Improved base-phi expansion method for Koblitz curves over optimal extension fields

Abstract

An improved base-phi expansion method is proposed, in which the bit-length of coefficients is shorter and the number of coefficients is smaller than in Kobayashi's expansion method. The proposed method meshes well with efficient multi-exponentiation algorithms. In addition, two efficient algorithms based on the proposed expansion method, named phi-wNAF and phi-SJSF, are presented which significantly reduce the computational effort involved in online precomputation by using the property of Frobenius endomorphism. The proposed algorithms noticeably accelerate computation of a scalar multiplication on Koblitz curves over optimal extension fields (OEFs). In particular, for OEFs where the characteristic is close to 32 bits or 64 bits, the required number of additions is reduced up to 50% in comparison with Kobayashi's base-phi scalar multiplication algorithm. Finally, a method that significantly reduces the memory usage of the precomputation table at the expense of slightly more computation is presented.