A comparison of Differential Addition and Doubling in Binary Edwards Curves for Elliptic Curve Cryptography

Binary Edwards curves (BEC) over finite fields can be used as an additive cyclic elliptic curve group to enable elliptic curve cryptography (ECC), where the most time consuming is scalar multiplication. This operation is computed by means of the group operation, either point addition or point doubling. The most notorious property of these curves is that their group operation is complete, which mitigates the need to verify for special cases. Different formulae for the group operation in BECs have been reported in the literature. Of particular interest are those designed to work with the differential properties of the Montgomery ladder, which offer constant time computation of the scalar multiplication as well as reduced field operations count. In this work, we review and compare the complexity of BEC differential addition and doubling in terms of field operations. We also provide software implementations of scalar multiplications which employ these formulae under a fair scenario. Our work provides insights on the advantages of using BECs in ECC. Our study of the different formulae for group addition in BEC also showcases the advantages and limitations of the different design strategies employed in each case.