Two new fast methods for simultaneous scalar multiplication in elliptic curve cryptosystems

This paper defines a signed factorial expansion of an integer, and proposes two new methods for simultaneous scalar multiplication based on the expansion when multiple bases are fixed in advance. Where the First method can be parallelized easily, and the Second method only requires a half of elliptic points stored of the First method, relatively. In addition, it greatly improves up the implementation speed of simultaneous scalar multiplication when using the vector key in the methods. The theoretical analyses and the implementation results show that our methods are faster than the current fast methods.