Jacobi Quartic Curves Revisited

Abstract

This paper provides new results about efficient arithmetic on (extended) Jacobi quartic form elliptic curves y 2 = dx 4 + 2ax 2 + 1. Recent works have shown that arithmetic on an elliptic curve in Jacobi quartic form can be performed solidly faster than the corresponding operations in Weierstrass form. 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-aprime 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 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 = ±1/2.

Exponentiating in Pairing Groups

“…We note that in the context of plain ECC these models have been studied with small curve constants; in pairing-based cryptography, however, we must put up with whatever constants we get under the transformation to the non-Weierstrass model. The only exception we found in this work is for the k = 12 BLS curves, where G 1 can be transformed to a Jacobi quartic curve with a = −1/2, which gives a worthwhile speedup [34].…”

Embedded implementation of Edwards curve- and extended Jacobi quartic curve-based cryptosystems

“…R(x 3 , y 3 ) is on the curve only if the coordinates (x 3 , y 3 ) are defined, i.e., when is not a square in F p and x 2 1 x 2 2 = 1 [13].…”

Duffing Oscillator and Elliptic Curve Cryptography

“…However, it is cryptology as a computational science that has been a driving force behind the arithmetic of algebraic curves and the other parts of algebraic geometry in the past few decades. As a result, in cryptology we have formulae, algorithms and even computer programs prepared for usage [4,[7][8][9]. For B > 0 and −Bα 2 < A < −2Bα 2 the periodic solution is…”