Versions on 3-ISD Method for Twisted Edwards Scalar Multiplication

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Jolan Lazim Theyab, Ruma Kareem K. Ajeena

Abstract

Twisted Edwards curve is a generalized of Edwards curves. These generalized curves are employed as an important tool to increase the security of encryption schemes. This work presents a new contribution of the 3-deminsion integer sub-decomposition (3-ISD) method to compute a scalar multiplication kP on the twisted Edwards curve defined over prime fields Fp that uses the efficiently computable endomorphisms of .The 3-ISD method depends on the randomization of generating the 3-ISD generators. The elements of these generators are vectors, their components are chosen from the range [1, p-1], where p is a prime number. In each vector, the elements are relatively prime to each other. Using the 3-ISD generators, a scalar t in [1, n-1] can be decomposed into t1, t2 and t3 with maxwhere n is a prime order of a point P that lies on .These scalars, namely t1, t2 and t3, are sub-decomposed again into sub-scalars t11, t12 , t13, t21 , t22, t23 and  t31, t32 , t33The scalar multiplication tP using the 3-ISD method is computed by


where and are six efficiently computable endomorphisms of Edwards curve defined over Fp. . On the 3-ISD method, fast computations are determined based on the randomized generating of 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 tP, the 3-ISD method considers as more secure communications using the twisted Edwards curve cryptography.

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