Improved algorithms for an efficient arithmetic on some categories of elliptic curves

The Frobenius endomorphism τ is known to be useful for an efficient scalar multiplication on elliptic curves E(Fqm) defined either over fields with small characteristics or over optimal extension fields. In this paper, we will present two techniques that aim to enhance the Frobenius-based methods for computing the scalar multiplication on these curves. The first method, called the generalised τ-adic method, is dedicated to improve the efficiency of the generalised τ-adic method when the elliptic curves are defined over fields of small characteristics. The generalised τ-adic with even digits improves substantially the computation time and the number of stored points whereas the generalised τ-adic with odd digits reduces only the number of stored points but it offers better resistance against the SPA attacks. The generalised τ-adic method is particularly efficient when the trace of the used curve is small. The second method allows to reduce by about 50% the number of the stored points by the Frobenius-based algorithm on elliptic curve defined over optimal extension fields. Finally, we show that there are a lot of curves which are well suited for cryptography, and for which the proposed methods can be applied.