Paper 2009/312

Jacobi Quartic Curves Revisited

Huseyin Hisil, Kenneth Koon-Ho Wong, Gary Carter, and Ed Dawson

Abstract

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, $y^2 = d x^4 + 2 a x^2 + 1$. With recent proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require $d = 1$ for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if $d = 1$. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when $d$ is arbitrary and $a = \pm1/2$.