Exact Value of the Nonmonotone Complexity of Boolean Functions

A Boolean function $f:A_1 \times A_2 \times \cdots \times A_n \to \{ 0,1 \}$ is t-private if there exists a protocol for computing f so that no coalition of size $\leqq t$ can infer any additional information from the execution, other than the value of the function. It is shown that f is $\lceil n/2 \rceil $-private if and only if it can be represented as \[ f ( x_1 ,x_2 , \cdots ,x_n ) = f_1 ( x_1 ) \oplus f_2 ( x_2 ) \oplus \cdots \oplus f_n ( x_n ), \] where the $f_i $ are arbitrary Boolean functions. It follows that if f is $\lceil n/2 \rceil $-private, then it is also n-private. Combining this with a result of Ben-Or, Goldwasser, and Wigderson, and of Chaum, Crepeau, and Damgard, [Proc. 20th Symposium on Theory of Computing, 1988, pp. 1–10 and pp. 11–19] an interesting “zero-one” law for private distributed computation of Boolean functions is derived: every Boolean function defined over a finite domain is either n-private, or it is $\lfloor ( n - 1 )/2 \rfloor $-private but not $\lceil n/2 \rceil $-private. A weaker notion of privacy is also investigated, where (a) coalitions are allowed to infer a limited amount of additional information, and (b) there is a probability of error in the final output of the protocol. It is shown that the same characterization of $\lceil n/2 \rceil $-private Boolean functions holds, even under these weaker requirements.In particular, this implies that for Boolean functions, the strong and the weak notions of privacy are equivalent.

In this paper, an approach to computing arbitrary Boolean functions using a continuous-state cellular neural/nonlinear/nanoscale network (CNN) architecture with local static memory is presented. We explain how any given Boolean function can be mapped to a CNN array and how the function is executed using a sequence of wave operations. Furthermore, we explain how gray-scale waves could reduce the number of CNN cells required to perform a certain logic operation. The main benefits of our approach are simple implementation of arbitrary Boolean functions, asynchronous operation, and applicability in multi-state computation.

Substitution boxes (S-boxes) are often used as the most important nonlinear components in many symmetric encryption algorithms. The cryptographic properties of an S-box directly affect the security of the whole cipher system. Recently, generalized global avalanche characteristics (GGAC) were introduced to measure the correlation between two arbitrary Boolean functions. In this paper, to better evaluate the security of an S-box, we present two cross-correlation indicators for it. In addition, by studying the related properties of the cross-correlation between two balanced Boolean functions, we propose the lower bounds on the sum-of-squares indicator related to GGAC for two balanced functions and also for an S-box.

This paper describes nanocircuits that draw on negative differential resistance and are capable of implementing complex threshold functions in a single gate structure. Due to nanometer dimensions, high operational frequencies and low power consumption these devices can be used for efficient hardware realisation of artificial neural networks (ANNs). We present state of the art in development of such circuit and focus on Generalised Threshold Gates (GTGs) that are capable of implementing arbitrary Boolean functions in a single gate structure. Algorithm for implementing Boolean functions outputs circuits with predefined weights and thresholds. This enables to construct application specific ANNs and eliminates the requirement for network learning when this kind of gates are used.

To measure the correlation between two arbitrary Boolean functions, we propose the two new criteria called the sum-of-squares indicator and the absolute indicator of the cross-correlation. The two indicators generalize the GAC criterion (which was proposed by X.M. Zhang and Y.L. Zheng, in GAC-the criterion for global avalanche characteristics of cryptographic functions, Journal for Universal Computer Science, 1(5): 316–333, 1995). Lower and upper bounds on the two indicators are obtained.

Robust template design for cellular neural networks (CNNs) implementing an arbitrary Boolean function is currently an active research area. If the given Boolean function is linearly separable, a single robust uncoupled CNN can be designed preferably as a maximal margin classifier to implement the Boolean function. On the other hand, if the linearly separable Boolean function has a small geometric margin or the Boolean function is not linearly separable, a popular approach is to find a sequence of robust uncoupled CNNs implementing the given Boolean function. In the past research works using this approach, the control template parameters and thresholds are usually restricted to assume only a given finite set of integers. In this study, we try to remove this unnecessary restriction. NXOR- or XOR-based decomposition algorithm utilizing the soft margin and maximal margin support vector classifiers is proposed to design a sequence of robust templates implementing an arbitrary Boolean function. Several illustrative examples are simulated to demonstrate the efficiency of the proposed method by comparing our results with those produced by other decomposition methods with restricted weights.

We consider a realization of Boolean functions by the circuits from unreliable elements in a full basis B, containing a special function. We assume that all elements of a circuit are exposed to the faults type 0 at the outputs with probability e ∈ (0, 1/2) independently of each other. We prove that almost any Boolean function can be implemented by an asymptotically optimal in reliability circuit functioning with the unreliability which is asymptotically equal to e with e → 0.

A systematic method of testing and realization of Boolean functions with a large number of unspecified states by a single threshold logic element has been presented in this paper. By this method, it is possible to determine many different threshold realizations of the given incomplete function. The method can be applied to switching functions containing any number of variables.

If the given Boolean function is linearly separable, a robust uncoupled cellular neural network can be designed as a maximal margin classifier. On the other hand, if the given Boolean function is linearly separable but has a small geometric margin or it is not linearly separable, a popular approach is to find a sequence of robust uncoupled cellular neural networks implementing the given Boolean function. In the past research works using this approach, the control template parameters and thresholds are restricted to assume only a given finite set of integers, and this is certainly unnecessary for the template design. In this study, we try to remove this restriction. Minterm- and maxterm-based decomposition algorithms utilizing the soft margin and maximal margin support vector classifiers are proposed to design a sequence of robust templates implementing an arbitrary Boolean function. Several illustrative examples are simulated to demonstrate the efficiency of the proposed method by comparing our results with those produced by other decomposition methods with restricted weights.

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.

MOS networks and fault-tolerant sequential machines

We consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates. We define the entropy of an operator f:{0,1}n →{0,1}m as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f. Our main result is that every depth-2 circuit for f requires at least entropy(f) wires. This is reminiscent of a classical lower bound of Nechiporuk on the formula size, and gives an information-theoretic explanation of why some operators require many wires. We use this to prove a tight estimate Θ(n 3) of the smallest number of wires in any depth-2 circuit computing the product of two n by n matrices over any finite field. Previously known lower bound for this operator was Ω(n 2log n).

We study the realization of monotone Boolean functions by networks. Our main result is a precise version of the following statement: the complexity of realizing a monotone Boolean function ofn arguments is less by the factor (2/πn)1/2, whereπ is the circular ratio, than the complexity of realizing an arbitrary Boolean function ofn arguments. The proof combines known results concerning monotone Boolean functions with new methods relating the computing abilities of networks and machines.

One of the main security criteria that mostly influence the strength of a conventional cipher system is the Nonlinearity criterion. A classical cipher system is considered strong if it is characterized by high nonlinearity. That is, the underlying Boolean function employed within the system is required to be far from being linear. The measure of a Boolean function's nonlinearity can be done with the help of the Walsh transform as a tool. This paper presents an analysis of the nonlinearity criterion from the Haar transform domain perspective. The paper proceeds by utilizing the connection between the Haar and the Walsh transforms. The work provides two main contributions in the form of the definition of nonlinearity given the Haar spectrum of an arbitrary Boolean function and a Haar-Walsh hybrid algorithm for measuring the nonlinearity given the Haar spectrum. The derived algorithm is then simulated along with the Walsh benchmark and the results are presented in a comparison study. The paper then follows up by presenting a discussion on the computational advantages of employing the Haar-Walsh hybrid method in terms of time and space complexities. It is shown that, the Haar-Walsh hybrid method performs better when the transformed function is characterized by high number of input variables (n > 11). The paper concludes with a summary of the presented work and suggestions for future work.

A decomposition and reconstruction approach for synthesizing an arbitrary Boolean function with a minimum number of threshold logic elements connected by feedforward paths only is presented. Attention is mainly focused on cascade-type realizations. The approach has the advantage that near-minimal solutions are readily derived. An estimate of how closely the minimality has been approached is obtainable in this method. The method has been successfully applied by the authors to Boolean functions of 5 and 6 variables.