Minimality of the Hamming Weight of the τ-NAF for Koblitz Curves and Improved Combination with Point Halving

In order to efficiently perform scalar multiplications on elliptic Koblitz curves, expansions of the scalar to a complex base associated with the Frobenius endomorphism are commonly used. One such expansion is the

τ

-adic NAF, introduced by Solinas. Some properties of this expansion, such as the average weight, are well known, but in the literature there is no proof of its

optimality

, i.e. that it always has minimal weight. In this paper we provide the first proof of this fact.

Point halving, being faster than doubling, is also used to perform fast scalar multiplications on generic elliptic curves over binary fields. Since its computation is more expensive than that of the Frobenius, halving was thought to be uninteresting for Koblitz curves. At PKC 2004, Avanzi, Ciet, and Sica combined Frobenius operations with one point halving to compute scalar multiplications on Koblitz curves using on average 14% less group additions than with the usual

τ

-and-add method without increasing memory usage. The second result of this paper is an improvement over their expansion. The new representation, called the

wide-double-NAF

, is not only simpler to compute, but it is also optimal in a suitable sense. In fact, it has minimal Hamming weight among all

τ

-adic expansions with digits {0,±1} that allow one halving to be inserted in the corresponding scalar multiplication algorithm. The resulting scalar multiplication requires on average 25% less group operations than the Frobenius method, and is thus 12.5% faster than the previously known combination.