13. Computational complexity comparison on the serial and graphic computations of GBR and GNAF methods

This section discusses first the experimental results of the GBR method that uses serial computations to calculate l-tuple of the scalar multiplications and the GBR method that depends directly on using the graphs. Selecting the scalars v1, v2,…. vl from the interval [1. n � 1] to represent using the GBR method which needs the cost 0.5tld, where t is the length of the string binary representation, l is the length of the tuple and d is a normal addition operation. The final computational cost as given in Eq. (8).

Whereas, the binary representing of the scalars v1, v2, … vl can be taken directly from graphs or subgraphs without need to extra cost. This saves the 0.5tld operations to compute l-tuple of the scalar multiplications h i vP : The total cost of the graphic GBR method has been determined previously in Eq. (12). The serial GBR and graphic GBR computational costs for several experimental results are given in Table 12. In this table, one can see the serial GBR method with various values of p is more costly compared to the graphic GBR method.

Also, the experimental results of the serial GNAF and graphic GNAF methods that are used to calculate l-tuple of the scalar multiplications are discussed in this section. Selecting the scalars v1, v2, … vl from the interval [1. n � 1] to represent using the GNAF method which needs the 1lI þ 2lM þ 2lS cost, l is the length of the tuple, M is a field multiplication, S is a field squaring, and I is a field inversion. So, the total computational cost as given in Eq. (10).

The graphic GNAF of the scalars v1, v2, …. vl can be taken directly from graphs. So it can save 1lI þ 2lM þ 2lS operations for computing l-tuple of the scalar multiplications h i vP : The total cost of the graphic GNAF method is determined

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


Table 12.

The computational costs of the serial GBR and graphic GBR with different values of p.


Table 13.

The computational costs of the serial GNAF and graphic GNAF with different values of p.

previously in Eq. (14). Several experimental results on the serial GNAF and graphic GNAF computational costs are given in Table 13. With various values of p as shown in Table 13, it can observe that the graphic GNAF method is less costly than the serial GNAF method.
