Chukhrov, I. P.
Published in
Journal of Applied and Industrial Mathematics

The problem of minimizing Boolean functions for additive complexity measures in a geometric interpretation, as covering a subset of vertices in the unit cube by faces, is a special type of a combinatorial statement of the weighted problem of a minimal covering of a set. Its specificity is determined by the family of covering subsets, the faces of t...

Kurbatskaia, V. K.
Published in
Moscow University Computational Mathematics and Cybernetics

Estimates are obtained for the Shannon function of the length of a diagnostic test with respect to cyclic shifts of scheme inputs, and for the Shannon function of fault detection and length of a diagnostic test with respect to a single stuck-at fault and a cyclic shift of scheme inputs.

Logachev, Oleg A. Fedorov, Sergey N. Yashchenko, Valerii V.
Published in
Discrete Mathematics and Applications

A new approach to the study of algebraic, combinatorial, and cryptographic properties of Boolean functions is proposed. New relations between functions have been revealed by consideration of an injective mapping of the set of Boolean functions onto the sphere in a Euclidean space. Moreover, under this mapping some classes of functions have extremel...

Liu, Zhuojun Wu, Baofeng
Published in
Journal of Systems Science and Complexity

Boolean functions with optimal algebraic immunity (OAI functions) are important cryptographic primitives in the design of stream ciphers. During the past decade, a lot of work has been done on constructing such functions, among which mathematics, especially finite fields, play an important role. Notably, the approach based on decompositions of addi...

Charpin, Pascale Peng, Jie

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper we consider more generally the codes associated with functions that have differential uniformity at least 4. We emphasize, for such a function F , the role of codewords of weight 3 and 4 and of some cosets of its associated code C F. We give som...

Couceiro, Miguel Mercuriali, Pierre Péchoux, Romain Saffidine, Abdallah

In this document, we consider a median-based calculus to represent monotone Boolean functions efficiently. We study an equa-tional specification of median forms and extend it from the domain of monotone Boolean functions to the domain of polynomial functions over distributive lattices. This specification is sound and complete. We illustrate its use...

Couceiro, Miguel Lehtonen, Erkko Mercuriali, Pierre Péchoux, Romain

A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as...

Grabovskaya, S. M. Alekhina, M. A.
Published in
Lobachevskii Journal of Mathematics

The implementation of Boolean functions by non-branching programs with a conditional stop operator is considered in an arbitrary complete finite basis. We assume that conditional stop operators of the program are absolutely reliable while all computational operators are prone to the output one-type constant faults of either type 0 or type 1. An upp...

Redkin, Nikolay P.
Published in
Discrete Mathematics and Applications

When investigating the complexity of implementing Boolean functions, it is usually assumed that the basis inwhich the schemes are constructed and the measure of the complexity of the schemes are known. For them, the Shannon function is introduced, which associates with each Boolean function the least complexity of implementing this function in the ...

Voronenko, A. A.
Published in
Computational Mathematics and Modeling

The following problem is considered: specify a Boolean function of n variables such that every bilinear (polylinear) function is reduced, on a certain number of tuples of the specified function, to a unique bilinear (polylinear) function that is identical with the specified function on these tuples. We show that this is feasible for bilinear functi...