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In Eurocrypt’18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over $\mathbb {F}_{2^{2m}}$ with $m$ odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined. [ABSTRACT FROM AUTHOR]

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for $x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$ being involutions over the finite field ${\mathbb F}_{q}$ , where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ from given involutions over some subgroups of ${\mathbb F}_{q}^{*}$ by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ are obtained. [ABSTRACT FROM AUTHOR]

In 2016, Chang et al. proposed a Format Preserving Encryption (FPE) scheme over a finite field and used an MDS matrix in the diffusion layer of the scheme for optimal diffusion. Later that year, Gupta et al. defined an algebraic structure named Format Preserving Set (FPS) is the diffusion layer of an FPE scheme. In 2018, Barua et al. showed that it is not possible to construct an FPS over a finite field in the diffusion layer of an FPE scheme if the cardinality of the set is not a power of prime. They extended the search of FPS over a finite commutative ring $\mathcal {R}$ and showed that if an FPS $S \subseteq \mathcal {R}$ is closed under addition then it gets module structure over some subring of $\mathcal {R}$. Moreover, in this case, the only possible cardinalities of FPS are some power of the cardinalities of subrings when the module is free. The purpose of this article is twofold. Firstly, we show that it is possible to construct format preserving sets over a finite commutative ring which are not closed under addition. Secondly, we search for format preserving sets and MDS matrices over torsion modules. We provide examples of format preserving sets of cardinalities 26 and 52 over torsion modules and rings. These cardinalities are interesting because they correspond to the set of English alphabets, without and with capitalization. By considering a finite Abelian group as a torsion module over a PID, we show that a matrix $M$ with entries from the PID is MDS if and only if $M$ is MDS under the projection map on the same Abelian group. [ABSTRACT FROM AUTHOR]

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria. [ABSTRACT FROM AUTHOR]

We study ${q}$ -ary linear codes ${C}$ obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of ${C}$ over all field extensions of $\mathbb {F}_{q}$. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code ${C}$. [ABSTRACT FROM AUTHOR]