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doi: 10.3934/amc.2021060
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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

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.

Keywords: Power mapping, Niho exponent, Differential uniformity, Differential spectrum, Walsh spectrum.
Mathematics Subject Classification: Primary: 94A60; Secondary: 11T06.
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:
[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.

show all references

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.

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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$ 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
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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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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