论文标题
关于最大弯曲组件数量的功能
On functions with the maximal number of bent components
论文作者
论文摘要
一个函数$ f：\ mathbb {f} _2^n \ rightarrow \ mathbb {f} _2^n $，$ n = 2m $，最多可以具有$ 2^n-2^m $ bent bent component函数。琐碎的示例以$ f（x）=（f_1（x），\ ldots，f_m（x），a_1（x），a_1（x），\ ldots，a_m（x））$，其中$ \ tilde {f}（f}（x）=（f_1（f_1（f_1（f_1（x），x），\ ldots，f_m（x），f_m（x） $ \ mathbb {f} _2^n $ to $ \ mathbb {f} _2^m $，$ a_i $，$ 1 \ le i \ le i \ le m $，是Aggine boolean函数。以单变量形式给出了一类非平凡示例，其功能$ f（x）= x^{2^rm} {\ rm tr^n_m}（λ（x））$，其中$λ$是$ \ m athbb {f} _ {2^m} $的线性化置换。在本文的第一部分中，显示出具有$ 2^n-2^m $弯曲组件的高原功能最多可以具有非线性，最多可以2^{n-1} -2} -2^{\ lfloor \ lfloor \ frac {n+m} {2} {2} {2} \ rfloor} $ tr^n_m}（x）$，$ 1 \ le r <m $（Pott等，2018）。这部分解决了Pott等人的问题5。 2018。然后，我们分析$ x^{2^r} {\ rm tr^n_m}（λ（x））$的功能。我们表明，对于奇数$ m $，只有$ x^{2^r} {\ rm tr^n_m}（x）$，$ 1 \ le r <m $，具有最大的非线性，而其中甚至更多的$ m $，我们再呈现一个无限级别的班级。详细研究，我们研究了WALSH频谱，差分光谱及其与功能的关系$ x^{2^r} {\ rm tr^n_m}（λ（x））$。我们的结果表明，此类包含许多非平凡的EA等价类别的功能类别，具有最大弯曲组件的数量，如果$ m $甚至是最大可能的非线性。
A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where $\tilde{F}(x)=(f_1(x),\ldots,f_m(x))$ is a vectorial bent function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, and $a_i$, $1\le i\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $F(x) = x^{2^r}{\rm Tr^n_m}(Λ(x))$, where $Λ$ is a linearized permutation of $\mathbb{F}_{2^m}$. In the first part of this article it is shown that plateaued functions with $2^n-2^m$ bent components can have nonlinearity at most $2^{n-1}-2^{\lfloor\frac{n+m}{2}\rfloor}$, a bound which is attained by the example $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$ (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form $x^{2^r}{\rm Tr^n_m}(Λ(x))$. We show that for odd $m$, only $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$, has maximal nonlinearity, whereas there are more of them for even $m$, of which we present one more infinite class explicitly. In detail, we investigate Walsh spectrum, differential spectrum and their relations for the functions $x^{2^r}{\rm Tr^n_m}(Λ(x))$. Our results indicate that this class contains many nontrivial EA-equivalence classes of functions with the maximal number of bent components, if $m$ is even, several with maximal possible nonlinearity.
