an elliptic curve point. We introduce a new family of radix 2 representations which use the same digits as the w-NAF but have the advantage that they result in a window method which uses less memory. This memory savings results from the fact that these new representations can be deduced using a very simple left-to-right algorithm. Further, we show that like the w-NAF, these new representations have a minimal number of nonzero digits.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
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)
Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography. Cambridge University Press, Cambridge (1999)
Cohen, H.: Analysis of the Flexible Window Powering Algorithm. To appear in Journal of Cryptology, Available from http://www.math.u-bordeaux.fr/~cohen/window.dvi
Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed Coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)
Gordon, D.M.: A Survey of Fast Exponentiation Methods. Journal of Algorithms 27, 129–146 (1998)
Grabner, P., Heuberger, C., Prodinger, H., Thuswaldner, J.: Analysis of Linear Combination Algorithms in Cryptography (Submitted) Available from http://www.opt.math.tu-graz.ac.at/~cheub/publications/
Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)
Heuberger, C., Katti, R., Prodinger, H., Ruan, X.: The Alternating Greedy Expansion and Applications to Left-To-Right Algorithms in Cryptography, Submitted Available from http://www.opt.math.tu-graz.ac.at/~cheub/publications/
Joye, M., Yen, S.: Optimal Left-to-Right Binary Signed-Digit Recoding. IEEE Transactions on Computers 49, 740–748 (2000)
Muir, J.A., Stinson, D.R.: New Minimal Weight Representations for Left-to- Right Window Methods (Extended Version) Available from http://www.math.uwaterloo.ca/~jamuir/papers.htm
Muir, J.A., Stinson, D.R.: Minimality and Other Properties of the Width-w Nonadjacent Form. To appear in Mathematics of Computation, Available from http://www.math.uwaterloo.ca/~jamuir/papers.htm
Müller, V.: Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two. Journal of Cryptology 11, 219–234 (1998)
Okeya, K., Schmidt-Samoa, K., Spahn, C., Takagi, T.: Signed Binary Representations Revisited. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 123–139. Springer, Heidelberg (2004)
Solinas, J.A.: Efficient arithmetic on Koblitz curves. Designs, Codes and Cryptography 19, 195–249 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Muir, J.A., Stinson, D.R. (2005). New Minimal Weight Representations for Left-to-Right Window Methods. In: Menezes, A. (eds) Topics in Cryptology – CT-RSA 2005. CT-RSA 2005. Lecture Notes in Computer Science, vol 3376. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30574-3_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-30574-3_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24399-1
Online ISBN: 978-3-540-30574-3
eBook Packages: Computer ScienceComputer Science (R0)