Fast simultaneous scalar multiplication on elliptic curve with montgomery form

Abstract

We propose a new methodto compute x-coordinate of kP + lQ simultaneously on the elliptic curve with Montgomery form over IFp without precomputedp oints. To compute x-coordinate of kP +lQ is requiredin ECDSA signature verification. The proposedmetho dis about 25% faster than the methodusing scalar multiplication andthe recovery of Y -coordinate of kP and lQ on the elliptic curve with Montgomery form over IFp, andalso slightly faster than the simultaneous scalar multiplication on the elliptic curve with Weierstrass form over IFp using NAF andmixedco ordinates. Furthermore, our methodis applicable to Montgomery methodon elliptic curves over IF2n.