Efficient Septuple Formula for Elliptic Curve and Efficient Scalar Multiplication Using a Triple-Base Chain Representation

Because of shorter key and higher security, elliptic curve cryptosystem has attracted people’s attention and it is widely used in various fields, such as wireless sensor networks. Scalar multiplication is one of the most basic and critical links in the realization of elliptic curve public key systems, and its operation efficiency directly affects the implementation efficiency of the entire cryptographic system. To improve speed up the efficiency of the scalar multiplication algorithm, we express k for scalar multiplication kP by using triple-base chain representation of the scalar using {2,3,7} as basis of the triple-base chain in this paper. However, the efficiency of scalar multiplication is not only related to the length of representation but also the numbers and costs of doubling, tripling, septupling and addition. Therefore, we improve septuple formula of the elliptic curve by using Co_Z operation, which costs decreased by 8.3%. Due to the high redundancy of the triple-base chain representation, the algorithm can resist side channel attacks. The experimental results show that the proposed algorithm compared with that of other scalar multiplication algorithms, it requires less cost.