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On the second-order zero differential properties of several classes of power functions over finite fields

Huan Zhou, Xiaoni Du, Xingbin Qiao, Wenping Yuan

TL;DR

By computing the number of solutions of specific equations over finite fields, the second-order zero differential spectra of power functions are determined and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails.

Abstract

Feistel Boomerang Connectivity Table (FBCT) is an important cryptanalytic technique on analysing the resistance of the Feistel network-based ciphers to power attacks such as differential and boomerang attacks. Moreover, the coefficients of FBCT are closely related to the second-order zero differential spectra of the function $F(x)$ over the finite fields with even characteristic and the Feistel boomerang uniformity is the second-order zero differential uniformity of $F(x)$. In this paper, by computing the number of solutions of specific equations over finite fields, we determine explicitly the second-order zero differential spectra of power functions $x^{2^m+3}$ and $x^{2^m+5}$ with $m>2$ being a positive integer over finite field with even characteristic, and $x^{p^k+1}$ with integer $k\geq1$ over finite field with odd characteristic $p$. It is worth noting that $x^{2^m+3}$ is a permutation over $\mathbb{F}_{2^n}$ and only when $m$ is odd, $x^{2^m+5}$ is a permutation over $\mathbb{F}_{2^n}$, where integer $n=2m$. As a byproduct, we find $F(x)=x^4$ is a PN and second-order zero differentially $0$-uniform function over $\mathbb{F}_{3^n}$ with odd $n$. The computation of these entries and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails.

On the second-order zero differential properties of several classes of power functions over finite fields

TL;DR

By computing the number of solutions of specific equations over finite fields, the second-order zero differential spectra of power functions are determined and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails.

Abstract

Feistel Boomerang Connectivity Table (FBCT) is an important cryptanalytic technique on analysing the resistance of the Feistel network-based ciphers to power attacks such as differential and boomerang attacks. Moreover, the coefficients of FBCT are closely related to the second-order zero differential spectra of the function over the finite fields with even characteristic and the Feistel boomerang uniformity is the second-order zero differential uniformity of . In this paper, by computing the number of solutions of specific equations over finite fields, we determine explicitly the second-order zero differential spectra of power functions and with being a positive integer over finite field with even characteristic, and with integer over finite field with odd characteristic . It is worth noting that is a permutation over and only when is odd, is a permutation over , where integer . As a byproduct, we find is a PN and second-order zero differentially -uniform function over with odd . The computation of these entries and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails.
Paper Structure (6 sections, 7 theorems, 63 equations, 1 table)

This paper contains 6 sections, 7 theorems, 63 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Feistel boomerang uniformity of $F(x)=x^{2^m+3}$ over $\mathbb{F}_{2^n}$
  4. Feistel boomerang uniformity of $F(x)=x^{2^m+5}$ over $\mathbb{F}_{2^n}$
  5. Second-order zero differential spectrum of $F(x)=x^{p^k+1}$ over $\mathbb{F}_{p^n}$
  6. Concluding remarks

Key Result

Lemma 1

SM2022 Let $F(x)=a_2x^{2}+a_1x+a_0\in \mathbb{F}_{p^{n}}[x]$ with $p$ odd and $a_2\neq0$. Then the equation $F(x)=0$ has two (resp. one) solutions in $\mathbb{F}_{p^{n}}$ if and only if the discriminant $\Delta=a_{1}^{2}-4a_{0}a_{2}$ is a nonzero (resp. zero) square in $\mathbb{F}_{p^{n}}$. That is

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • proof
  • Remark 1
  • ...and 12 more