Asymptotic Bound for Multiplication Complexity in the Extensions of Small Finite Fields

In 1986, D. V. Chudnovsky and G. V. Chudnovsky first employed algebraic curves over finite fields to construct bilinear multiplication algorithms implicitly through supercodes introduced by Shparlinski-Tsfasman-Vladuţ, or equivalently, multiplication-friendly codes that we will introduce in this paper. This idea was further developed by Shparlinski-Tsfasman-Vladuţ in order to study the asymptotic behavior of multiplication complexity in extension fields. Later on, Ballet et al. further investigated the method and obtained some improvements. Recently, Ballet and Pieltant made use of curves over an extension field of to obtain an improvement on the complexity of multiplications in extensions of the binary field. In this paper, we develop the multiplication-friendly splitting technique and then apply this technique to study asymptotic behavior of multiplications in extension fields. By combining this with the idea of using algebraic function fields, we are able to improve further the asymptotic results of multiplication complexity. In particular, the improvement for small fields such as the binary and ternary fields is substantial.