Algorithms on Elliptic Curves over Fields of Characteristic Two with Non-adjacent Forms

Let IFq be a finite field of characteristic two and let φ be the Frobenius endomorphism of an elliptic curve. To find or improve efficient algorithms for scalar multiplication sP of point P in the elliptic curve cryptography, it is always an important subject. If IFq = IF2, Solinas [5] has developed an algorithm for computing the φ-NAF. In this note, we extend Solinas’ φ-NAF algorithm to IFq, where q is a power of two, and give another efficient algorithms for φ-NAF, thereby show that the length of φ-NAF is at most two bits longer than the length of φ-expansion.