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Differential spectra of a class of power permutations with Niho exponents

  • *Corresponding author: Haode Yan

    *Corresponding author: Haode Yan

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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  • 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.

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

    Citation:

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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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