Elliptic Curve Scalar Multiplication with a Bijective Transform

Abstract

It is important to speed up scalar multiplication in elliptic curve cryptosystems and then various speeding-up techniques for that have been proposed. This paper proposes a method for computing a scalar multiplication, in which, first, we map a point we would like to compute a scalar multiplication into another point on another curve so that it has a special coordinate, second, we compute a scalar multiplication on the another curve, finally, we map the computed point on original curve. In fact, when we use the proposed method, the cost of scalar multiplication is reduced by about from 2 to 5% in projective, Jacobian, and modified Jacobian coordinate systems.