Constructing commutative semifields of square order

TL;DR

This paper generalizes the projection construction method to create new commutative semifields of odd order using two twisted semifields, also generating new planar functions with strong algebraic connections.

Contribution

It introduces a novel projection construction using two twisted semifields, expanding the methods for constructing semifields and related planar functions.

Findings

Successfully constructs new semifields of odd order

Develops new planar functions with strong algebraic properties

Extends previous projection construction techniques

Abstract

The projection construction has been used to construct semifields of odd characteristic using a field and a twisted semifield [Commutative semifields from projection mappings, Designs, Codes and Cryptography, 61 (2011), 187--196]. We generalize this idea to a projection construction using two twisted semifields to construct semifields of odd characteristic. Planar functions and semifields have a strong connection so this also constructs new planar functions.