The Stochastic Boolean Function Evaluation problem for symmetric Boolean functions

The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

Two important classes of symmetric Boolean functions are the equal-weight Boolean functions and the elementary (or homogeneous) symmetric Boolean functions. In this paper we studied the equal-weight symmetric Boolean functions. First the Walsh spectra of the equal-weight symmetric Boolean functions are given. Second the sufficient and necessary condition on correlation-immunity of the equal-weight symmetric Boolean function is derived and other cryptology properties such as the nonlinearity, balance and propagation criterion are taken into account. In particular, the nonlinearity of the equal-weight symmetric Boolean functions with n (n ≥ 10) variables is determined by their Hamming weight. Considering these properties will be helpful in further investigations of symmetric Boolean functions.

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In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize symmetrical Boolean functions in the main and polynomial bases. Prospective reserves in the analytical method were identified, such as simplification of polynomial function conjuncterms using the created equivalent transformations based on the method of inserting the same conjuncterms followed by the operation of super-gluing the variables. The method of figurative transformations was extended to the process of minimizing the symmetrical Boolean functions with the help of algebra in terms of rules for simplifying the functions of the main and polynomial bases and developed equivalent transformations of conjuncterms. It was established that the simplification of symmetric Boolean functions by the method of figurative transformations is based on a flowchart with repetition, which is the actual truth table of the assigned function. This is a sufficient resource to minimize symmetrical Boolean functions that makes it possible to do without auxiliary objects, such as Karnaugh maps, cubes, etc. The perfect normal form of symmetrical functions can be represented by binary matrices that would represent the terms of symmetrical Boolean functions and the OR or XOR operation for them. The experimental study has confirmed that the method of figurative transformations that employs the 2-(n, b)-design, and 2-(n, x/b)-design combinatorial systems improves the efficiency of minimizing symmetrical Boolean functions. Compared to analogs, this makes it possible to enhance the productivity of minimizing symmetrical Boolean functions by 100‒200 %. There are grounds to assert the possibility of improving the effectiveness of minimizing symmetrical Boolean functions in the main and polynomial bases by the method of figurative transformations. This is ensured, in particular, by using the developed equivalent transformations of polynomial function conjuncterms based on the method of inserting similar conjuncterms followed by the operation of super-gluing the variables.

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

Symmetric polynomials over [formula omitted] and simultaneous communication protocols

Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

Optimal Depth, Very Small Size Circuits for Symmetrical Functions in AC0

We study deterministic one-way communication complexity of functions with Hankel communication matrices. In this paper some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.

We study the negabent Boolean functions which are symmetric. The Boolean function which has equal absolute spectral values under the nega-Hadamard transform is called a negabent function. For a bent function, the absolute spectral values are the same under the Hadamard-Walsh transform. Unlike bent functions, negabent functions can exist on odd number of variables. Moreover, all the affine functions are negabent. We prove that a symmetric Boolean function is negabent if and only if it is affine.

Boolean function is one of the most fundamental computation models in theoretical computer science. The two most common representations of Boolean functions are Fourier transform and real polynomial form. Applying analytic tools under these representations to the study Boolean functions has led to fruitful research in many areas such as complexity theory, learning theory, inapproximability, pseudorandomness, metric embedding, property testing, threshold phenomena, social choice, etc. In this thesis, we focus on \emph{sparse representations} of Boolean function in both Fourier transform and polynomial form, and obtain the following new results. A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the Fourier support of $f$, then $f$ must be the indicator function of some affine subspace of dimension $n-k$. Here we slightly generalize their result, and show that Boolean functions whose Fourier coefficients take values in the set $\{-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k\}$ are indicator functions of two disjoint affine subspaces of dimension $n-k$ or four disjoint affine subspaces of dimension $n-k-1$. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of $\mathbb{F}_2^{n}$ when the doubling constant of the subset is small. For polynomial representation of Boolean function, we study the distribution of the number of non-zero coefficients of \emph{random} Boolean functions. For a random Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, i.e., a function whose value at each point on the Boolean cube is chosen independently and uniformly at random from $\{0,1\}$, in real polynomial representation, we give several bounds and concentration results about the distribution of the sparsity of $f$.

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence of functions (Fvi,Y)i, and (c) a word w = (w1,…,wk), where each wi is a vertex in Y. The function Fvi,Y updates the state of vertex vi as a function of the state of vi and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map [Formula: see text]. In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circ n), and all the maps Fvi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ 4. Finally, we analyze invertible ACA and offer an interpretation of S35 as the group of an SDS over the three-dimensional cube with local functions induced by nor3 + nand3.

We study how hard is to determine some fundamental properties of dynamics of certain types of network automata. We address the computational complexity of determining how many different possible dynamic evolutions can arise from some structurally very simple, deterministic and sparsely connected network automata. In this as well as our prior, related work, we try to push the limits on the underlying simplicity of two structural aspects of such network automata: (i) the uniform sparseness of their topologies, and (ii) severely restricted local behaviors of the individual agents (that is, the local update rules of the network nodes). In this endeavor, we prove that counting the Fixed Point (FP) configurations and the predecessor and ancestor configurations in two classes of network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems. Moreover, this intractability is shown to hold when each node in such a network is required to update according to (i) a monotone Boolean function, (ii) a symmetric Boolean function, or even (iii) a simple threshold function that is both monotone and symmetric. Furthermore, the hardness of the exact enumeration of FPs and other types of configurations of interest remains to hold even in some severely restricted cases with respect to both the network topology and the diversity (or lack thereof) of individual node's local update rules. Namely, we show that the counting problems of interest remain hard even when the nodes of an SDS or SyDS use at most two different update rules from a given restricted class, and, additionally, when the network topologies are constrained so that each node has only $c = O(1)$ neighbors for small values of constant $c$. Our results also have considerable implications for other discrete dynamical system models studied in applied mathematics, physics, biology and computer science, such as Hopfield networks and spin glasses. In particular, one corollary of our results is that determining the memory capacity of sparse discrete Hopfield networks (viewed as associative memories) remains computationally intractable even when the interconnection and dependence structure among the nodes of a Hopfield network is severely restricted.

The properties of the 2m-variable symmetric Boolean functions with maximum algebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtained. At last, some necessary conditions for a symmetric Boolean function on even number variables to have maximum algebraic immunity are introduced.