Low c-differential uniformity for functions modified on subfields

Abstract

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 ${\beta }_i$), $1\le i\le n$, then $F(x)={\sum }_{i=1}^n{\beta }_i{f}_i(L_i(x))$ has $c$-differential uniformity equal to ${\prod }_{i=1}^n{\delta }_i$.