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2014•Unpublished venueRequires access

Elliptic Curve Scalar Multiplication with a Bijective Transform

Yoshitaka Nagai, Masaaki Shirase, Tetsuya Izu

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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.

About this research paper

What this paper is about

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.

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Available 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.

Key concepts: Scalar multiplication, Elliptic curve point multiplication, Hessian form of an elliptic curve, Scalar (mathematics), Edwards curve, Jacobian matrix and determinant, Mathematics, Tripling-oriented Doche–Icart–Kohel curve

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