Algorithm 8.1 The graphic binary representation of a subgraph from a given graph

Input: A graph G(V, E), where V = (v1, v2, …, vl) and E = (e1, e2,…, em). Output: The BRsubgraph(v).

1. Determine (v1, v2, …, vk) and (e1, e2,…, em) in any subgraph H of G.

 2. i 0.

3. For j = 0: k, where k ≤ l.

4. If there is an edge between vs and vt, where s, t ∈ j


10. Return BRsubgraph = (em-1, …, e1, e0)2.

Figure 4. The subgraphs Hi, for i = 1, 2, 3, for a graph G.


Table 2.

The experimental results of the binary representations of scalars using subgraphs.


#### Table 3.

The experimental results for computing of the scalar multiplications based on using the binary representation of the subgraphs.

The small numerical results based on Figure 4 can be shown in Table 2.

On the binary representations which are found directly from the subgraphs, the scalar multiplications HiP on elliptic curve E defined over a prime field Fp can be computed using Algorithm (4.1) or (4.2). Some experimental results for computing the scalar multiplications based on using the subgraphs to represent the scalars are given in Table 3.

### 9. The signed digit representations

Suppose G is a digraph and Hi, for i = 1, 2, 3, are directed subgraphs as shown in Figure 5. Algorithm (8.2) can be used to find the signed digit representation of any subgraph from a given graph.

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

Figure 5. The directed subgraphs Hi, for i = 1, 2, 3, 4, for a digraph G.
