Two Classes of Crooked Multinomials Inequivalent to Power Functions

TL;DR

This paper introduces two new infinite classes of quadratic crooked multinomials over fields of order 2^{2m}, expanding the known constructions and establishing their inequivalence to power functions.

Contribution

It presents two novel classes of quadratic crooked multinomials and proves their EA inequivalence to power functions, advancing the understanding of crooked functions.

Findings

Two infinite classes of quadratic crooked multinomials are constructed.

One class generalizes a previously known APN function.

The classes are proven EA inequivalent to power functions.

Abstract

It is known that crooked functions can be used to construct many interesting combinatorial objects, and a quadratic function is crooked if and only if it is almost perfect nonlinear (APN). In this paper, we introduce two infinite classes of quadratic crooked multinomials on fields of order $2^{2m}$. One class of APN functions constructed in [7] is a particular case of the one we construct in Theorem 1. Moreover, we prove that the two classes of crooked functions constructed in this paper are EA inequivalent to power functions and conjecture that CCZ inequivalence between them also holds.