Equivalence for generalized Boolean functions

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

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.

For functions over the finite field $$\mathbb {F}_{p^n}$$ , we study a generalization of affine equivalence. Two functions of algebraic degree d are equivalent if, when ignoring monomials of algebraic degree $$d-1$$ or less, they coincide with two affine equivalent functions. This equivalence appears naturally for functions defined by the properties of their derivatives of order $$d-1$$ . In order to test this equivalence, as usual, invariants can be used as a first approach. We describe several such invariants, based on the kernel/image of the derivatives of order $$d-1$$ and on the set of vectors orthogonal to those images (similar to the orthoderivatives used as invariants for quadratic APN functions). We also define a canonical form with respect to left composition with linear transformations, which decreases the computational effort of testing for equivalence. We then apply these techniques and computer search to classify all GAPN (generalized APN) functions of algebraic degree 3 over $$\mathbb {F}_{3^3}$$ . We determined that there are exactly 31 equivalence classes; for each class we list, firstly, a representative in multivariate algebraic normal form which is in canonical form with respect to left composition with linear transformations; secondly, we also list a representative with a shortest univariate representation. Moreover, our computations show that all of the GAPN functions of algebraic degree 3 over $$\mathbb {F}_{3^3}$$ have optimal second-order differential uniformity. While this is not always the case for other values of p and n , we show that the GAPN functions of algebraic degree p over $$\mathbb {F}_{p^n}$$ of the form $$x^{p-1+p^{\ell }}$$ with $$\gcd (\ell ,n)=1$$ do have optimal order- $$(p-1)$$ differential uniformity.

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.

We show that, for a positive integer r, every minimal 1-saturating set in ${PG}(r-1,2)$ of size at least $\frac{11}{36}\,2^r+3$ either is a complete cap or can be obtained from a complete cap S by fixing some $s\in S$ and replacing every point $s'\in S\setminus\{s\}$ by the third point on the line through s and $s'$. Since, conversely, every set obtained in this way is a minimal 1-saturating set and the structure of large sum-free sets in an elementary abelian 2-group is known, this provides a complete description of large minimal 1-saturating sets. An algebraic restatement is as follows. Suppose that G is an elementary abelian 2-group and a subset $A\subseteq G\setminus\{0\}$ satisfies $A\cup2A=G$ and is minimal subject to this condition. If $|A|\ge\frac{11}{36}\,|G|+3$, then either A is a maximal sum-free set or there are a maximal sum-free set $S\subseteq G$ and an element $s\in S$ such that $A=\{s\}\cup\bigl(s+(S\setminus\{s\})\bigr)$. Our approach is based on characterizing those large sets A in elementary abelian 2-groups such that, for every proper subset B of A, the sumset $2B$ is a proper subset of $2A$.

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.

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

We first give a proof of an isomorphism between the group of affine equivalent maps and the automorphism group of Sylvester Hada-mard matrices. Secondly, we prove the existence of new nonlinearity preserving bijective mappings without explicit construction. Continuing the study of the group of nonlinearity preserving bijective mappings acting on \(n\)-variable Boolean functions, we further give the exact number of those mappings for \(n\,\le \,6\). Moreover, we observe that it is more beneficial to study the automorphism group of bijective mappings as a subgroup of the symmetric group of the \(2^{n}\) dimensional \(\mathbb {F}_{2}\)-vector space due to the existence of non-affine mapping classes.

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is on the Bieberbach groups with elementary abelian 2-group point group, The central subgroup of the nonabelian tensor square of a group is generated by for all in The purpose of this paper is to compute the central subgroups of the nonabelian tensor squares of two Bieberbach groups with elementary abelian 2-point group of dimension three.

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Elementary proof that [formula omitted] is a DCI-group

Finite Groups Admitting Automorphisms Whose Fixed Point Set is an Elementary Abelian 2-Group

Perfect Isometries for Principal Blocks with Abelian Defect Groups and Elementary Abelian 2-Inertial Quotients

For $2 \leqq c \leqq \infty $, the c-linear complexity $L_c (A)$ of a complex matrix A is defined as the minimal number of additions, subtractions, and multiplications by complex constants of absolute value $ \leqq c$, sufficient to evaluate A at a generic input vector. It is shown that if A is a Fourier transform on the finite group G, then $| L_\infty (A) - L_\infty (A^{ - 1} ) | \leqq | G |$. $L_c (G): = \min \{ L_c (A)|A{\text{ a Fourier transform for }}G\} $ is called the c-linear complexity of the finite group G. It is proved that $L_2 (G) > \frac{1}{4}| G |\log | G |$ for any finite group G, and two infinite classes of non-abelian groups G with $L_2 (G) \leqq 0.6| G |\log | G |$ and $L_2 (G) \leqq 0.8| G |\log | G |$, respectively, are presented. Thus there are non-abelian groups with even faster Fourier transforms than elementary abelian 2-groups (for which $L_2 (G) \leqq | G |\log | G |$)!

Let K be a quadratic extension field of the rational numbers Q. Let C K be the 2-class group of K in the narrow sense. It is a classical result that rank C K= t l , where t is the number of primes that ramify in K/Q. Now let C~ = {ai: a e CK}, and let R K denote the 4-class rank of K in the narrow sense; i.e., RK= rank ~Kc z = dimv2 ttc'Z/t~41~K/~KJ" Here F 2 is the finite field with two elements, and 2 4 CK/C K is an elementary abelian 2-group which we are viewing as a vector space o v e r F 2. Given a quadratic field K, one can compute R K by computing the rank (over F2) of a certain matrix of Legendre symbols (cf. [11]). Now assume K is imaginary quadratic. So K=Q(1/~mm), where m is a square-free positive integer. For each positive integer t, each nonnegative integer e, and each positive real number x, we define