In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In partic...

We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function and computing its evaluation vector requires a worst-case complexity of O(n2n) integer operations. This way, with a different approach, we reach the same complex...

We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.

Vectorial Boolean functions are usually called Substitution Boxes (S-Boxes) and are used as basic component of block ciphers in Cryptography. The ciphers that are Substitution-permutation networks use bijective S-Boxes i.e., Boolean permutations. S-Boxes with low differential uniformity and high nonlinearity are considered as cryptographically strong. In this paper we study some properties of S...

A quantified Boolean formula is true, if for any existentially quantified variable there exists a Boolean function depending on the preceding universal variables, such that substituting the existential variables by the Boolean functions results in a true formula. We call a satisfying set of Boolean functions a model. In this paper, we investigate for various classes of quantified Boolean formul...

We de ne the notions of reducibility and completeness in (two party and multi-party) private computations. Let g be an n-argument function. We say that a function f is reducible to a function g if n honest-but-curious players can compute the function f n-privately, 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 c...

Linear algebra is introduced by Faugère in F4 to speed up the reduction procedure during Gröbner basis computations. Linear algebra has also been used in fast implementations of F5 and other signature-based Gröbner basis algorithms. To use linear algebra for reductions, an important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function...

This paper develops theoretical foundations and presents a practical algorithm for optimizing multi-output Boolean function M(x) whose outputs are combined using Boolean OR operator into a single-output Boolean function S(x). The proposed algorithm simplifies the logic structure of function M(x) and may change or remove some of its outputs, while preserving the functionality of function S(x). A...

– This paper gives a new and efficient method of determining the equivalence of two given Boolean functions. We define a characteristic polynomial directly from the sum-of-product form of the logic function. The polynomial contains a real variable corresponding to each Boolean variable. Logical operations on the Boolean function correspond to arithmetic operations on the polynomial. We show tha...

A lot of mathematical approaches are used in importance analysis, which permits to investigate influence of system component state changes on the system reliability or availability. One of these approaches is Logical Differential Calculus, in particular Direct Partial Boolean Derivatives. A new algorithm for the calculation of Importance Measures with application of Direct Partial Boolean Deriv...