Vectorial Functions Research Articles

A Note on APN Permutations and Their Derivatives

Prior to the discovery of an APN permutation in six dimension it was conjectured that such functions do not exist in even dimension, as none had been found at that time. However, finding APN permutations in even dimension ≥8 remains a significant challenge. Understanding and determining more properties of these functions is a crucial approach to discovering them. In this note, we study the properties of vectorial Boolean functions based on the weights of the first-order and second-order derivatives of their components. We show that a function is an APN permutation if and only if the sum of the squares of the weights of the first-order derivatives of its components is exactly 22n−1(2n−1+1)(2n−1). Additionally, we determined that the sum of the weights of the second-order derivatives of the components of any vectorial Boolean function is at most 22n−1(2n−1)(2n−2). This bound is achieved if and only if a function is APN.

On Second-Order Derivatives of Boolean Functions and Cubic APN Permutations in Even Dimension

The big APN problem is one of the most important challenges in the theory of Boolean functions, i.e. finding a new APN permutation in even dimension. Among this class of functions, those with the lowest possible degree are cubic. Yet, none has been found so far. In this paper, we introduce new parameters for Boolean functions and for vectorial Boolean functions, mostly derived from the behavior of their second-order derivatives. These parameters are invariant under extended affine equivalence, and they are particularly relevant for small-degree functions. They allow studying bent, semi-bent and APN functions of degrees two and three. In particular, they allow tackling the big APN problem for cubic permutations. Notably, we focus on the case of dimension 8, providing some computational results.

Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations

The Hilbert spaces H(curl) and H(div) are employed in various variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Seeing as the well-posedness follows automatically from the continuous setting to the discrete setting in the presence of commuting interpolants as per Fortin’s criterion, the construction of conforming subspaces becomes a crucial step in the formulation of stable numerical schemes. This work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference element via template vectors associated with the geometric polytopes of the element and an underlying H1-conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying H1-conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. We prove a linearly independent construction of Nédélec elements of the first and second type, Brezzi–Douglas–Marini elements, and Raviart–Thomas elements on triangulations and tetrahedralizations. The application of the method is demonstrated with two examples in the relaxed micromorphic model.

Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

AbstractWe consider the quasilinear diffusion problem of for an open set , , for , and any . Here, denotes an operator which may involve the distributional Riesz fractional gradient of order , with , the classical gradient or/and nonlocal derivatives , with . We show global existence results for various quasilinear diffusion systems in nondivergence form for linear elliptic operators , including classical elliptic systems, anisotropic fractional equations and systems, and anisotropic local and nonlocal operators of the following type: for coercive, invertible matrices and suitable vectorial functions .

Distance between vectorial Boolean functions and affine analogues (following the Eighth International Olympiad in Cryptography)

Изучается расстояние Хэмминга от векторной булевой функции до множества аффинных отображений (нелинейность векторной функции). Получены новые верхняя граница нелинейности векторных функций и нижняя граница нелинейности для отображений с заданной дифференциальной равномерностью, которые уточняют ранее известные. Найдена зависимость расстояния Хэмминга между векторной функцией и аффинным отображением от коэффициентов Уолша - Адамара ненулевых линейных комбинаций координат векторной функции, позволяющая дать оценки нелинейности через указанные коэффициенты.

The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions.

Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (such as few-weight codes or minimal codes for secret sharing, locally recoverable codes for storage, etc.). On the other hand, various criteria and tables have been introduced to analyse the security of S-boxes that are related to vectorial Boolean functions, such as the Differential Distribution Table (DDT), the Boomerang Connectivity Table (BCT), and the Differential-Linear Connectivity Table (DLCT). In previous years, two new tables have been proposed for which the literature was pretty abundant: the c-DDT to extend the DDT and the c-BCT to extend the BCT. In the same vein, we propose extended concepts to study further the security of vectorial Boolean functions, especially the c-Walsh transform, the c-autocorrelation, and the c-differential-linear uniformity and its accompanying table, the c-Differential-Linear Connectivity Table (c-DLCT). We study the properties of these novel functions at their optimal level concerning these concepts and describe the c-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, we draw new ideas for future research toward linear code designs.

Parameterization of Boolean functions by vectorial functions and associated constructions

<p style='text-indent:20px;'>Too few general constructions of Boolean functions satisfying all cryptographic criteria are known. We investigate the construction in which the support of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> equals the image set of an injective vectorial function <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula>, that we call a parameterization of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Every balanced Boolean function can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes:</p><p style='text-indent:20px;'>- the sums of indicators of disjoint graphs of <inline-formula><tex-math id="M4">\begin{document}$ (k,n-k $\end{document}</tex-math></inline-formula>)-functions,</p><p style='text-indent:20px;'>- functions parameterized through <inline-formula><tex-math id="M5">\begin{document}$ (n-1,n) $\end{document}</tex-math></inline-formula>-functions due to Beelen and Leander.</p><p style='text-indent:20px;'>We study the cryptographic parameters of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula>, that we relate to the algebraic degrees of <inline-formula><tex-math id="M7">\begin{document}$ F $\end{document}</tex-math></inline-formula> and of its graph indicator, the nonlinearity of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula>, that we relate by a bound to the nonlinearity of <inline-formula><tex-math id="M9">\begin{document}$ F $\end{document}</tex-math></inline-formula>, and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be approached (in two ways) by using the graph indicator of <inline-formula><tex-math id="M10">\begin{document}$ F $\end{document}</tex-math></inline-formula>. We revisit each of the five classes for each criterion. The fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen <inline-formula><tex-math id="M11">\begin{document}$ (k,n-k $\end{document}</tex-math></inline-formula>)-functions. The sub-class of the sums of indicators of affine functions, for which we prove an upper bound and a lower bound on the nonlinearity, seems also interesting. The fifth class includes functions with an optimal algebraic degree, good nonlinearity and good AI.</p>

Vectorial bent functions with non-weakly regular components

Vectorial bent functions with non-weakly regular components

Equivalence for generalized Boolean functions

Equivalence plays a key-role for the classification of functions between elementary abelian groups $ {{\mathbb V}}_n^{(p)} $ and $ {{\mathbb V}}_k^{(p)} $. One distinguishes between affine equivalence, extended affine (EA) equivalence, and the most general CCZ-equivalence. Recently, there has been an increased interest in functions from elementary abelian groups $ {{\mathbb V}}_n^{(p)} $ to cyclic groups $ {{\mathbb Z}}_{p^k} $. We initiate the study of equivalence for functions from $ {{\mathbb V}}_n^{(p)} $ to $ {{\mathbb Z}}_{p^k} $. We show that CCZ-equivalence is more general than EA-equivalence. For some classes of functions, CCZ-equivalence reduces to EA-equivalence. We show that CCZ-equivalence between two functions from $ {{\mathbb V}}_n^{(p)} $ to $ {{\mathbb Z}}_{p^k} $ implies CCZ-equivalence of two associated vectorial functions from $ {{\mathbb V}}_n^{(p)} $ to $ {{\mathbb V}}_k^{(p)} $.

Nonlinearity of vectorial functions over finite fields

Нелинейность векторной функции определена как расстояние Хэмминга до множества аффинных отображений. Установлена связь параметров, характеризующих нелинейность, с коэффициентами Фурье характеров векторной функции. На ее основе показана возможность нахождения параметров нелинейности отображения через аналогичные параметры его составляющих при различных видах декомпозиции. Представлена универсальная верхняя граница нелинейности, получены выражения границ нелинейности через коэффициенты Фурье характеров, позволяющие уточнить известные ранее границы для некоторых классов отображений. Найдена зависимость нижней границы нелинейности от дифференциальной равномерности.

On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property

On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property

Correction of the article "On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property"

Correction of the article "On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property"

Identification of postnatal development dependent genes and proteins in porcine epididymis

BackgroundThe epididymis is a highly regionalized tubular organ possesses vectorial functions of sperm concentration, maturation, transport, and storage. The epididymis-expressed genes and proteins are characterized by regional and developmental dependent pattern. However, a systematic and comprehensive insight into the postnatal development dependent changes in gene and protein expressions of porcine epididymis is still lacking. Here, the RNA and protein of epididymis of Duroc pigs at different postnatal development stages were extracted by using commercial RNeasy Midi kit and extraction buffer (7 M Urea, 2 M thiourea, 3% CHAPS, and 1 mM PMSF) combined with sonication, respectively, which were further subjected to transcriptomic and proteomic profiling.ResultsTranscriptome analysis indicated that 198 and 163 differentially expressed genes (DEGs) were continuously up-regulated and down-regulated along with postnatal development stage changes, respectively. Most of the up-regulated DEGs linked to functions of endoplasmic reticulum and lysosome, while the down-regulated DEGs mainly related to molecular process of extracellular matrix. Moreover, the following key genes INSIG1, PGRMC1, NPC2, GBA, MMP2, MMP14, SFRP1, ELN, WNT-2, COL3A1, and SPARC were highlighted. A total of 49 differentially expressed proteins (DEPs) corresponding to postnatal development stages changes were uncovered by the proteome analysis. Several key proteins ACSL3 and ACADM, VDAC1 and VDAC2, and KNG1, SERPINB1, C3, and TF implicated in fatty acid metabolism, voltage-gated ion channel assembly, and apoptotic and immune processes were emphasized. In the integrative network, the key genes and proteins formed different clusters and showed strong interactions. Additionally, NPC2, COL3A1, C3, and VDAC1 are located at the hub position in each cluster.ConclusionsThe identified postnatal development dependent genes and proteins in the present study will pave the way for shedding light on the molecular basis of porcine epididymis functions and are useful for further studies on the specific regulation mechanisms responsible for epididymal sperm maturation.

Further Results on Autocorrelation of Vectorial Boolean Functions

In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound of $\sum_{u\in \mathbb{F}_2^n,v\in \mathbb{F}_2^m}\mathcal{W}_F^3(u,v)$. Secondly, based on the Walsh-Hadamard transform, we give a secondary construction of vectorial bent functions. Further, three kinds of sum-of-squares indicators of vectorial Boolean functions are defined by autocorrelation function and the lower and upper bounds of the sum-of-squares indicators are derived. Finally, we study the sum-of-squares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.

Value Distributions of Perfect Nonlinear Functions

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.

Vectorial bent functions and linear codes from quadratic forms

Vectorial bent functions and linear codes from quadratic forms

Automatic refinement of ephemeris‐derived range‐Doppler model through the least‐squares matching of Sentinel‐1 Single‐Look Complex images and DEM in mountainous areas

AbstractRange‐Doppler model is a transformation that relates ground and Single‐Look Complex SAR image spaces by knowing the instantaneous position and velocity of the sensor in its trajectory. Its instantaneous position and velocities are determined with the help of vectorial functions, which their precisions directly influence the accuracy of the range‐Doppler model. Ephemeris data is one of the sources used for the estimation of temporal functions. Some reasons can cause uncertainty in the temporal function directly derived from the satellite navigation data. Ground control points are other alternatives for calibration of the range‐Doppler model whose visual measurements are difficult in the mountainous areas of Sentinel‐1 Single‐Look Complex images due to their resolutions and distortions. The temporal functions of the range‐Doppler model are refined through numerous control points extracted from the automatic matching between Single‐Look Complex images and Digital Elevation Model. For this purpose, the radiometric and geometric similarities of the Single‐Look Complex images and Digital Elevation Model in mountainous areas are first increased based on the initial range‐Doppler model, and then their geometric discrepancies are compensated with the least‐squares image matching method. The results indicated that the proposed method in mountainous areas could automatically compensate for up to 30 pixels errors and reach sub‐pixel accuracies.

Characteristics of nonlinearity of vectorial functions over finite fields

Рассмотрены характеристики нелинейности векторной функции, определенной на векторном пространстве над конечным полем, а именно нелинейность (расстояние Хэмминга между множеством нетривиальных линейных комбинаций ее координатных функций и множеством аффинных функций), дифференциальная равномерность и другое понятие нелинейности (расстояние Хэмминга от векторной функции до множества аффинных отображений). Продемонстрирован метод построения векторных функций с высокой нелинейностью по всем характеристикам. Найдены значения этих характеристик для подстановок, заданных степенными функциями, и подстановок, определенных ГОСТ Р 34.11-94.

Minimal Binary Linear Codes From Vectorial Boolean Functions

Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, by studying deeply the construction of linear codes in this paper, we find a necessary and sufficient condition of the linear codes to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared the parameters with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.

Differential and Linear properties of vectorial boolean functions based on chi

To evaluate the security of a cryptographic primitive, investigating its resistance against differential and linear cryptanalysis is required. Many modern cryptographic primitives repeatedly apply similar round functions alternated with the addition of round keys or constants. A round function usually consists of a non-linear mapping and a number of linear mappings. The non-linear mapping χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} is used in different cryptographic primitives such as Keccak and Subterranean. An alternative version of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} is used in Ascon and the non-linear layer of Simon has the same differential and linear properties of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document}. The mapping χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} can be applied to strings with different lengths. For instance, it can be parallelly applied to small-length strings as in Keccak, where it works on 5-bit strings, or it can be applied to big-length strings as in Subterranean, where it works on a string of length 257. Investigating the differential and linear properties of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} working on alternative lengths of strings, provides useful information to designers to make a better choice for the non-linear layer. Some differential properties of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} have been analyzed in [8] and in this work we provide a revised presentation of them. We then extend this study and we analyze linear propagation properties of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document}. Thanks to these additional results, we extend the comparison between the application of parallel instances of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} on small-length strings and the application of a single instance of χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document} on a big-length string. We show how we can apply the results of this study also to the non-linear layers of Ascon and Simon thanks to their affine-equivalence with χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document}.