11. The computational complexity for serial computing l-tuple of the scalar multiplications

### 11.1 The computational complexity of the serial GBR

On l-tuple of the scalar multiplications h i vP ¼ hv1P; v2P; …; vlPi, the computations of v1P, v2P, …, vlP without using the graphs or subgraphs can be done serially. So, the computational cost of these computations using the binary representations of v1, v2, …, vl is given by

$$\frac{t}{2}lA + tlD + 0.5tld.\tag{7}$$

In other words, the running time can be expressed in terms of field operations by

$$\text{3tlS} + \text{3tlM} + \text{1.5tlI} + \text{0.5tld.} \tag{8}$$

Table 8 displays some small experimental results for computational complexities for serial computations of l-tuples h i vP using the generalized binary method.

#### 11.2 The computational complexity of the serial GNAF

The computational complexity for computing l-tuple of the scalar multiplications using GNAF representations in serial way is given by

$$lD + \frac{t}{3}lA + tlD = \frac{t}{3}lA + (t+1)lD. \tag{9}$$

Using the field operations, the formula in Eq. (9) can be rewritten by.

$$(\mathbf{(t/3)} + \mathbf{t} + \mathbf{1})l\mathbf{I} + ((2/3)\mathbf{t} + 2\,\mathbf{t} + 2)l\mathbf{M} + ((\mathbf{t/3}) + 2\,\mathbf{t} + 2)l\mathbf{S}.\tag{10}$$

The computational complexity results for serial computations of l-tuples h i vP using the GNAF method are given in Table 9.


#### Table 8.

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

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


Table 9.

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