# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021060
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## Differential spectra of a class of power permutations with Niho exponents

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

*Corresponding author: Haode Yan

Received  August 2021 Revised  October 2021 Early access December 2021

Fund Project: 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)

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.

Citation: Zhen Li, Haode Yan. Differential spectra of a class of power permutations with Niho exponents. Advances in Mathematics of Communications, doi: 10.3934/amc.2021060
##### References:
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##### References:
 [1] E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.  doi: 10.1007/BF00630563. [2] C. Blondeau, A. Canteaut and P. Charpin, Differential properties of power functions, Int. J. Inf. Coding Theory., 1 (2010), 149-170.  doi: 10.1504/IJICOT.2010.032132. [3] C. Blondeau, A. Canteaut and P. Charpin, Differential properties of ${x\mapsto x^{2^{t}-1}}$, IEEE Trans. Inf. Theory., 57 (2011), 8127-8137.  doi: 10.1109/TIT.2011.2169129. [4] C. Blondeau and L. Perrin, More differentially 6-uniform power functions, Des. Codes Cryptogr., 73 (2014), 487-505.  doi: 10.1007/s10623-014-9948-2. [5] P. Charpin, Cyclic codes with few weights and Niho exponents, J. Combinat. Theory. Ser. A., 108 (2004), 247-259.  doi: 10.1016/j.jcta.2004.07.001. [6] H. Dobbertin, Almost perfect nonlinear power functions on GF($2^n$): The Niho case, Inform. and Comput., 151 (1999), 57-72.  doi: 10.1006/inco.1998.2764. [7] J. Daemen and V. Rijmen, The Design of Rijndael: AES- The Advanced Encryption Standard, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04722-4. [8] H. Dobbertin, Almost perfect nonlinear power functions on GF($2^n$): The Welch case, IEEE Trans. Inf. Theory., 45 (1999), 1271-1275.  doi: 10.1109/18.761283. [9] T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X. [10] T. Helleseth, C. Rong and D. Sandberg, New families of almost perfect nonlinear power mappings, IEEE Trans. Inf. Theory., 45 (1999), 474-485.  doi: 10.1109/18.748997. [11] H. Hollmann and Q. Xiang, A proof of the Welch and Niho conjectures on cross-correlations of binary $m$-sequences, Finite Fields Appl., 7 (2001), 253-286.  doi: 10.1006/ffta.2000.0281. [12] N. Li, T. Helleseth, A. Kholosha and X. Tang, On the walsh transform of a class of functions from Niho exponents, IEEE Trans. Inf. Theory., 59 (2013), 4662-4667.  doi: 10.1109/TIT.2013.2252053. [13] N. Li, Y. Wu, X. Zeng and X. Tang, On the differential spectrum of a class of power functions over finite fields, Computer Science, 2020, arXiv: 2012.04316v1. [14] N. Li and X. Zeng, A survey on the applications of Niho exponents, Cryptogr. Commun., 11 (2019), 509-548.  doi: 10.1007/s12095-018-0305-6. [15] Y. Niho, Multivalued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequence, PhD Thesis, Univ. of Southern California, Los Angle, 1972. [16] K. Nyberg, Differentially uniform mappings for cryptography, Advances in Cryptology–EUROCRYPT'93, 765 (1993), 55-64.  doi: 10.1007/3-540-48285-7_6. [17] A. Pott, Almost perfect and planar functions, Des. Codes Cryptogr., 78 (2016), 141-195.  doi: 10.1007/s10623-015-0151-x. [18] M. Xiong, N. Li, Z. Zhou and C. Ding, Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes, Des. Codes Cryptogr., 78 (2016), 713-730.  doi: 10.1007/s10623-014-0027-5. [19] M. Xiong and H. Yan, A note on the differential spectrum of a 4-uniform power function, Finite Fields and Appl., 48 (2017), 117-125.  doi: 10.1016/j.ffa.2017.07.008. [20] M. Xiong, H. Yan and P. Yuan, On a conjecture of differentially 8-uniform power function, Des. Codes Cryptogr., 86 (2018), 1601-1621.  doi: 10.1007/s10623-017-0416-7.
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
 $d$ 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
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 $d=131$ $\mathbb S=\left\{ {\omega_0=9600, \omega_2=6111, \omega_6=672, \omega_{130}=1} \right\}$
 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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