On a classification of planar functions in characteristic three

TL;DR

This paper classifies planar functions over finite fields of characteristic three, exploring their equivalence relations, and discovers new sporadic functions through computational methods, enhancing understanding of their structure and diversity.

Contribution

It provides a comprehensive survey of known planar functions up to CCZ-equivalence, resolves isotopic equivalence cases for certain fields, and introduces new sporadic functions and simplified representatives.

Findings

Complete resolution for fields of order 3^n with n ≤ 11.

Discovery of seven new sporadic planar functions over GF(3^6).

Introduction of simple quadrinomial representatives for Dickson planar functions.

Abstract

Planar functions are functions over a finite field that have optimal combinatorial properties and they have applications in several branches of mathematics, including algebra, projective geometry and cryptography. There are two relevant equivalence relations for planar functions, that are isotopic equivalence and CCZ-equivalence. Classification of planar functions is performed via CCZ-equivalence which arises from cryptographic applications. In the case of quadratic planar functions, isotopic equivalence, coming from connections to commutative semifields, is more general than CCZ-equivalence and isotopic transformations can be considered as a construction method providing up to two CCZ-inequivalent mappings. In this paper, we first survey known infinite classes and sporadic cases of planar functions up to CCZ-equivalence, aiming to exclude equivalent cases and to identify those with the potential to provide additional functions via isotopic equivalence. In particular, for fields of order $3^n$ with $n\le 11$, we completely resolve if and when isotopic equivalence provides different CCZ-classes for all currently known planar functions. Further, we perform an extensive computational investigation on some of these fields and find seven new sporadic planar functions over $\mathbb{F}_{3^6}$ and two over $\mathbb{F}_{3^9}$. Finally, we give new simple quadrinomial representatives for the Dickson family of planar functions.