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

Authors: Mustapha Hedabou

Addresses: Department Computer Science, ENSA de Safi, University of Marrakech, Morocco

Abstract: The Frobenius endomorphism τ is known to be useful for an efficient scalar multiplication on elliptic curves E(F_q^m) 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.

Keywords: elliptic curves; scalar multiplication; Frobenius map; normal basis; tau-adic expansion; precomputed tables; Frobenius endomorphism; cryptography; security.

DOI: 10.1504/IJCCIA.2016.077465

International Journal of Computational Complexity and Intelligent Algorithms, 2016 Vol.1 No.1, pp.54 - 67

Received: 27 Feb 2013
Accepted: 03 Apr 2014
Published online: 02 Jul 2016 *