A normal form for elliptic curves

A Normal Form for Elliptic Curves

The normal form x2+y2 = a2+a2x2y2 for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be

Twisted μ4-Normal Form for Elliptic Curves

We introduce the twisted μ4-normal form for elliptic curves, deriving in particular addition algorithms with complexity 9M+ 2S and doubling algorithms with complexity 2M + 5S + 2m over a binary field. Every ordinary elliptic curve over a finite field of characteristic 2 is isomorphic to one in this family. This improvement to the addition algorithm, applicable to a larger class of curves, is co...

On Silverman's conjecture for a family of elliptic curves

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...

The Hermitian Intersection Form for a Family of Elliptic Curves *

Speeding up Huff Form of Elliptic Curves

This paper presents faster inversion-free point addition formulas for the curve y(1 + ax) = cx(1 + dy). The proposed formulas improve the point doubling operation count record from 6M + 5S to 8M and mixed addition operation count record from 10M to 8M. Both sets of formulas are shown to be 4-way parallel, leading to an effective cost of 2M per either of the group operations.