Differential properties of functions x -> x^{2^t-1} -- extended version

TL;DR

This paper thoroughly investigates the differential properties of functions of the form x -> x^{2^t-1} over finite fields, revealing their spectra depend on roots of specific linear polynomials and establishing connections between different exponents.

Contribution

It introduces a novel relationship between the differential spectra of x -> x^{2^t-1} and x -> x^{2^{s}-1}, and determines the spectra for specific cases using Kloosterman sums.

Findings

Differential spectra are linked to roots of certain linear polynomials.

Established a relationship between spectra of functions with exponents t and n-t+1.

Computed spectra for x^7 and specific t values using Kloosterman sums.

Abstract

We provide an extensive study of the differential properties of the functions $x\mapsto x^{2^t-1}$ over $\F$, for $2 \leq t \leq n-1$. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials $x^{2^t}+bx^2+(b+1)x$ where $b$ varies in $\F$.We prove a strong relationship between the differential spectra of $x\mapsto x^{2^t-1}$ and $x\mapsto x^{2^{s}-1}$ for $s= n-t+1$. As a direct consequence, this result enlightens a connection between the differential properties of the cube function and of the inverse function. We also determine the complete differential spectra of $x \mapsto x^7$ by means of the value of some Kloosterman sums, and of $x \mapsto x^{2^t-1}$ for $t \in \{\lfloor n/2\rfloor, \lceil n/2\rceil+1, n-2\}$.