The c-differential uniformity and boomerang uniformity of two classes of permutation polynomials

The article of record as published may be found at http://dx.doi.org/10.1109/TIT.2021.3123104The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the func- tion’s resistance against differential cryptanalysis. This concept was extended recently to c-DDT and c-differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the c-DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its c-differential and boomerang uniformities.The research of Sartaj Ul Hasan is partially supported by MATRICS grant MTR/2019/000744 from the Science and Engineering Research Board, Government of India. Pantelimon Stănică acknowledges the sabbatical support from Naval Postgraduate School from September 2020 to July 2021

P℘\wpN functions, complete mappings and quasigroup difference sets

We investigate pairs of permutations $F,G$ of $\mathbb{F}_{p^n}$ such that $F(x+a)-G(x)$ is a permutation for every $a\in\mathbb{F}_{p^n}$. We show that necessarily $G(x) = \wp(F(x))$ for some complete mapping $-\wp$ of $\mathbb{F}_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear (P$\wp$N) function. If $\wp(x) = cx$, then $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on $\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain a quasigroup, and show that the graph of a P$\wp$N function $F$ is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P$\wp$N functions, respectively, the difference sets in the corresponding quasigroup

The c−c-differential uniformity and boomerang uniformity of three classes of permutation polynomials over F2n\mathbb{F}_{2^n}

Permutation polynomials with low $c$-differential uniformity and boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over $\mathbb{F}_{2^n}$, we determine the $c$-differential uniformity and boomerang uniformity of these permutation polynomials: (1) $f_1(x)=x+\mathrm{Tr}_1^n(x^{2^{k+1}+1}+x^3+x+ux)$, where $n=2k+1$, $u\in\mathbb{F}_{2^n}$ with $\mathrm{Tr}_1^n(u)=1$; (2) $f_2(x)=x+\mathrm{Tr}_1^n(x^{{2^k}+3}+(x+1)^{2^k+3})$, where $n=2k+1$; (3) $f_3(x)=x^{-1}+\mathrm{Tr}_1^n((x^{-1}+1)^d+x^{-d})$, where $n$ is even and $d$ is a positive integer. The results show that the involutions $f_1(x)$ and $f_2(x)$ are APcN functions for $c\in\mathbb{F}_{2^n}\backslash \{0,1\}$. Moreover, the boomerang uniformity of $f_1(x)$ and $f_2(x)$ can attain $2^n$. Furthermore, we generalize some previous works and derive the upper bounds on the $c$-differential uniformity and boomerang uniformity of $f_3(x)$

Differentially low uniform permutations from known 4-uniform functions

Functions with low differential uniformity can be used in a block cipher as S-boxes since they have good resistance to differential attacks. In this paper we consider piecewise constructions for permutations with low differential uniformity. In particular, we give two constructions of differentially 6-uniform functions, modifying the Gold function and the Bracken–Leander function on a subfield.publishedVersio

Relation between o-equivalence and EA-equivalence for Niho bent functions

Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.publishedVersio