Abstract
In pairing-based cryptosystems, radix-r signed-digit representations are used to speed up point multiplication over supersingular elliptic curves or hyper-elliptic curves in characteristic r. We propose a left-to-right radix-r signed-digit recoding algorithm, which can obtain a new signed-digit representation from left to right. It is proved that its average non-zero density is asymptotically \(\frac{1}{2}-\frac{2r+3}{2r(r+1)^2}\), which is reduced by 20%-50% compared with the previous left-to-right radix-r signed-digit representations. The proposed algorithm can be applied to efficient implementations of pairing-based cryptosystems over supersingular elliptic curves or hyper-elliptic curves.
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References
Joux, A.: A one-round protocol for tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)
Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)
Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001)
Miller, V.S.: Short programs for functions on curves. Unpublished manuscript (1986), Available at: http://crypto.stanford.edu/miller/miller.pdf
Barreto, P., et al.: Efficient Algorithms for Pairing-Based Cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–368. Springer, Heidelberg (2002)
Galbraith, S., Harrison, K., Soldera, D.: Implementing the Tate pairing. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)
Smart, N., Westwood, J.: Point Multiplication on Ordinary Elliptic Curves over Fields of Characteristic Three. Applicable Algebra in Engineering, Communication and Computing 13(6), 485–497 (2003)
Duursma, I.M., Lee, H.-S.: Tate Pairing Implementation for Hyperelliptic Curves y 2 = x p − x + d. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 111–123. Springer, Heidelberg (2003)
Booth, A.D.: A Signed Binary Multiplication Technique. Q. J. Mech. Appl. Math. 4(2), 236–240 (1951)
Reitwiesner, G.W.: Binary arithmetic. Advances in Computers 1, 231–308 (1960)
Clark, W., Liang, J.: On Arithmetic Weight for a General Radix Representation of Integers. IEEE Transaction on IT 19, 823–826 (1973)
Arno, S., Wheeler, F.S.: Signed digit representations of minimal hamming weight. IEEE Transactions on Computers 42(8), 1007–1010 (1993)
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)
Müller, V.: Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two. Journal of Cryptology 11, 219–234 (1998)
Joye, M., Yen, S.-M.: Optimal left-to-right binary signed-digit recoding. IEEE Trans. on Comp. 49(7), 740–748 (2000)
Avanzi, R.M.: A Note on the Signed Sliding Window Integer Recoding and a Left-to-Right Analogue. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 130–143. Springer, Heidelberg (2004)
Takagi, T., et al.: Signed Binary Representations Revisited. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 123–139. Springer, Heidelberg (2004)
Muir, J.A., Stinson, D.R.: New Minimal Weight Representations for Left-to-Right Window Methods. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 366–383. Springer, Heidelberg (2005)
Joye, M., Yen, S.-M.: New Minimal Modified Radix-r Representation with Applications to Smart Cards. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 375–384. Springer, Heidelberg (2002)
Muir, J.A.: A Simple Left-to-Right Algorithm for Minimal Weight Signed Radix-r Representations. IEEE Transactions on Information Theory 53(3), 1234–1241 (2007)
Kong, F., et al.: Left-to-right Generalized Non-adjacent Form Recoding for Elliptic Curve Cryptosystems. In: The First International Conference on Hybrid Information Technology, pp. 299–303. IEEE Computer Society Press, Los Alamitos (2006)
Takagi, T., Yen, S.-M., Wu, B.-C.: Radix-r Non-Adjacent Form. In: Zhang, K., Zheng, Y. (eds.) ISC 2004. LNCS, vol. 3225, pp. 99–110. Springer, Heidelberg (2004)
Kong, F., Li, D.: A Note on Signed Binary Window Algorithm for Elliptic Curve Cryptosystems. In: Desmedt, Y.G., et al. (eds.) CANS 2005. LNCS, vol. 3810, pp. 223–235. Springer, Heidelberg (2005)
Gordon, D.M.: A Survey of Fast Exponentiation Methods. Journal of Algorithms 27, 129–146 (1998)
Phillips, B., Burgess, N.: Minimal Weight Digit Set Conversions. IEEE Transactions on Computers 53(6), 666–677 (2004)
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Kong, F., Yu, J., Cai, Z., Li, D. (2007). New Left-to-Right Radix-r Signed-Digit Recoding Algorithm for Pairing-Based Cryptosystems. In: Cai, JY., Cooper, S.B., Zhu, H. (eds) Theory and Applications of Models of Computation. TAMC 2007. Lecture Notes in Computer Science, vol 4484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72504-6_17
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DOI: https://doi.org/10.1007/978-3-540-72504-6_17
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