Affine Equivalency and Nonlinearity Preserving Bijective Mappings over $$\mathbb {F}_2$$

The multiplicative complexity of an S-box over a finite field is the minimum number of multiplications needed to implement the S-box as an arithmetic circuit. In this paper we fully characterize bijective S-boxes with multiplicative complexity 1 up to affine equivalence over any finite field. We show that under affine equivalence in odd characteristic there are two classes of bijective functions and in even characteristic there are three classes of bijective functions with multiplicative complexity 1. Moreover, in (Jeon et al., Cryptogr. Commun., 14(4), 849-874 (2022)) A-boxes where introduced to lower bound the differential uniformity of an S-box over F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2}^{n}$$\\end{document} via its multiplicative complexity. We generalize this concept to arbitrary finite fields. In particular, we show that the differential uniformity of a (n, m)-S-box over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{q}$$\\end{document} is at least qn-l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^{n - l}$$\\end{document}, where ⌊n-12⌋+l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lfloor \\frac{n - 1}{2} \\rfloor + l$$\\end{document} is the multiplicative complexity of the S-box.

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

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)} $.

The Goldreich–Levin algorithm was originally proposed for a cryptographic purpose and then applied to learning. The algorithm is to find some larger Walsh coefficients of an n variable Boolean function. Roughly speaking, it takes a \(poly(n,\frac{1}{\epsilon }\log \frac{1}{\delta })\) time to output the vectors w with Walsh coefficients \(S(w)\ge \epsilon \) with probability at least \(1-\delta \). However, in this paper, a quantum algorithm for this problem is given with query complexity \(O(\frac{\log \frac{1}{\delta }}{\epsilon ^4})\), which is independent of n. Furthermore, the quantum algorithm is generalized to apply for an n variable m output Boolean function F with query complexity \(O(2^m\frac{\log \frac{1}{\delta }}{\epsilon ^4})\).

Boolean functions can be used in Cryptography (especially, the global avalanche characteristics of one Boolean function is an important property in symmetric Cipher). In this paper, when an n-variable balanced Boolean function satisfies the minimum the sum-of-squares indicator, we give some new properties of \({(n-1)}\)-variable decomposition Boolean functions. Meanwhile, we derive a new condition on the sum-of-squares indicator, if the sum-of-squares indicator of a balanced Boolean function with n-variable is greater than \({2^{2n}+2^{n+3}}\) for \({n\ge 3}\).

S-boxes, typically the only nonlinear part of a block cipher, are the heart of symmetric cryptographic primitives. They significantly impact the cryptographic strength and the implementation characteristics of an algorithm. Due to their simplicity, quadratic vectorial Boolean functions are preferred when efficient implementations for a variety of applications are of concern. Many characteristics of a function stay invariant under affine equivalence. So far, all 6-bit Boolean functions, 3- and 4-bit permutations have been classified up to affine equivalence. At FSE 2017, Bozoliv et al. presented the first classification of 5-bit quadratic permutations. In this work, we propose an adaptation of their work resulting in a highly efficient algorithm to classify n x m functions for n ≥ m. Our algorithm enables for the first time a complete classification of 6-bit quadratic permutations as well as all balanced quadratic functions for n ≤ 6. These functions can be valuable for new cryptographic algorithm designs with efficient multi-party computation or side-channel analysis resistance as goal. In addition, we provide a second tool for finding decompositions of length two. We demonstrate its use by decomposing existing higher degree S-boxes and constructing new S-boxes with good cryptographic and implementation properties.

Differential cryptanalysis is a method of attacking iterated mappings based on differences known as characteristics. The probability of a given characteristic is derived from the XOR tables associated with the iterated mapping. If π is a mapping π: Z m2 , then for each Δ, X, ΔY ε Z m2 the XOR table for π gives the number of input pairs of difference ΔX=X+X′ for which gp(X)+π(X′)=ΔY.The complexity of a differential attack depends upon two properties of the XOR tables: the density of zero entries in the table, and the size of the largest entry in the table. In this paper we present the first results on the expected values of these properties for a general class of mappings π. We prove that if π: Z m2 → Z m2 is a bijective mapping, then the expected size of the largest entry in the XOR table for π is bounded by 2m, while the fraction of the XOR table that is zero approaches e −1/2=0.60653. We are then able to demonstrate that there are easily constructed classes of iterated mappings for which the probability of a differential-like attack succeeding is very small.

Geometrically, curves and surfaces are usually considered to be sets of points with some special properties, living in a space consisting of “points.” Typically, one is also interested in geometric properties invariant under certain transformations, for example, translations, rotations, projections, etc. One could model the space of points as a vector space, but this is not very satisfactory for a number of reasons. One reason is that the point corresponding to the zero vector (0), called the origin, plays a special role, when there is really no reason to have a privileged origin. Another reason is that certain notions, such as parallelism, are handled in an awkward manner. But the deeper reason is that vector spaces and affine spaces really have different geometries. The geometric properties of a vector space are invariant under the group of bijective linear maps, whereas the geometric properties of an affine space are invariant under the group of bijective affine maps, and these two groups are not isomorphic. Roughly speaking, there are more affine maps than linear maps.

Affine equivalence for cubic rotation symmetric Boolean functions with [formula omitted] variables

Let T be a countable complete theory and C(T) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C(T). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C(T) determine T?”There is some obvious limitation: for example let T0 be the theory of infinite sets (in a language containing only =) and T1 the theory, in the language ( =, U(ν0),f(ν0)) stating that:(1) U is infinite.(2)f is a bijective map from U onto its complement.It is quite easy to see that C(T0) is equivalent to C(T1). But, in this case, T0 and T1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T, defined as follows:(1) The objects are the formulas in the given language.(2) The morphisms from into are the formulas such that(i.e. f defines a map from ϕ into ϕ; two morphisms defining the same map in all models of T should be identified).

In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map can be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices

Let F q be a finite field with q elements, n(≥2) a positive integer, and T n (q) the semigroup of all n × n upper triangular matrices over F q . The generalized Cayley graph GCay(T n (q)) of T n (q) is a directed graph with vertex set T n (q), in which there is a directed edge from a vertex A to a distinct vertex B if and only if B = XAY for some X, Y ∈ T n (q). The main result of this article proves that a bijective map σ is an automorphism of GCay(T n (q)) if and only if, for any vertex A of GCay(T n (q)), either σ(A) = P A AQ A or σ(A) = P A JA t JQ A , where A t denotes the transpose of A, , and P A and Q A are invertible upper triangular matrices depending on A.

We undertake a systematic study of the so-called 2-adic ring C^* -algebra \mathcal Q_2 . This is the universal C^* -algebra generated by a unitary U and an isometry S_2 such that S_2U=U^2S_2 and S_2S_2^*+US_2S_2^*U^*=1 . Notably, it contains a copy of the Cuntz algebra \mathcal O_2=C^*(S_1, S_2) through the injective homomorphism mapping S_1 to US_2 . Among the main results, the relative commutant C^*(S_2)'\cap \mathcal Q_2 is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion \mathcal O_2\subset\mathcal Q_2 , namely the endomorphisms of \mathcal Q_2 that restrict to the identity on \mathcal O_2 are actually the identity on the whole \mathcal Q_2 . Moreover, there is no conditional expectation from \mathcal Q_2 onto \mathcal O_2 . As for the inner structure of \mathcal Q_2 , the diagonal subalgebra \mathcal D_2 and C^*(U) are both proved to be maximal abelian in \mathcal Q_2 . The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of \mathcal Q_2 . In particular, the semigroup of the endomorphisms fixing U turns out to be a maximal abelian subgroup of Aut (\mathcal Q_2) topologically isomorphic with C(\mathbb{T},\mathbb{T}) . Finally, it is shown by an explicit construction that Out (\mathcal Q_2) is uncountable and non-abelian.

Landmark constrained genus-one surface Teichmüller map applied to surface registration in medical imaging.