Division polynomials on the Hessian model of elliptic curves

Abstract

In this paper we derive formulas for the scalar multiplication by n map, denoted [n], on the Hessian model of elliptic curve. This enables to characterize n-torsion points on this curve. The computation involves three families of polynomials $$P_n$$ , $$Q_n$$ and $$V_n$$ and we show some properties on the coefficients and degrees of these polynomials. We also show some functional equations satisfied by these polynomials. As application we provide a type of mean-value theorem for the Hessian elliptic curve.