Coordinate system for elliptic curve cryptosystem on twisted Edwards curve

Abstract

This paper proposes a coordinate system in which a point (x, y) is represented as (X, Y, Z, W), where x = X/Z and y = Y/W are satisfied. In the proposed coordinate system, the cost of scalar multiplication, which is the dominant operation of elliptic curve cryptosystems, on the twisted Edwards curve ±x² + y² = 1 + dx²y² is the same as the existing lowest cost in a mixed coordinate system, and implementation of scalar multiplication becomes easier.