Computational Explorations on Semifields

TL;DR

This paper investigates the multiplication complexity in small semifields and finite field extensions, establishing tensor ranks, proposing bounds on additive complexity, and introducing improved algorithms for efficient computation.

Contribution

It determines tensor ranks for certain semifields and finite fields, and develops new algorithms with better overall complexity for finite field operations.

Findings

Tensor ranks for semifields of characteristics 2 and 3 determined.

New upper and lower bounds on additive complexity proposed.

Algorithms with improved overall complexity introduced.

Abstract

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance. In this paper, we establish the tensor rank of some semifields and finite fields of characteristics 2 and 3. We also propose new upper and lower bounds on their additive complexity, and give new associated algorithms improving on the state-of-the-art in terms of overall complexity. We achieve this by considering short straight line programs for encoding linear codes with given parameters.