Abstract

The scalar multiplication on elliptic curves defined over finite fields is a core operation in elliptic curve cryptography (ECC). Several different methods are used for computing this operation. One of them, the binary method, is applied depending on the binary representation of the scalar v in a scalar multiplication vP, where P is a point that lies on elliptic curve E defined over a prime field Fp. On the binary method, two methodologies are performed based on the implementation of the binary string bits from the right to the left (RLB) [or from the left to the right (LRB)]. Another method is a nonadjacent form (NAF) which depended on the signed digit representation of a positive integer v. In this chapter, the graphs and subgraphs are employed for the serial computations of elliptic scalar multiplications defined over prime fields. This work proposed using the subgraphs H of the graphs G or the (simple, undirected, directed, connected, bipartite, and other) graphs to represent a scalar v directly. This usage speeds up the computations on the elliptic scalar multiplication algorithms. The computational complexities of the proposed algorithms and previous ones are determined. The comparison results of the computational complexities on all these algorithms are discussed. The experimental results show that the proposed algorithms which are used the sub-graphs H and graphs G need to the less costs for computing vP in compare to previous algorithms which are employed the binary representations or NAF expansion. Thus, the proposed algorithms that use the subgraphs or the graphs to represent the scalars v are more efficient than the original ones.

Keywords: ECC, scalar multiplication, BRL, BLR, NAF, graphs, subgraphs, computational complexity
