Versions on 3-Dimension Integer Sub-Decomposition Method Using the Edwards Curves

Abstract

cIn this work, another version of the integer sub-decomposition (ISD) method is presented to compute a scalar multiplication  on Edwards curves Eds which are defined over prime fields Fp. This version depended on applying a 3-dimension of the ISD generators. The elements of these generators are chosen randomly from the range [1, p-1], where p is a prime number. In each vector, the elements are relatively prime to each other. Using these generators, a scalar k can be decomposed into k1 , k2 and k3 with max  These scalars are sub-decomposed again into sub-scalars  and t31, t32, t33. The scalar multiplication tP using the 3-ISD method is computed using the sub-scalars and the efficiently computable endomorphisms of Edwards curve Ed defined over Fp. On the 3-ISD method, fast computations are determined based on the randomization choices of the elements that form the 3-ISD generators in comparison with the previous version that is depended on the 2-ISD generators. In comparison with the 2-ISD computation method to compute  the 3-ISD method considers as more secure communications using the Edwards curve cryptography.