Abstract
The joint sparse form (JSF) is a representation of a pair of integers, which is famous for accelerating a multi-scalar multiplication in elliptic curve cryptosystems. Solinas’ original paper showed three unsolved problems on the enhancement of JSF. Whereas two of them have been solved, the other still remains to be done. The remaining unsolved problem is as follows: To design a representation of a pair of integers using a larger digit set such as a set involving ±3, while the original JSF utilizes the digit set that consists of 0, ±1 for representing a pair of integers. This paper puts an end to the problem; width-3 JSF. The proposed enhancement satisfies properties that are similar to that of the original. For example, the enhanced representation is defined as a representation that satisfies some rules. Some other properties are the existence, the uniqueness of such a representation, and the optimality of the Hamming weight. The non-zero density of the width-3 JSF is 563/1574( = 0.3577) and this is ideal. The conversion algorithm to the enhanced representation takes O(logn) memory and O(n) computational cost, which is very efficient, where n stands for the bit length of the integers.
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References
Avanzi, R.: On multi-exponentiation in cryptography, Cryptology ePrint Archive: Report 2002/154 (2002)
Booth, A.: A signed binary multiplication technique. Quarterly Journal of Mechanics and Applied Mathematics 4(2), 236–240 (1951)
Dahmen, E., Okeya, K., Takagi, T.: An Advanced Method for Joint Scalar Multiplications on Memory Constraint Devices. In: Molva, R., Tsudik, G., Westhoff, D. (eds.) ESAS 2005. LNCS, vol. 3813, pp. 189–204. Springer, Heidelberg (2005)
Gallant, R.P., Lambert, J.L., Vanstone, S.A.: Faster Point Multiplication on Elliptic Curves with Efficient Endomorphism. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001)
Heuberger, C., Katti, R., Prodinger, H., Ruan, X.: The Alternating Greedy Expansion and Applications to Left-To-Right Algorithms in Cryptography. Theoret. Comput. Sci. 341, 55–72 (2005)
Koblitz, N.: Elliptic Curve Cryptosystems. Math. Comp. 48, 203–209 (1987)
Kuang, B., Zhu, Y., Zhang, Y.: An Improved Algorithm for uP+vQ using JSF 3. In: Jakobsson, M., Yung, M., Zhou, J. (eds.) ACNS 2004. LNCS, vol. 3089, pp. 467–478. Springer, Heidelberg (2004)
Lim, C., Lee, P.: More flexible exponentiation with precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)
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)
Morain, F., Olivos, J.: Speeding up the computations on an elliptic curve using addition-subtraction chains. Inform. Theor. Appl. 24, 531–543 (1990)
Shoup, V.: NTL: A Library for doing Number Theory (version 5.5.2), http://www.shoup.net/ntl/
Proos, J.: Joint Sparse Forms and Generating Zero Columns when Combing, Technical Report of the Centre for Applied Cryptographic Research, University of Waterloo - CACR, CORR 2003-23 (2003), http://www.cacr.math.uwaterloo.ca
Reitwiesner, G.W.: Binary arithmetic. Advances in Computers 1, 231–308 (1960)
Solinas, J.A.: Efficient Arithmetic on Koblitz Curves. Designs, Codes and Cryptography 19, 195–249 (2000)
Solinas, J.A.: Low-weight binary representations for pairs of integers, Technical Report of the Centre for Applied Cryptographic Research, University of Waterloo - CACR, CORR 2001-41 (2001), http://www.cacr.math.uwaterloo.ca
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Okeya, K., Kato, H., Nogami, Y. (2010). Width-3 Joint Sparse Form. In: Kwak, J., Deng, R.H., Won, Y., Wang, G. (eds) Information Security, Practice and Experience. ISPEC 2010. Lecture Notes in Computer Science, vol 6047. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12827-1_6
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DOI: https://doi.org/10.1007/978-3-642-12827-1_6
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