论文标题
一类局部函数的差分光谱和回旋镖光谱
The differential spectrum and boomerang spectrum of a class of locally-APN functions
论文作者
论文摘要
在本文中，我们研究了电源映射$ f（x）= x^{k（q-1）} $ over $ {\ mathbb f} _ {q^2} $，其中$ q = p^m $，$ p $是prime，$ m $是一个正integer和$ \ gcd（kcd（kcd（kcd））= 1 $ 1）= 1 $ 1）= 1 = 1 $ 1 = 1 = 1）= 1 = 1 $ 1 = 1 = 1 = 1 = 1 = 1 = 1 $ 1 = 1 = 1 = 1 $ 1）的动力映射的回旋镖光谱。我们首先确定$ f（x）$的差异光谱，并表明$ f（x）$是本地apn。这扩展了[IEEE Trans的结果。 inf。理论57（12）：8127-8137，2011]从$（p，k）=（2,1）$ to General $（p，k）$。然后，我们通过使用其差异频谱来确定$ f（x）$的回旋镖频谱，这表明$ f（x）$的回旋镖均匀性为4，如果$ p = 2 $ and $ m $是奇数，否则是2。我们的结果不仅将结果概括为[des。代码加密。 89：2627-2636，2021]和[arxiv：2203.00485，2022]，但也扩展了$ x^{45} $ over $ {\ mathbb f} _ {2^8} $ in [des。代码加密。 89：2627-2636，2021]进入带有回旋镖均匀性2的无限的功率映射2。
In this paper, we study the boomerang spectrum of the power mapping $F(x)=x^{k(q-1)}$ over ${\mathbb F}_{q^2}$, where $q=p^m$, $p$ is a prime, $m$ is a positive integer and $\gcd(k,q+1)=1$. We first determine the differential spectrum of $F(x)$ and show that $F(x)$ is locally-APN. This extends a result of [IEEE Trans. Inf. Theory 57(12):8127-8137, 2011] from $(p,k)=(2,1)$ to general $(p,k)$. We then determine the boomerang spectrum of $F(x)$ by making use of its differential spectrum, which shows that the boomerang uniformity of $F(x)$ is 4 if $p=2$ and $m$ is odd and otherwise it is 2. Our results not only generalize the results in [Des. Codes Cryptogr. 89:2627-2636, 2021] and [arXiv:2203.00485, 2022] but also extend the example $x^{45}$ over ${\mathbb F}_{2^8}$ in [Des. Codes Cryptogr. 89:2627-2636, 2021] into an infinite class of power mappings with boomerang uniformity 2.
