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

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... tic 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. Reference to this paper should be made as follows: Hedabou, M. (2016) 'Improved algorithms for an efficient arithmetic on some categories of elliptic curves', Int.