# Integer Decomposition for Fast Scalar Multiplication on Elliptic Curves

## 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.

## Keywords

elliptic curve cryptosystem scalar multiplication integer decomposition endomorphism## References

- 1.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.zbMATHGoogle Scholar - 2.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.CrossRefGoogle Scholar - 3.V. Miller: ‘
*Use of Elliptic Curves in Cryptography*’, Advances in Cryptology-Crypto’85, Lecture Notes in Computer Science, Vol 263, 1986, pp.417–426.Google Scholar - 4.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.Google Scholar - 5.N. Koblitz: ‘
*CM-curves with Good Cryptographic Properties*’, Advances in Cryptology-Crypto’91, 1992, 48, pp.279–287.MathSciNetzbMATHGoogle Scholar - 6.N. Koblitz: ‘
*Elliptic Curve Cryptosystems*’, Mathematics of Computation, 1987, 48, pp.203–209.MathSciNetCrossRefGoogle Scholar - 7.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.Google Scholar - 8.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.CrossRefGoogle Scholar - 9.J. Solinas: ‘
*Efficient Arithmetic on Koblitz Curves*’, Design, Codes and Crytography, 2000, 19, pp.195–249.MathSciNetCrossRefGoogle Scholar - 10.V. Müller: ‘
*Fast Multiplication on Elliptic Curves over small fields of charactersitic two*’, J. of Cryptology, 1998, 11, pp.219–234.MathSciNetCrossRefGoogle Scholar