Computational Explorations on Semifields

2026-02-10

AI Summary

This study investigates the computational complexity of multiplication in finite semifields and finite field extensions, with a focus on precisely characterizing tensor rank—equivalently, multiplicative complexity—and additive complexity. By constructing short straight-line programs that encode linear codes with specific parameters, and integrating tensor rank analysis with additive complexity modeling, the work systematically addresses small-scale structures in characteristics 2 and 3. The main contributions include the first exact determinations of tensor ranks for several small-order semifields and finite fields, the derivation of new upper and lower bounds on additive complexity, and the design of efficient algorithms whose overall computational complexity improves upon existing methods, yielding significant gains in multiple concrete instances.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• semifields
• tensor rank
• multiplicative complexity
• additive complexity
• finite fields

Innovation

• tensor rank
• additive complexity
• semifields
• straight line programs
• finite fields