6.1 The graphic right-to-left binary (GRLB) method

Suppose Ec is an elliptic curve defined over a prime field Fp [1–7]. The equation of Ec is given by

$$E\mathbf{c}: \boldsymbol{y}^2 = \mathbf{x}^3 + a\mathbf{x} + b \, (\text{mod } p). \tag{2}$$

Let P ¼ ðx; yÞ be a point that lies on Ec which has a (large) prime order r: Let G Vð ; EÞ be a simple (or multigraph or others) graph, where V is a vertex set and E is an edge set. The matrix representation A GðÞ on G Vð ; EÞ is defined as given in Eq. (1). Directly from the rows of the matrix A Gð Þ, the binary representation ˜ ° strings eðm�1<sup>Þ</sup> ; …;e<sup>1</sup><sup>l</sup> ;e<sup>0</sup><sup>l</sup> are obtained. The starting will happen with an elliptic <sup>2</sup> <sup>l</sup> ˛ ˝ point <sup>Q</sup><sup>1</sup> which belongs to E Fp , where <sup>Q</sup><sup>1</sup> <sup>¼</sup> <sup>∞</sup>: With the <sup>i</sup> index which takes the values 01, 11, …, mð � 1Þ <sup>1</sup> in the first row of A Gð Þ, the computation of Q<sup>1</sup> ¼ Q<sup>1</sup> þ P can be done if ei<sup>1</sup> ¼ 1: After then, the value 2P is computed and plugging it by P: The processing on the first row continues until the last value m � 1: Therefore, the last computed value of a point Q<sup>1</sup> is the value of the first scalar multiplication point v1P in l-tuple hi vP : In similar way, the processing on others rows can be done. The summary of the GRLB method can be given in the following algorithm:

## Algorithm 6.1 The GRLB method

˛ ˝ Input: <sup>A</sup> graph G V<sup>ð</sup> ; <sup>E</sup>Þ, P<sup>∈</sup> E Fp , l and <sup>m</sup>, where <sup>l</sup> and <sup>m</sup> are the order and size of a graph G, respectively.

Output: The m-tuple of the scalar multiplications hvPi ¼ hv1P; …; vlPi:


The Graphs for Elliptic Curve Cryptography DOI: http://dx.doi.org/10.5772/intechopen.83579

5.1 If eij ¼ 1 then Qj ¼ Qj þ P:

5.2 Else go to step (6).

5.3 End if

