1. Introduction

The scalar multiplication on elliptic curves defined over finite fields is considered as a central and most time-consuming operation in elliptic curve cryptography (ECC) [1–7]. Different methods are used for computing the scalar multiplication such as the binary method, nonadjacent form, and others [8–15]. 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)], whereas the nonadjacent form (NAF) depends on the signed digit representation of a positive integer v [1].

In this chapter, the computation of the scalar multiplication vP on elliptic curve E defined over a prime field Fp has been done using the (undirected or directed) graph and (undirected or directed) subgraph. These graph and subgraph are used to represent the scalar v in two ways. The first one is the binary representation and the second one is the sign digit representation.

Also, the l-tuple of the elliptic scalar multiplications is computed using the proposed generalized binary methods (GRLB) and (GLRB) and GNAF. 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. Several experimental results showed that the proposed algorithms which are used the graphs G need to the less costs for computing vP in compare to previous algorithms which are employed the binary representations or NAF expansion. Therefore, the proposed algorithms that use the subgraphs or the graphs to represent the scalars v are more efficient than the original ones.

This chapter is organized as follows: Section 2 presents the vector representation of the graph. Section 3 discusses the matrix representation of the graph. Section 4 includes the binary methods of the elliptic scalar multiplication which are the rightto-left binary and left-to-right binary representations. Section 5 explains the non-adjacent form method, whereas Section 6 discusses the graphic binary methods of the elliptic scalar multiplications. Section 7 displays the digraphic NAF method. Section 8 presents the subgraphs for computing the elliptic scalar multiplication. Section 9 determines the computational complexities on the original elliptic scalar multiplication methods. Section 10 shows the computational complexity for serial computing l-tuple of the scalar multiplications. The computational complexity of the graphic elliptic scalar multiplication methods is explained in Section 11. Section 12 illustrates the computational complexity comparison on the serial and graphic computation methods. Finally, Section 13 draws the conclusions.
