论文标题
在NIHO类型的本地功率功能及其回旋镖频谱上
On the Niho type locally-APN power functions and their boomerang spectrum
论文作者
论文摘要
在本文中，我们专注于本地性质的概念（``Apn'是Blondeau，Canteaut和Charpin引入的众所周知概念的缩写），这使得S-box的语料体使S-boxes的语料体变得更大，因此在某种程度上较大的既有均匀性，并且更合适的是不同的候选者（因此，他们都具有不同的派遣），或者是不同的，或者，他们的差异化（或者）具有不同的不同，或者是他们的不同，或者是他们的差异化攻击（或者，他们的差异都如此，或者是他们的不同，或者是他们的不同，或者，他们的差异化（或者都具有不同的不同），或者是他们的差异化。整数$ m $和$ k $使得$ \ gcd（2^m+1,2^k+1）= 1 $，我们调查了无限的NIHO类型功率家族的局部apn-属性，以$ f（x）= x^{s（2^m-1）+1} $ a $ f（x）= x^{s（2^m-1）$ co { $ s =（2^k+1）^{ - 1} $，其中$（2^k+1）^{ - 1} $表示乘法逆Modulo $ 2^m+1 $。 通过对有限场（甚至具有特征）的某些方程式解决方案的解决方案数量以及对求解某些方程式的一些微妙的操作进行更精细的研究，我们证明$ f（x）$是本地APN并确定其差异频谱。值得注意的是，计算机实验表明，这类本地APN功率功能以$ 2 \ leq M \ leq10 $覆盖所有NIHO类型本地功能功能。此外，我们还通过使用其差异光谱来确定$ f（x）$的回旋镖光谱，这特别概括了YAN，Zhang和Li的最新结果。
In this article, we focus on the concept of locally-APN-ness (``APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger regarding their differential uniformity and, therefore, possibly, more suitable candidates against the differential attack (or their variants). Specifically, given two coprime positive integers $m$ and $k$ such that $\gcd(2^m+1,2^k+1)=1$, we investigate the locally-APN-ness property of an infinite family of Niho type power functions in the form $F(x)=x^{s(2^m-1)+1}$ over the finite field ${\mathbb F}_{2^{2m}}$ for $s=(2^k+1)^{-1}$, where $(2^k+1)^{-1}$ denotes the multiplicative inverse modulo $2^m+1$. By employing finer studies of the number of solutions of certain equations over finite fields (with even characteristic) as well as some subtle manipulations of solving some equations, we prove that $F(x)$ is locally APN and determine its differential spectrum. It is worth noting that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for $2\leq m\leq10$. In addition, we also determine the boomerang spectrum of $F(x)$ by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang, and Li.
