On the c-differential uniformity of certain maps over finite fields

Low c-differential uniformity for functions modified on subfields

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 $β_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $δ_i$, and $L_i$ (specific linearized polynomials defined in terms of $β_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^nβ_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{…

Characters, Weil sums and $c$-differential uniformity with an application to the perturbed Gold function

Building upon the observation that the newly defined~\cite{EFRST20} concept of $c$-differential uniformity is not invariant under EA or CCZ-equivalence~\cite{SPRS20}, we showed in~\cite{SG20} that adding some appropriate linearized monomials increases the $c$-differential uniformity of the inverse function, significantly, for some~$c$. We continue that investigation here. First, by analyzing the involved equations, we find bounds for the uniformity of the Gold function perturbed by a single monomial, exhibiting the discrepancy we previously observed on the inverse function. Secondly, to treat …

Some new classes of (almost) perfect $c$-nonlinear permutations

The concept of differential uniformity was recently extended to the $c$-differential uniformity. An interesting problem in this area is the construction of functions with low $c$-differential uniformity and a lot of research has been done in this direction in the recent past. Here, we present three classes of (almost) perfect $c$-nonlinear permutations over finite fields of even characteristic.

On the Boomerang Uniformity of Permutations of Low Carlitz Rank

Finding permutation polynomials with low differential and boomerang uniformityis an important topic in S-box designs of many block ciphers. For example, AES chooses the inverse function as its S-box, which is differentially 4-uniform and boomerang 6-uniform. Also there has been considerable research on many non-quadratic permutations which are obtained by modifying certain set of points from the inverse function. In this paper, we give a novel approach that shows that plenty of existing modifications of the inverse function are in fact affine equivalent to permutations of low Carlitz rank and …

有限域上某些置换多项式的微分性质

The differential properties of certain permutation polynomials over finite fields

该文研究有限域上几类置换多项式的c-差分均匀度，新增四类偶特征域完美c-非线性置换多项式及一类特征3域低差分均匀度置换多项式，运用数论技术求出相关Walsh系数与Weil和。

On construction and (non)existence of $c$-(almost) perfect nonlinear functions

Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low $c$-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results.