Metrical properties of the set of bent functions in view of duality

In the paper, we give a review of metrical properties of the entire set of bent functions and its significant subclasses of self-dual and anti-self-dual bent functions. We present results for iterative construction of bent functions in n + 2 variables based on the concatenation of four bent functions and consider related open problem proposed by one of the authors. Criterion of self-duality of such functions is discussed. It is explored that the pair of sets of bent functions and affine functions as well as a pair of sets of self-dual and anti-self-dual bent functions in n > 4 variables is a pair of mutually maximally distant sets that implies metrical duality. Groups of automorphisms of the sets of bent functions and (anti-)self-dual bent functions are discussed. The solution to the problem of preserving bentness and the Hamming distance between bent function and its dual within automorphisms of the set of all Boolean functions in n variables is considered

О некоторых свойствах самодуальных обобщенных бент-функций

Бент-функции вида FI) ^ , где q ^ 2 — натуральное число, называются обобщёнными бент-функциями. Обобщённые бент-функции, для которых можно определить дуальную бент-функцию, называются регулярными. Регулярная обобщённая бент-функция называется самодуальной, если она совпадает со своей дуальной. Получены необходимые и достаточные условия самодуальности обобщённых бент-функций из класса Елисеева — Мэйорана — МакФарланда. Представлен полный спектр расстояний Ли между данными функциями. Доказано несуществование аффинных самодуальных обобщённых бент-функций. Приведён класс изомет- ричных отображений, сохраняющих самодуальность обобщённой бент-функции. С помощью данных отображений получена уточнённая классификация самодуальных бент-функций вида F| ^ Z4. Bent functions of the form Fn ^ , where q ^ 2 is a positive integer, are known as generalized bent (gbent) functions. A gbent function for which it is possible to define a dual gbent function is called regular. A regular gbent function is said to be self-dual if it coincides with its dual. We obtain the necessary and sufficient conditions for the self-duality of gbent functions from Eliseev — Maiorana — McFarland class. We find the complete Lee distance spectrum between all self-dual functions in this class and obtain that the minimal Lee distance between them is equal to q ■ 2n-3. For Boolean case, there are no affine bent functions and self-dual bent functions, while it is known that for generalized case affine bent functions exist, in particular, when q is divisible by 4. We prove the non-existence of affine self-dual gbent functions for any natural even q. A new class of isometries preserving self-duality of a gbent function is presented. Based on this, a refined classification of self-dual gbent functions of the form F2 ^ is given

The group of automorphisms of the set of self-dual bent functions

A bent function is a Boolean function in even number of variables which is on the maximal Hamming distance from the set of affine Boolean functions. It is called self-dual if it coincides with its dual. It is called anti-self-dual if it is equal to the negation of its dual. A mapping of the set of all Boolean functions in n variables to itself is said to be isometric if it preserves the Hamming distance. In this paper we study isometric mappings which preserve self-duality and anti-self-duality of a Boolean bent function. The complete characterization of these mappings is obtained for n>2. Based on this result, the set of isometric mappings which preserve the Rayleigh quotient of the Sylvester Hadamard matrix, is characterized. The Rayleigh quotient measures the Hamming distnace between bent function and its dual, so as a corollary, all isometric mappings which preserve bentness and the Hamming distance between bent function and its dual are described

Extended Formulations in Mixed-integer Convex Programming

We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201

Fourier duality for fractal measures with affine scales

For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in \br^d, and they both have the same matrix scaling. But the two use different translation vectors, one by a subset $B$ in \br^d, and the other by a related subset $L$. Among other things, we show that there is then a pair of infinite discrete sets $\Gamma(L)$ and $\Gamma(B)$ in \br^d such that the $\Gamma(L)$-Fourier exponentials are orthogonal in $L^2(\mu_B)$, and the $\Gamma(B)$-Fourier exponentials are orthogonal in $L^2(\mu_L)$. These sets of orthogonal "frequencies" are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line. Our duality pairs do not always yield orthonormal Fourier bases in the respective $L^2(\mu)$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.Comment: v

On metric complements and metric regularity in finite metric spaces

This review deals with the metric complements and metric regularity in the Boolean cube and in arbitrary finite metric spaces. Let A be an arbitrary subset of a finite metric space M, and A be the metric complement of A — the set of all points of M at the maximal possible distance from A. If the metric complement of the set A coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets was posed by N. Tokareva in 2012 when studying metric properties of bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this paper, main known problems and results concerning the metric regularity are overviewed, such as the problem of finding the largest and the smallest metrically regular sets, both in the general case and in the case of fixed covering radius, and the problem of obtaining metric complements and establishing metric regularity of linear codes. Results concerning metric regularity of partition sets of functions and Reed — Muller codes are presented

Quantum algorithms for highly non-linear Boolean functions

Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups--which generalize Shor's factoring algorithm perhaps most faithfully--only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of the paper contains a new exponential separation between classical and quantum query complexit