12. The computational complexity of the graphic elliptic scalar multiplication methods

Suppose h i vP ¼ hv1P; v2P; …; vlPi is an l-tuple of the scalar multiplications. The graphic computations of v1P, v2P, …, vlP can be done using the graphs or subgraphs in two ways. One of them is using the graphs directly to find the binary representations of the scalars v1, v2, …, vl, whereas another one uses the digraphs to represent these scalars. The computational costs of these computations can be discussed as follows.

#### 12.1 The computational complexity of the graphic binary representation (GBR)

Using the graphs to compute l-tuple of the scalar multiplications costs

$$\frac{t}{2}lA + tlD.\tag{11}$$

In terms of field operations, the computational complexity of GBR can be expressed by

$$\text{3tlS} + \text{3tlM} + \text{1.5tlI.} \tag{12}$$

Table 10 displays some small experimental results for computational complexities for the graphic representations of l-tuples h i vP using the generalized binary method.


#### Table 10.

The experimental results for computational complexities for graphic computations of l-tuples h i vP using the generalized binary method.


Table 11.

The experimental results for computational complexities for graphic computations of l-tuples of h i vP using the GNAF method.

### 12.2 The computational complexity of the digraphic NAF

The computational complexity for computing l-tuple of the scalar multiplications using the digraphs is given by

$$\frac{t}{3}lA + tlD.\tag{13}$$

Eq. (13) can be rewritten using field operations by:

$$\frac{1}{2}(\mathbf{(t/3)} + \mathbf{t})\mathbf{l} + ((2/3)\mathbf{t} + 2\,\mathbf{t})l\mathbf{M} + ((\mathbf{t/3}) + 2\,\mathbf{t})l\mathbf{S}.\tag{14}$$

Several experimental results for computational complexities for digraph representations of l-tuples h i vP are given in Table 11.
