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Relative Difference sets from Almost Perfect Nonlinear Functions

Zeying Wang

Abstract

In this paper we explore a connection between certain Almost Perfect Nonlinear Functions (APN functions) and relative difference sets. In particular, we show that the image set of certain 2-to-1 APN functions is a relative difference set. Through a result of Pott this further provides a connection between APN functions and bent functions.

Relative Difference sets from Almost Perfect Nonlinear Functions

Abstract

In this paper we explore a connection between certain Almost Perfect Nonlinear Functions (APN functions) and relative difference sets. In particular, we show that the image set of certain 2-to-1 APN functions is a relative difference set. Through a result of Pott this further provides a connection between APN functions and bent functions.
Paper Structure (6 sections, 6 theorems, 17 equations)

This paper contains 6 sections, 6 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. The APN map $f(x)= x^3+a^{-1}\mathrm{Tr}(a^3x^9)$
  4. The APN map $f(x)=x^{2^k-1}+x^{2^k}$
  5. On bent functions derived from the image of some APN functions
  6. Open problems

Key Result

Lemma 2.1

Let $a$ be a non-zero element in $\mathbb{F}_{2^n}$. If $n$ is odd, then the map $x \to x+a^{-1}\mathrm{Tr}(a^3x^3)$ is a 2-to-1 function.

Theorems & Definitions (6)

  • Lemma 2.1
  • Corollary 2.3
  • Corollary 2.5
  • Lemma 2.6
  • Corollary 2.8
  • Lemma 2.9