Investigations on cc-Boomerang Uniformity and Perfect Nonlinearity

We defined in~\cite{EFRST20} a new multiplicative $c$-differential, and the corresponding $c$-differential uniformity and we characterized the known perfect nonlinear functions with respect to this new concept, as well as the inverse in any characteristic. The work was continued in~\cite{RS20}, investigating the $c$-differential uniformity for some further APN functions. Here, we extend the concept to the boomerang uniformity, introduced at Eurocrypt '18 by Cid et al.~\cite{Cid18}, to evaluate S-boxes of block ciphers, and investigate it in the context of perfect nonlinearity and related functions.Comment: 31 pages, 1 figur

Characters, Weil sums and cc-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 the general case of perturbations via any linearized polynomial, we use characters in the finite field to express all entries in the $c$-Differential Distribution Table (DDT) of an $(n,n)$-function on the finite field \F_{p^n}, and further, we use that method to find explicit expressions for all entries of the $c$-DDT of the perturbed Gold function (via an arbitrary linearized polynomial).Comment: 22 page

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

Low cc-differential and cc-boomerang uniformity of the swapped inverse function

Modifying the binary inverse function in a variety of ways, like swapping two output points has been known to produce a $4$-differential uniform permutation function. Recently, in \cite{Li19} it was shown that this swapped version of the inverse function has boomerang uniformity exactly $10$, if $n\equiv 0\pmod 6$, $8$, if $n\equiv 3\pmod 6$, and 6, if $n\not\equiv 0\pmod 3$. Based upon the $c$-differential notion we defined in \cite{EFRST20} and $c$-boomerang uniformity from \cite{S20}, in this paper we characterize the $c$-differential and $c$-boomerang uniformity for the $(0,1)$-swapped inverse function in characteristic~$2$: we show that for all~$c\neq 1$, the $c$-differential uniformity is upper bounded by~$4$ and the $c$-boomerang uniformity by~$5$ with both bounds being attained for~$n\geq 4$.Comment: 25 page

A Degree Bound For The c-Boomerang Uniformity Of Permutation Monomials

Let $\mathbb{F}_q$ be a finite field of characteristic $p$. In this paper we prove that the $c$-Boomerang Uniformity, $c \neq 0$, for all permutation monomials $x^d$, where $d > 1$ and $p \nmid d$, is bounded by $d^2$. Further, we utilize this bound to estimate the $c$-boomerang uniformity of a large class of Generalized Triangular Dynamical Systems, a polynomial-based approach to describe cryptographic permutations, including the well-known Substitution-Permutation Network

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 βi (a basis of Fqn over Fq), some functions fi of c-differential uniformities δi , and Li (specific linearized polynomials defined in terms of βi), 1 ≤ i ≤ n, then F(x) = Pn i=1 βifi(Li(x)) has c-differential uniformity equal to Qn i=1 δi

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 βi (a basis of Fqn over Fq), some functions fi of c-differential uniformities δi , and Li (specific linearized polynomials defined in terms of βi), 1 ≤ i ≤ n, then F(x) = Pn i=1 βifi(Li(x)) has c-differential uniformity equal to Qn i=1 δi