Exact relations between nonlinearity and algebraic immunity

We sharpen some lower bounds on the higher order nonlinearity of a Boolean function in terms of the value of its algebraic immunity and obtain new tight bounds. We prove a universal tight lower bound, which enables us to reduce the problem of estimating higher order nonlinearity to finding the dimension of certain linear subspaces in the space of Boolean functions. As a simple corollary of this result, we obtain all previously known estimates in this area. For polynomials with disjoint terms, finding the dimension of those linear subspaces reduces to a simple combinatorial inspection. We prove a tight lower bound on the second order nonlinearity of a Boolean function in terms of the value of its algebraic immunity.     

Binary linear codes from vectorial boolean functions and their weight distribution

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contain the all-one codeword, and determine their weight distribution. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.’s generic construction.     

The asymptotically best method for synthesizing limited-depth Boolean recursive schemes

A model of limited-depth recursive schemes for the functions of Boolean algebra (Boolean functions), constructed from multi-output functional elements, is considered. A lower estimate of the Shannon function for the complexity of schemes of this class is derived. Upper estimates for the complexity of some specific functions and systems of functions in this class of schemes are obtained. A method is proposed for synthesizing schemes of this class for arbitrary functions that allow us (using the derived lower estimate) to determine the asymptotics of the Shannon function for their complexity.     

Generating functions for real character degree sums of finite general linear and unitary groups

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius–Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall–Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius–Schur indicator 1 or ?1.     

A note on APN permutations in even dimension

APN permutations in even dimension are vectorial Boolean functions that play a special role in the design of block ciphers. We study their properties, providing some general results and some applications to the low-dimension cases. In particular, we prove that none of their components can be quadratic. For an APN vectorial Boolean function (in even dimension) with all cubic components we prove the existence of a component having a large number of balanced derivatives. Using these restrictions, we obtain the first theoretical proof of the non-existence of APN permutations in dimension 4. Moreover, we derive some constraints on APN permutations in dimension 6.     

Reachability analysis of linear systems using support functions

This work is concerned with the algorithmic reachability analysis of continuous-time linear systems with constrained initial states and inputs. We propose an approach for computing an over-approximation of the set of states reachable on a bounded time interval. The main contribution over previous works is that it allows us to consider systems whose sets of initial states and inputs are given by arbitrary compact convex sets represented by their support functions. We actually compute two over-approximations of the reachable set. The first one is given by the union of convex sets with computable support functions. As the representation of convex sets by their support function is not suitable for some tasks, we derive from this first over-approximation a second one given by the union of polyhedrons. The overall computational complexity of our approach is comparable to the complexity of the most competitive available specialized algorithms for reachability analysis of linear systems using zonotopes or ellipsoids. The effectiveness of our approach is demonstrated on several examples.     

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

Linear bilevel programming with interval coefficients

In this paper, we address linear bilevel programs when the coefficients of both objective functions are interval numbers. The focus is on the optimal value range problem which consists of computing the best and worst optimal objective function values and determining the settings of the interval coefficients which provide these values. We prove by examples that, in general, there is no precise way of systematizing the specific values of the interval coefficients that can be used to compute the best and worst possible optimal solutions. Taking into account the properties of linear bilevel problems, we prove that these two optimal solutions occur at extreme points of the polyhedron defined by the common constraints. Moreover, we develop two algorithms based on ranking extreme points that allow us to compute them as well as determining settings of the interval coefficients which provide the optimal value range.     

Boolean-rank-preserving operators and Boolean-rank-1 spaces

We study the extent to which certain theorems on linear operators on field-valued matrices carry over to linear operators on Boolean matrices. We obtain analogues and near analogues of several such theorems. One of these leads us to consider linear spaces of m × n Boolean matrices whose nonzero members all have Boolean rank 1. We obtain a structure theorem for such spaces that enables us to determine the maximum Boolean dimension of such spaces and their maximum cardinality.     

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

So far there is no systematic attempt to construct Boolean functions with maximum annihilator immunity. In this paper we present a construction keeping in mind the basic theory of annihilator immunity. This construction provides functions with the maximum possible annihilator immunity and the weight, nonlinearity and algebraic degree of the functions can be properly calculated under certain cases. The basic construction is that of symmetric Boolean functions and applying linear transformation on the input variables of these functions, one can get a large class of non-symmetric functions too. Moreover, we also study several other modifications on the basic symmetric functions to identify interesting non-symmetric functions with maximum annihilator immunity. In the process we also present an algorithm to compute the Walsh spectra of a symmetric Boolean function with O(n²) time and O(n) space complexity. We use the term “Annihilator Immunity” instead of “Algebraic Immunity” referred in the recent papers [3–5, 9, 18, 19]. Please see Remark 1 for the details of this notational change     

Characterization of Linear Structures

We study the notionof linear structure of a function defined from F ^mto F ⁿ, and in particular of a Boolean function.We characterize the existence of linear structures by means ofthe Fourier transform of the function. For Boolean functions,this characterization can be stated in a simpler way. Finally,we give some constructions of resilient Boolean functions whichhave no linear structure.     

On the security of the Feng–Liao–Yang Boolean functions with optimal algebraic immunity against fast algebraic attacks

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g, such that fg = 0. There is an increasing interest in construction of Boolean functions that possess optimal algebraic immunity, combined with other characteristics, like balancedness, high nonlinearity, and high algebraic degree. In this paper, we investigate a recently proposed infinite class of balanced Boolean functions with optimal algebraic immunity, optimum algebraic degree and much better nonlinearity than all the previously introduced classes of Boolean functions with maximal algebraic immunity. More precisely, we study the resistance of the functions against one of the new algebraic attacks, namely the fast algebraic attacks (FAAs). Using the special characteristics of the family members, we introduce an efficient method for the evaluation of their behavior against these attacks. The new algorithm is based on the well studied Berlekamp–Massey algorithm.     

On Some Special Classes of Sequential Dynamical Systems

Sequential Dynamical Systems (SDSs) are mathematical models for analyzing simulation systems. We investigate phase space properties of some special classes of SDSs obtained by restricting the local transition functions used at the nodes. We show that any SDS over the Boolean domain with symmetric Boolean local transition functions can be efficiently simulated by another SDS which uses only simple threshold and simple inverted threshold functions, where the same threshold value is used at each node and the underlying graph is d-regular for some integer d. We establish tight or nearly tight upper and lower bounds on the number of steps needed for SDSs over the Boolean domain with 1-, 2- or 3-threshold functions at each of the nodes to reach a fixed point. When the domain is a unitary semiring and each node computes a linear combination of its inputs, we present a polynomial time algorithm to determine whether such an SDS reaches a fixed point. We also show (through an explicit construction) that there are Boolean SDSs with the NOR function at each node such that their phase spaces contain directed cycles whose length is exponential in the number of nodes of the underlying graph of the SDS.AMS Subject Classification: 68Q10, 68Q17, 68Q80.     

m值逻辑函数的多分块仿射逼近

文章定义了ｍ值逻辑函数在Ｄｚｎｍ上的Ｃｈｒｅｓｔｅｎｓｏｎ变换，并考察了这类变换的性质，在此基础上提出了对ｍ值逻辑函数进行多分块仿射逼近的方法，并分析了这种方法的优越性。特别地，重点给出了布尔函数的多分块仿射逼近，并用此方法得到了文献［２］所给出的最大相关子。     

A unified framework for testing linear‐invariant properties

In the study of property testing, a particularly important role has been played by linear invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed‐Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear‐invariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following:

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An algorithm for the estimation of a regression function by continuous piecewise linear functions

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L ₂ risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.     

Recursions associated to trapezoid,symmetric and rotation symmetric functions over Galois fields

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of ${\mathbb{F}}_2$, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field ${\mathbb{F}}_q$. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences.     

Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.     

Complexity of the Realization of a Linear Boolean Function in the Class of <Emphasis Type="Italic">π</Emphasis>-Schemes

Using Khrapchenko’s method, we obtain the exact lower bound of 40 for the complexity in the class of π-schemes of a linear Boolean function depending substantially on 6 variables. We give a simplified proof of several lower bounds for the complexity of linear Boolean functions which are previously obtained on the basis of the same method.