In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential uniformity and show several results. For example, we prove that given $\beta_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $\delta_i$, and $L_i$ (specific linearized polynomials defined in terms of $\beta_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^n\beta_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{i=1}^n \delta_i$.
翻译:在本文中, 我们构建了某些按片段定义的函数, 并研究它们的美元差异统一性。 作为副产品, 我们改进了先前的几项结果 。 此外, 我们审视了功能的组合, 差异性差强, 并展示了几项结果 。 例如, 我们证明给$\beta_ i 美元( 基数为$\\ mathbb{ F ⁇ q ⁇ n} 超过$\ mathbb{ F ⁇ Q ⁇ qqqq 美元), 一些函数 $_ i 美元( 美元) 和 $_ i ( 以 $\ delta_ 美元为定义的特定线性多元), $\leq i\ leq n美元, 然后$F (x)\ sup ⁇ i= 1\\\\\ beta_ i ( L_ (x) 美元) 等价美元( =\ prod ⁇ = 1\\\\ delta_ i) 美元) 。