Trisymmetric Multiplication Formulae in Finite Fields

Abstract

Multiplication is an expensive arithmetic operation, therefore there has been extensive research to find Karatsuba-like formulae reducing the number of multiplications involved when computing a bilinear map. The minimal number of multiplications in such formulae is called the bilinear complexity, and it is also of theoretical interest to asymptotically understand it. Moreover, when the bilinear maps admit some kind of invariance, it is also desirable to find formulae keeping the same invariance. In this work, we study trisymmetric, hypersymmetric, and Galois invariant multiplication formulae over finite fields, and we give an algorithm to find such formulae. We also generalize the result that the bilinear complexity and symmetric bilinear complexity of the twovariable multiplication in an extension field are linear in the degree of the extension, to trisymmetric bilinear complexity, and to the complexity of t-variable multiplication for any t ≥ 3.