# Differential uniformity and second order derivatives for generic polynomials

## TL;DR

This paper introduces a new measure called second order differential uniformity for polynomials over finite fields, extending existing concepts, and proves a density theorem related to this measure.

## Contribution

It defines the second order differential uniformity for polynomials over finite fields and establishes a density theorem analogous to Voloch's theorem for differential uniformity.

## Key findings

- Defines second order differential uniformity for polynomials over finite fields
- Proves a density theorem related to this new measure
- Extends the understanding of polynomial derivatives in cryptography

## Abstract

For any polynomial $f$ of ${\mathbb F}\_{2^n}[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$$\delta^2(f):=\max\_{\substack{\alpha \in {\mathbb F}\_{2^n}^{\ast} ,\alpha' \in {\mathbb F}\_{2^n}^{\ast} ,\beta \in {\mathbb F}\_{2^n} \alpha\not=\alpha'}} \sharp\{x\in{\mathbb F}\_{2^n} \mid D\_{\alpha,\alpha'}^2f(x)=\beta\}$$where $D\_{\alpha,\alpha'}^2f(x):=D\_{\alpha'}(D\_{\alpha}f(x))=f(x)+f(x+\alpha)+f(x+\alpha')+f(x+\alpha+\alpha')$ is the second order derivative.Our purpose is to prove a density theorem relative to this quantity,which is an analogue of a density theorem proved by Voloch for the differential uniformity.