In this paper, we present two efficient algorithms computing scalar multiplications of a point in an elliptic curve defined over a small finite field, the Frobenius map of which has small trace. Both methods use the identity which expresses multiplication-by-m maps by polynomials of Frobenius maps. Both are applicable for a large family of elliptic curves and more efficient than any other methods applicable for the family. More precisely, by Algorithm 1(Frobenius k-ary method), we can compute mP in at most 21/5+28 elliptic additions for arbitrary I bit integer m and a point P on some elliptic curves. For other curves, the number of elliptic additions required is less than l. Algorithm 2(window method) requires at average 21/3 elliptic additions to compute mP for I bit integer m and a point P on a family of elliptic curves. For some 'good' elliptic curves, it requires 51/12 + 11 elliptic additions at average.
['Jung Hee Cheon', 'Sung-Mo Park', 'Sangwoo Park', 'Daeho Kim']
Two Efficient Algorithms for Arithmetic of Elliptic Curves Using Frobenius Map