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Low c-Differential Uniformity of the Swapped Inverse Function in Odd Characteristic

Jaeseong Jeong, Namhun Koo, Soonhak Kwon

Abstract

The study of Boolean functions with low $c$-differential uniformity has become recently an important topic of research. However, in odd characteristic case, there are not many results on the ($c$-)differential uniformity of functions that are not power functions. In this paper, we investigate the $c$-differential uniformity of the swapped inverse functions in odd characteristic, and show that their $c$-differential uniformities are at most 6 except for some special case.

Low c-Differential Uniformity of the Swapped Inverse Function in Odd Characteristic

Abstract

The study of Boolean functions with low -differential uniformity has become recently an important topic of research. However, in odd characteristic case, there are not many results on the (-)differential uniformity of functions that are not power functions. In this paper, we investigate the -differential uniformity of the swapped inverse functions in odd characteristic, and show that their -differential uniformities are at most 6 except for some special case.
Paper Structure (8 sections, 15 theorems, 32 equations)

This paper contains 8 sections, 15 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Some Previous Results
  3. The $c$-differential uniformity of $Inv\circ (0,1)$
  4. The $c$-differential uniformity of $Inv\circ (1,\gamma)$
  5. The differential uniformity
  6. The $c$-differential uniformity with $c\ne 0,1$
  7. Concluding Remark
  8. Proof of Lemma \ref{['panot6_lemma']}

Key Result

Proposition 1.2

We call an affine permutation of the form $A(x)=a_1x+a_0$ where $a_0\in \mathbb{F}_{p^n}$ and $a_1\in\mathbb{F}_{p^n}^*$ an affine permutation of degree one. We call two functions $F$ and $F'$ are affine equivalent of degree one if $F=A_1 \circ F' \circ A_2$ where $A_1$ and $A_2$ are affine permutat

Theorems & Definitions (30)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 20 more