Jacobi Quartic Curves Revisited

This paper provides new results about efficient arithmetic on (extended) Jacobiquartic form elliptic curves y = dx + 2ax + 1. Recent works have shown thatarithmetic on an elliptic curve in Jacobi quartic form can be performed solidly fasterthan the corresponding operations in Weierstrass form. These proposals use up to 7coordinates to represent a single point. However, fast scalar multiplication algorithmsbased on windowing techniques, precompute and store several points which requiremore space than what it takes with 3 coordinates. Also note that some of theseproposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in extended Jacobi quartic form if d = 1.Even worse the contemporary formulae may fail to output correct coordinates for someinputs. This paper provides improved speeds using fewer coordinates without causingthe above mentioned problems. For instance, our proposed point doubling algorithmtakes only 2 multiplications, 5 squarings, and no multiplication with curve constantswhen d is arbitrary and a = ±1/2.