Optimal ordering of transmissions for computing Boolean threshold functions

In [Proceedings of the Second Symposium on Switching Circuit Theory and Logical Design (FOCS), 1961, pp. 34–38], Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow parameters. Providing an algorithmic version of Chow's theorem—i.e., efficiently constructing a representation of a threshold function given its Chow parameters—has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized polynomial-time approximation scheme with the following behavior: Given the Chow parameters of a Boolean threshold function f over n bits and any constant $\epsilon>0$, the algorithm runs in time $O(n^2\log^2n)$ and with high probability outputs a representation of a threshold function $f'$ which is $\epsilon$-close to f. Along the way we prove several new results of independent interest about Boolean threshold functions. In addition to various structural results, these include $\tilde{O}(n^2)$-time learning algorithms for threshold functions under the uniform distribution in the following models: (i) the restricted focus of attention model, answering an open question of Birkendorf et al.; (ii) an agnostic-type model. This contrasts with recent results of Guruswami and Raghavendra who show NP-hardness for the problem under general distributions; (iii) the PAC model, with constant $\epsilon$. Our $\tilde{O}(n^2)$-time algorithm substantially improves on the previous best known running time and nearly matches the $\Omega(n^2)$ bits of training data that any successful learning algorithm must use.

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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

A set S of Boolean points is a specifying set for a threshold function f if the only threshold function consistent with f on S is f itself. The minimal cardinality of a specifying set for f is the specification number of f and it is never smaller than n+1 for a function with n relevant variables. In the present paper, we develop an inductive approach to describing the set of Boolean threshold functions with minimum specification number by means of operations that allow us to extend functions of n variables in this set to functions of n+1 variables.

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.

Dimension reduction and classification of data is an important issue in the data processing. Reduct in the rough set is a minimal subset of features, which has the same discernible power as the entire features in the higher dimensional scheme. Boolean threshold function was of fundamental interest for classification function. The nearest neighbor relation with minimal distance proposed here has a fundamental information for classification. Further, the nearest neighbor relation plays a crucial role for generation of reducts and threshold functions using the Boolean reasoning on the discernibility and in discernibility matrices, in which the indiscernibility matrix is proposed here to test the sufficient condition for reduct and threshold function. Then, a genetation method for the threshold functions based on the nearest neighbor relation are proposed here using Boolean operations.

Boolean functions with a variety of secure cipher properties are the key factors to design cryptosystem with the ability to resist multiple cipher attacks and good safety performance. In this paper, using the derivative of the Boolean functions and the e-derivative defined by ourselves as the main research tools, we study algebraic immunity, correlation immunity and other cryptographic properties of the quadratic rotation symmetric Boolean functions. We determine the quadratic rotation symmetric Boolean functions which are H Boolean functions, and the range of weight distribution of the quadratic rotation symmetry H Boolean functions. Besides, we get the compatibility among propagation, balance, correlation immunity and algebraic immunity of the quadratic rotation symmetry H Boolean functions, and also focus on the relationship of balance, correlation immunity and dimension. Furthermore, we check the existence of the cubic rotation symmetry H Boolean functions, and obtain the relationship between existence and dimension of the cubic rotation symmetry H Boolean functions. Moreover, we obtain a more convenient method for solving annihilator. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.

As soft-hardware-logic circuits had been proposed in the literature as an alternative for digital circuits taking advantage the fact that any Boolean function could be implemented with the same cell, just configuring external signals, this work shows a methodology that could be followed particularly for the design of a four bits logic gate, using the so-called neuron MOS transistor (ν-MOS). Simulation results show the feasibility of the design for performing as XNOR, NOR, OR, XOR, AND or NAND logic gates, for instance. In order to extrapolate the design to a higher number of bits, the key issue is to properly consider the weight of the input capacitances in correlation with the number of input bits. A D/A converter can be used as the input stage of the configuration. This design considers the D/A converter-less version, since it helps to increase device integration as the number of transistors used is reduced with no difference in its performance. The design should be based on the theoretical floating potential diagram (FPD) of the desired logic gate.

Optimal ordered binary decision diagrams for read-once formulas

Given a Boolean function f, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions -- functions invariant to the reordering of all but a constant number of their variables -- are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

The number and probability of canalizing 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.

Results on highly nonlinear Boolean functions with provably good immunity to fast algebraic attacks

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.