# Fast Simultaneous Scalar Multiplication on Elliptic Curve with Montgomery Form

Conference paper

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## Abstract

We propose a new method to compute x-coordinate of *kP + lQ* simultaneously on the elliptic curve with Montgomery form over IFp without precomputed points. To compute *x*-coordinate of *kP +lQ* is required in ECDSA signature verification. The proposed method is about 25% faster than the method using scalar multiplication and the recovery of *Y*-coordinate of *kP* and *lQ* on the elliptic curve with Montgomery form over \(
\mathbb{F}_p
\) and also slightly faster than the simultaneous scalar multiplication on the elliptic curve with Weierstrass form over \(
\mathbb{F}_p
\) using NAF and mixed coordinates. Furthermore, our methodis applicable to Montgomery method on elliptic curves over \(
\mathbb{F}_{2^n }
\).

## Keywords

Computational Cost Elliptic Curve Signature Scheme Elliptic Curf Scalar Multiplication
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© Springer-Verlag Berlin Heidelberg 2001