14. Conclusions

The present chapter was concerned with presenting new graphic elliptic scalar multiplication algorithms for speeding up the computations of the scalar multiplication defined on elliptic curves over a prime field in different ways. These ways employed the undirected graphs and subgraphs to construct the binary representations of the scalars v in the scalar multiplications vP. Also, the sign digit representation of v has been obtained directly from using the digraphs or di-subgraphs. These representations are used to compute one scalar multiplication vP and l-tuple <vP> of the scalar multiplications. The computational complexities of the proposed graphic elliptic scalar multiplication algorithms have been determined. The computational complexity comparison of the proposed algorithms and original ones is discussed based on the elliptic curve and field operations. The experiment results of the computational complexities show that the proposed algorithms are less costly for computing the scalar multiplication or l-tuple of the scalar multiplications than original algorithms which are dependent on the computations of the binary representations or NAF expansions. The new propositions with graphic representations speed up the computations on elliptic scalar multiplication algorithms. Also, it gives the generalized cases with the computations of the l-tuples <vP> using (undirected or directed) graphs or subgraphs. This insight makes the working with graphic elliptic scalar multiplication algorithms more efficient in comparison with the serial original ones.

Applied Mathematics
