Abstract
Since Miller and Koblitz applied elliptic curves to cryptographic system in 1985 [3],[6], a lot of researchers have been interested in this field and various speedup techniques for the scalar multiplication have been developed. Recently, Gallant et al. published a method that accelerates the scalar multiplication and is applicable to a larger class of curves [4]. In the process of their method, they assumed the existence of a special pair of two short linearly independent vectors. Once a pair of such vectors exists, their decomposition method improves the efficiency of the scalar multiplication roughly about 50%. In this paper, we state and prove a necessary condition for the existence of a pair of desired vectors and we also present an algorithm to find them.
This work was supported by R&D project 2002-s-073 of KISA
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References
D. Bailey and C. Paar: ‘Optimal extention fields for fast arithmetic in public-key algorithms’, Advances in Cryptology-Crypto’98, Lecture Notes in Computer Science, Vol 1462, 1998, pp.472–485.
H. Cohen, A. Miyaji, and T. Ono: ‘Efficient Elliptic Curve Exponentiation using Mixed Coordinates’, Advances in Cryptology-Asiacrypt’98, Lecture Notes in Computer Science, Vol 1514, 1998, pp.51–65.
V. Miller: ‘Use of Elliptic Curves in Cryptography’, Advances in Cryptology-Crypto’85, Lecture Notes in Computer Science, Vol 263, 1986, pp.417–426.
R. Gallant, R. Lambert, and L. Vanstone: ‘Faster Point Multiplication on Elliptic Curves with Efficient Endomorphism’, Advances in Cryptology-Crypto’2001, Lecture Notes in Computer Science, Vol 2139, 2001, pp.190–201.
N. Koblitz: ‘CM-curves with Good Cryptographic Properties’, Advances in Cryptology-Crypto’91, 1992, 48, pp.279–287.
N. Koblitz: ‘Elliptic Curve Cryptosystems’, Mathematics of Computation, 1987, 48, pp.203–209.
C. Lim and P. Lee: ‘More Flexible Exponentiation with Precomputation’, Advances in Cryptology-Crypto’94, Lecture Notes in Computer Science, Vol 839, 1994, pp.95–107.
J. Solinas: ‘An Improved Algorithm for Arithmetic on a Family of Elliptic Curves’, Advances in Cryptology-Crypto’97, Lecture Notes in Computer Science, Vol 1294, 1997, pp.357–371.
J. Solinas: ‘Efficient Arithmetic on Koblitz Curves’, Design, Codes and Crytography, 2000, 19, pp.195–249.
V. Müller: ‘Fast Multiplication on Elliptic Curves over small fields of charactersitic two’, J. of Cryptology, 1998, 11, pp.219–234.
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Kim, D., Lim, S. (2003). Integer Decomposition for Fast Scalar Multiplication on Elliptic Curves. In: Nyberg, K., Heys, H. (eds) Selected Areas in Cryptography. SAC 2002. Lecture Notes in Computer Science, vol 2595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36492-7_2
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DOI: https://doi.org/10.1007/3-540-36492-7_2
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