Edwards model of elliptic curves de ned over any elds

In this paper, we present a generalization of Edwards model for elliptic curve which is de ned over any eld and in particular for eld of characteristic 2. This model generalize the well known Edwards model of [10] over characteristic zero eld, moreover it de ne an ordinary elliptic curve over binary elds. For this, we use the theory of theta functions and an intermediate model embed in P 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 uni ed, i.e., formulas using for addition and for doubling are the same. Over binary elds we have very e cient arithmetics on ordinary elliptic curve, but over odd eld our explicit addition laws are not competitives. Nevertheless, we give e cient di erential addition laws on level 4-theta model and on Edwards model de ned over any elds.