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...

Wang, Libo Wu, Baofeng Liu, Zhuojun Lin, Dongdai
Published in
Science China Information Sciences

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They are closely related to some interesting combinatorial objects and also have important applications in coding, cryptography and sequence design. In this paper, we firstly give a necessary and sufficient condition for a type of Boolean functions, which obt...

Charpin, Pascale Kyureghyan, Gohar M.

The differential uniformity of a mapping $F : F 2 n → F 2 n$ is defined as the maximum number of solutions $x$ for equations $F (x+a)+F (x) = b$ when $a ̸ = 0$ and $b$ run over $F 2 n$. In this paper we study the question whether it is possible to determine the differential uniformity of a mapping by considering not all elements $a ̸ = 0$, but only...

Alekseychuk, A. N.
Published in
Cybernetics and Systems Analysis

A theorem is proved that improves a previously upper bound for the relative distance between a Boolean function of n variables and the set of k-dimensional functions, k

Kolomeec, N. A.
Published in
Journal of Applied and Industrial Mathematics

Let f be a Boolean function of n variables such that, for every affine subspace L of dimension [n/2], f is either affine on all shifts of L or affine on none of the shifts of L. We prove that either the degree of f is at most 2 or there is no affine subspace of dimension [n/2] on which f is affine.

Filyuzin, S. Yu.
Published in
Journal of Applied and Industrial Mathematics

The algebraic immunity of a Boolean function of n variables is known to be at most ⌈n/2⌉. We prove the upper bound ⌈n/4⌉ + 1 for the algebraic immunity of the Dillon bent functions constructed with the use of linear functions which is almost two times less than the available maximum bound.

Wu, Baofeng Liu, Zhuojun Jin, Qingfang Zhang, Xiaoming
Published in
Journal of Systems Science and Complexity

Tu and Deng proposed a class of bent functions which are of optimal algebraic immunity under the assumption of a combinatorial conjecture. In this paper, the authors compute the dual of the Tu-Deng functions and then show that they are still of optimal algebraic immunity under the assumption of the same conjecture. For another class of Boolean func...

Frolova, A. A.
Published in
Journal of Applied and Industrial Mathematics

Kasami bent functions have the most complicated properties in the class of algebraic constructions of bent functions. We prove that the degree t Kasami functions have nonzero order t − 2 derivatives for 4 ≤ t ≤ (n + 3)/3 and nonzero order t − 3 derivatives for (n + 3)/3

Kolomeets, N. A.
Published in
Journal of Applied and Industrial Mathematics

We study a construction of the bent functions of least deviation from a quadratic bent function, describe all these bent functions of 2k variables, and show that the quantity of them is 2k(21 + 1) ... (2k + 1). We find some lower bound on the number of the bent functions of least deviation from a bent function of the Maiorana-McFarland class.

Tokareva, N. N.
Published in
Journal of Applied and Industrial Mathematics

Bent functions (Boolean functions with extreme nonlinearity properties) are actively studied for their numerous applications in cryptography, coding theory, and other fields. New statements of problems lead to a large number of generalizations of the bent functions many of which remain little known to the experts in Boolean functions. In this artic...