We define the notions of reducibility and completeness in multi-party private computations. Let g be an n-argument function. We say that a function f is reducible to g if n honest-but-curious players can compute the function f nprivately, given a black-box for g (for which they secretly give inputs and get the result of operating g on these inputs). We say that g is complete (for multi-party pr...

We say a polynomial P over ZZM strongly M -represents a Boolean function F if F (x) ≡ P (x) (mod M) for all x ∈ {0, 1} . Similarly, P one-sidedly M -represents F if F (x) = 0 ⇐⇒ P (x) ≡ 0 (mod M) for all x ∈ {0, 1} . Lower bounds are obtained on the degree and the number of monomials of polynomials over ZZM , which strongly or one-sidedly M -represent the Boolean function deciding if a given nb...

The paper presents a family of new expansions of Boolean functions called Function-driven Linearly Independent (fLI) expansions. On the basis of this expansion a new kind of a canonical representation of Boolean functions is constructed: Function-driven Linearly Independent Binary Decision Diagrams (fLIBDDs). They generalize both Function-driven Shannon Binary Decision Diagrams (fShBDDs) and Li...

In this paper, we introduce and define Probabilistic Boolean Logic, whose logical operators are “correct” with a probability 0 < p ≤ 1. Analogous to conventional Boolean logic, we define well-formed probabilistic Boolean formulae (pbf). Every pbf is associated with two attributes, the underlying Boolean function it computes, and a for a specific input, the probability that this boolean function...

In this paper a new approach for designing Boolean functions that satisfy the Strict Avalanche Criterion (SAC) is presented. The advantage of the suggested approach is the simplicity of its realization and the significant greater number of the generated functions compared to the known methods. The formalized procedure for construction of nonbalanced and balanced SAC-functions is described in de...

A pseudo-Boolean function is a function from a 0/1-vector to the reals. Minimizing pseudo-Boolean functions is a very general problem with many applications. In image analysis, the problem arises in segmentation or as a subroutine in task like stero estimation and image denoising. Recent years have seen an increased interest in higher-degree problems, as opposed to quadratic pseudo-Boolean func...

In this article, a noble approach is presented to classify n-variable Boolean functions in logical way such that each function belonging to a particular class can be traced with respect to a single base function. In the present study, two different methods have been proposed for this classification. The first one is done through the Hamming distance with regards to base 0 (2^n bits of zeros) Bo...

The universal approximation theorem for neural networks says that any reasonable function is well-approximated by a two-layer neural network with sigmoid gates but it does not provide good bounds on the number of hidden-layer nodes or the weights. However, robust concepts often have small neural networks in practice. We show an efficient analog of the universal approximation theorem on the bool...

This paper deals with the representation of Boolean functions using artificial neural networks and points out three important results. First, using a polynomial as transfer function, a single neuron is able to represent a non-monotonous Boolean function. Second, the number of inputs in the neural network can be decreased if the binary values of the Boolean variables are encoded. This approach s...