International Association
for Cryptologic Research

In this paper, we present a generalization of Edwards model for elliptic curve which is defined over any field and in particular for field of characteristic 2. This model generalize the well known Edwards model of \\cite{Edw07} over characteristic zero field, moreover it define an ordinary elliptic curve over binary fields.

For this, we use the theory of theta functions and an intermediate model embed in $\\mathbb{P}^3$ that we call a level $4$-theta model. We then present an arithmetic of this level $4$-theta model and of our Edwards model using Riemann relations of theta functions. The group laws are complete, i.e. none exceptional case for adding a pair of points; their are also unified, i.e. formulas using for addition and for doubling are the same. Over binary fields we have very efficient arithmetics on ordinary elliptic curve, but over odd field our explicit addition laws are not competitives. Nevertheless, we give efficient differential addition laws on level $4$-theta model and on Edwards model defined over any fields.