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Trinomials with high differential uniformity

Yves Aubry, Fabien Herbaut, Ali Issa

Abstract

Comparisons of arithmetic and geometric monodromy groups coupled with the Chebotarev density theorem enable to obtain families of trinomials defined over finite fields of even characteristic with high differential uniformity when the base field is large enough.

Trinomials with high differential uniformity

Abstract

Comparisons of arithmetic and geometric monodromy groups coupled with the Chebotarev density theorem enable to obtain families of trinomials defined over finite fields of even characteristic with high differential uniformity when the base field is large enough.
Paper Structure (6 sections, 4 theorems, 7 equations)

This paper contains 6 sections, 4 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of the main result
  4. Distinct critical values
  5. Application of the Chebotarev density theorem
  6. On the exceptional APN conjecture

Key Result

Theorem 2.2

Let $m\geq 8$ be an integer such that $m\equiv{0}\pmod{4}$ and $m-1\in \mathcal{M}$. For $n$ sufficiently large, if $f(x)=a_0x^m+a_1x^{m-1}+a_{2}x^{m-2}\in {\mathbb F}_{2^n}[x]$ is a polynomial of degree $m$ such that $a_1\neq 0$ then $\delta_{\mathbb{F}_{2^n}}(f)$ is maximal, that is $\delta_{\math

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 4.1
  • proof