6.2 The implementation results on the GRLB method

With different kinds of graphs which are given in Figure 2, the matrix representations of the graphs have been computed by A GðÞðÞ <sup>a</sup> ,A Gb ,A GðÞ <sup>c</sup> , and A GðÞ <sup>d</sup> , respectively.

� � � � � � � � � <sup>2</sup> <sup>3</sup> <sup>v</sup><sup>1</sup> 101 1001 � � <sup>2</sup> <sup>3</sup> � <sup>6</sup> <sup>7</sup> � <sup>v</sup><sup>1</sup> <sup>101001</sup> <sup>v</sup><sup>2</sup> <sup>6</sup> 1 100000 <sup>7</sup> � <sup>6</sup> <sup>7</sup> � <sup>v</sup><sup>2</sup> 1 10100 A Ga � <sup>v</sup><sup>3</sup> : 6 6 6 0010100 7 7 <sup>7</sup> ð Þ¼ � � � ð Þ¼ , A Gb :<sup>6</sup> <sup>7</sup> <sup>6</sup> <sup>7</sup> � <sup>v</sup><sup>3</sup> <sup>4</sup> <sup>0001</sup> <sup>1</sup> <sup>1</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> � � <sup>v</sup><sup>4</sup> <sup>4</sup> <sup>0001010</sup> <sup>5</sup> � � <sup>v</sup><sup>4</sup> <sup>011010</sup> v<sup>5</sup> 0001 1 1 1 <sup>2</sup> <sup>3</sup> � <sup>v</sup><sup>1</sup> 101 10000 <sup>2</sup> <sup>3</sup> � <sup>v</sup><sup>1</sup> 1 10000010 � <sup>6</sup> <sup>7</sup> <sup>v</sup><sup>2</sup> <sup>6</sup> 1 1 000000 <sup>7</sup> � <sup>v</sup><sup>2</sup> � <sup>6</sup> <sup>7</sup> 6 7 � <sup>6</sup> 001 1 001 0 0 <sup>7</sup> � � <sup>6</sup> <sup>7</sup> � <sup>v</sup><sup>3</sup> <sup>6</sup> <sup>001</sup> <sup>1</sup> <sup>1000</sup> <sup>7</sup> <sup>v</sup><sup>3</sup> <sup>6</sup><sup>00001</sup> <sup>1001</sup> <sup>7</sup> A Gð Þ¼ and <sup>A</sup> Gd : <sup>c</sup> � :<sup>6</sup> <sup>7</sup> ð Þ¼ � � <sup>6</sup> <sup>7</sup> <sup>6</sup> <sup>7</sup> <sup>6</sup> <sup>7</sup> � <sup>v</sup><sup>4</sup> <sup>6</sup> 01001 1 1 0 <sup>7</sup> � <sup>v</sup><sup>4</sup> <sup>6</sup> 0 0 1 0 1 0 0 1 0 7 � � <sup>6</sup> <sup>7</sup> � <sup>6</sup> <sup>4</sup> <sup>7</sup> <sup>5</sup> � <sup>v</sup><sup>5</sup> <sup>4</sup><sup>0100001</sup> 0 1 <sup>5</sup> <sup>v</sup><sup>5</sup> <sup>00000101</sup> � � � <sup>v</sup><sup>6</sup> <sup>100101000</sup> <sup>v</sup><sup>6</sup> 0000001 1

The l-tuple computations of the scalar multiplications that correspond to these graphs are shown in Table 1.

### 6.3 The graphic left-to-right binary method

� � � � With the same parameters p, E, P, G, and V which are used in the GRLB method, the computations of l-tuple hi vP using the GLRB method can be done easily. The scalars v1, …, vn can be written in the binary representation strings <sup>e</sup>ðm�1<sup>Þ</sup> ; …;e1<sup>j</sup>;e<sup>0</sup><sup>j</sup> , for <sup>j</sup> <sup>=</sup> 1, 2, …, <sup>l</sup>, directly from the matrix representation <sup>A</sup>(G) <sup>j</sup> <sup>2</sup> of G. Let us start with a point Q<sup>1</sup> in E Fp , where Q<sup>1</sup> ¼ ∞: With the i index which takes the values ðm � 1Þ1, …, 11, 01, then the computation of 2Q<sup>1</sup> can be done and plugged into Q1: After then, the value Q<sup>1</sup> ¼ Q<sup>1</sup> þ P is computed. The processing continues until the last value 01: Therefore, the last computed value of a point Q<sup>1</sup> is the first scalar multiplication point in an l-tuple hi vP : Similarly, the processing on others rows can be computed. The GLRB method can be summarized in Algorithm (6.2).

#### Figure 2. Different kinds of graphs [16].


#### Table 1.

The experimental results of the l-tuple of the scalar multiplications that correspond to the graphs Ga, Gb, Gc, and Gd.
