An alternative proof of the minimality of the Hamming weight of width-3 non-adjacent form

Abstract

In elliptic curve cryptography (ECC for short), point multiplication (or scalar multiplication) is the dominant operation. It is a very important matter to improve the efficiency of point multiplication for practical use. In ECC, recoding methods of the scalars play an important role in the performance of the algorithm used. One such example is the width-ω non-adjacent form (ω-NAF). Bosma (2001) proved that the Hamming weight of the NAF (2-NAF) is minimal, and Muir and Stinson (2005) proved that the Hamming weight of the ω-NAF is minimal. As other examples, the generalized non-adjacent form (GNAF) and the τ-adic non-adjacent form (τ -NAF) are known. Clark and Liang (1973) proved that the Hamming weight of the GNAF is minimal by constructing an injective map. A similar strategy was adopted by Hakuta, Sato, and Takagi (2010) to prove the minimality of the Hamming weight of the τ -NAF. In this paper, we shall give an alternative proof of the minimality of the Hamming weight of the 3-NAF (ω-NAF with ω = 3). We also discuss that our alternative proof may not work in the case ω ≥ 4.