Two kinds of division polynomials for twisted Edwards curves
Division polynomials for elliptic curves are polynomials which have as roots the \(n\)-torsion points. The authors give two kinds of division polynomials for the twisted Edwards \(E_{a,d}\) curves defined as \[ ax^2+y^2=1+dx^2y^2. \] The first approach uses the birational transformation from elliptic curves in Weierstrass form to elliptic curves in Edwards form and the computations for the \(n\)-th power in Weierstrass form, in order to obtain an analogous computation for the \(n\)-th power of a point on an elliptic curve in Edwards form. From this computation the authors arrive at polynomials \(\tilde{\psi}\in \mathbb{Z}[a,d][y]\), which have as roots the \(n\)-th torsion points of the Edwards curve. Their degree is bounded by \(n^2/2\). In the second approach the authors give polynomials that can be used to give formulas for multiplication. The method is motivated by the work of Abel and Eisenstein on lemniscape sine functions. The \(n\)-th such polynomial is recursively expressed in terms of the \((n-1)\)th and \((n-2)\)th polynomial. Section 6 is devoted to the comparison of the \(E_{1,-1}\) curves, which were also studied by Gauss.
scientific article; zbMATH DE number 6048435
scientific article