Differential spectra of a class of power permutations with Niho exponents

Zhen Li; Haode Yan

doi:10.3934/amc.2021060

Article Contents

2023, Volume 17, Issue 6: 1468-1475. Doi: 10.3934/amc.2021060

This issue Previous Article Generic constructions of MDS Euclidean self-dual codes via GRS codes Next Article Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling

Differential spectra of a class of power permutations with Niho exponents

Zhen Li and
Haode Yan^,

School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China

^*Corresponding author: Haode Yan
^*Corresponding author: Haode Yan

Received: August 2021

Revised: October 2021

Early access: December 2021

Published: December 2023

H. Yan's research was supported by the National Natural Science Foundation of China (Grant No.11801468) and the Fundamental Research Funds for the Central Universities of China (Grant No.2682021ZTPY076).

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Abstract

Let $ m\geq3 $ be a positive integer and $ n = 2m $. Let $ f(x) = x^{2^m+3} $ be a power permutation over $ {\mathrm {GF}}(2^n) $, which is a monomial with a Niho exponent. In this paper, the differential spectrum of $ f $ is investigated. It is shown that the differential spectrum of $ f $ is $ \mathbb S = \{\omega_0 = 2^{2m-1}+2^{2m-3}-1,\omega_2 = 2^{2m-2}+2^{m-1}, \omega_4 = 2^{2m-3}-2^{m-1},\omega_{2^m} = 1\} $ when $ m $ is even, and $ \mathbb S = \{\omega_0 = \frac{7\cdot2^{2m-2}+2^m}3, \omega_2 = 3\cdot2^{2m-3}-2^{m-2}-1, \omega_6 = \frac{2^{2m-3}-2^{m-2}}3, \omega_{2^m+2} = 1\} $ when $ m $ is odd.

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Mathematics Subject Classification: Primary: 94A60; Secondary: 11T06.

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Table 1. Power functions $ f(x) = x^d $ over $ {\mathrm {GF}}(2^n) $ with known differential spectra

$ d $	Conditions	$ \delta_f $	Reference
$ 2^n-2 $	$ n $ is even	4	[2]
$ 2^{2t}-2^t+1 $	$ \mathrm{gcd}(t,n)=2 $	4	[2]
$ 2^t+1 $	$ \mathrm{gcd}(t,n)=2 $	4	[2]
$ 2^{n/2}+2^{n/4}+1 $	$ 4\mid n $	4	[2,19]
$2^{n/2}-1;$ $2^{n/2+1}-1$	$ n\geq6 $ is even	$2^{n/2}-2$; $2^{n/2}$	[3]
$ 2^t-1 $	$ t=3,n-2 $	6	[3]
$ 2^t-1 $	$t=(n-1)/2$, $t=(n+3)/2$, $n$ is odd	6 or 8	[4]
$2^{n/2}+2^{(n+2)/4}+1;$ $2^{n/2+1}+3$	$ n\equiv 2(\mathrm{mod}\; 4) $, $ n\geq10 $	8	[20]
$ 2^{3n/4}+2^{n/2}+2^{n/4}-1 $	$ 4\mid n $	$ 2^{n/2} $	[13]
$ 2^{n/2}+3 $	$ n\geq6 $ is even	$ 2^{n/2} $ or $ 2^{n/2}+2 $	This paper

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Table 2. Differential spectrum of $ f(x) = x^{2^{n/2}+3} $ over $ {\mathrm {GF}}(2^n) $ for some values of $ n $

n	$ d=2^{n/2}+3 $	Differential spectra
8	$ d=19 $	$ \mathbb S=\left\{ {\omega_0=159, \omega_2=72, \omega_4=24, \omega_{16}=1} \right\} $
10	$ d=35 $	$ \mathbb S=\left\{ {\omega_0=608, \omega_2=375, \omega_6=40, \omega_{34}=1} \right\} $
12	$ d=67 $	$ \mathbb S=\left\{ {\omega_0=2559, \omega_2=1056, \omega_4=480, \omega_{64}=1} \right\} $
14	$ d=131 $	$ \mathbb S=\left\{ {\omega_0=9600, \omega_2=6111, \omega_6=672, \omega_{130}=1} \right\} $

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