In the article, we use the subset sum formula over a finite abelian group on the product of finite groups to derive the number of restricted partitions of elements in the group and to count the number of compositions over finite abelian groups. Later, we apply the formula for the multisubset sum problem on a group $\mathbb{Z}_n$ to produce a new technique for studying restricted partitions of positive integers. 2020 Mathematics Subject Classification. 05A17, 11P81

We present an application of generalized strong complete mappings to construction of a family of mutually orthogonal Latin squares. We also determine a cycle structure of such mapping which form a complete family of MOLS. Many constructions of generalized strong complete mappings over an extension of finite field are provided.

To prove that a fuzzy dependency follows from a set of fuzzy dependences can be a very demanding task. As far as we know, an algorithm or an application that generally and automatically solves the problem, does not exist. The main goal of this paper is to offer such an algorithm. In order to achieve our goal we consider fuzzy dependences as fuzzy formulas. In particular, we fix fuzzy logic operators: conjunction, disjunction and implication, and allow only these operators to appear within fuzzy formulas. Ultimately, we prove that a fuzzy dependency follows from a set of fuzzy dependences if and only if the corresponding fuzzy formula is a logical consequence of the corresponding set of fuzzy formulas. To prove an implication of the last type, one usually uses the resolution principle, i.e., the steps that can be fully automated. Our methodology assumes the use of soundness and completeness of fuzzy dependences inference rules as well as the extensive use of active fuzzy multivalued dependences fulfillment

In this paper, we show that the maximum number of bent component functions of a vectorial function <inline-formula> <tex-math notation="LaTeX">$F:GF(2)^{n}\to GF(2)^{n}$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$2^{n}-2^{n/2}$ </tex-math></inline-formula>. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form <inline-formula> <tex-math notation="LaTeX">$F\in GF(2^{n})[x]$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> has only a few terms. The only known power functions having such a large number of bent components are <inline-formula> <tex-math notation="LaTeX">$x^{d}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$d=2^{n/2}+1$ </tex-math></inline-formula>. In this paper, we show that the binomials <inline-formula> <tex-math notation="LaTeX">$F^{i}(x)=x^{2^{i}}(x+x^{2^{n/2}})$ </tex-math></inline-formula> also have such a large number of bent components, and these binomials are inequivalent to the monomials <inline-formula> <tex-math notation="LaTeX">$x^{2^{n/2}+1}$ </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">$0<i<n/2$ </tex-math></inline-formula>. In addition, the functions <inline-formula> <tex-math notation="LaTeX">$F^{i}$ </tex-math></inline-formula> have differential properties much better than <inline-formula> <tex-math notation="LaTeX">$x^{2^{n/2}+1}$ </tex-math></inline-formula>. We also determine the complete Walsh spectrum of our functions when <inline-formula> <tex-math notation="LaTeX">$n/2$ </tex-math></inline-formula> is odd and <inline-formula> <tex-math notation="LaTeX">$\gcd (i,n/2)=1$ </tex-math></inline-formula>.

Integer linear programming is a popular method of generating school timetables. Although computationally simpler, school timetabling is less developed area than university timetabling, because the models which resolve timetabling problems proposed thus far have been adjusted to individual cases differing from country to country. A proposed model meets most of constraints appeared in different school timetabling systems.

To identify and specify trace bent functions of the form T r n 1 (P (x)), where P (x) ∈ GF (2 n)[x], has been an important research topic lately. We show that an infinite class of quadratic vectorial bent functions can be specified in the univariate polynomial form as F (x) = T r^n_k (αx^2^i (x + x^k)), where n = 2k, i = 0,n-1, and α \notin GF(2^k). Most notably apart from the cases i \in {0,k} for which the polynomial x^2^i (x+x^2^k) is affinely inequivalent to the monomial x^{2^k+1}, for the remaining indices i the function x^2^i (x+x^2^k) seems to be affinely inequivalent to x^2^k+1, as confi rmed by computer simulations for small n. It is well-known that Tr^n_1(x^2^k+1) is Boolean bent for exactly 2^{2k}-2^k values (this is at the same time the maximum cardinality possible) of α \in GF(2n) and the same is true for our class of quadratic bent functions of the form T r^n_k (αx^2^i (x + x^k)) though for i > 0 the associated functions F : GF(2^n) -> GF(2^n) are in general CCZ inequivalent and also have dierent dierential distributions.

For any given polynomial $f$ over the finite field $\mathbb{F}_q$ with degree at most $q-1$, we associate it with a $q\times q$ matrix $A(f)=(a_{ik})$ consisting of coefficients of its powers $(f(x))^k=\sum_{i=0}^{q-1}a_{ik} x^i$ modulo $x^q -x$ for $k=0,1,\ldots,q-1$. This matrix has some interesting properties such as $A(g\circ f)=A(f)A(g)$ where $(g\circ f)(x) = g(f(x))$ is the composition of the polynomial $g$ with the polynomial $f$. In particular, $A(f^{(k)})=(A(f))^k$ for any $k$-th composition $f^{(k)}$ of $f$ with $k \geq 0$. As a consequence, we prove that the rank of $A(f)$ gives the cardinality of the value set of $f$. Moreover, if $f$ is a permutation polynomial then the matrix associated with its inverse $A(f^{(-1)})=A(f)^{-1}=PA(f)P$ where $P$ is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequence $\bar{a} = \{a_0, a_1, a_2, ... \}$ generated by $a_n = f^{(n)}(a_0)$ with initial value $a_0$, in terms of the order of the associated matrix. Finally we show that $A(f)$ is diagonalizable in some extension field of $\mathbb{F}_q$ when $f$ is a permutation polynomial over $\mathbb{F}_q$.

To identify and specify trace bent functions of the form Tr(P(x)), where P(x) ∈ F(2ⁿ)[x], has been an important research topic lately. We characterize a class of vectorial (hyper)bent functions of the form F(x) = Tr_kⁿ (Σ_i=0(2^k) a_ixⁱ⁽(2^k)^-1)), where n = 2k, in terms of finding an explicit expression for the coefficients a_i so that F is vectorial hyperbent. These coefficients only depend on the choice of the interpolating polynomial used in the Lagrange interpolation of the elements of U and some prespecified outputs, where U is the cyclic group of (2^n/2 + 1)th roots of unity in F(2ⁿ). We show that these interpolation polynomials can be chosen in exactly (2^k + 1)!2^k-1 ways and this is the exact number of vectorial hyperbent functions of the form Tr_kⁿ (Σ_i=0^2k a_ixⁱ⁽(2^k)^-1)). Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients.

In this paper, we provide necessary and sufficient conditions for a function of the form F(x)=Trk^2k(Σi=1^taix^ri(2k-1)) to be bent. Three equivalent statements, all of them providing both the necessary and sufficient conditions, are derived. In particular, one characterization provides an interesting link between the bentness and the evaluation of F on the cyclic group of the (2^k+1)th primitive roots of unity in GF(2^2k). More precisely, for this group of cardinality 2^k+1 given by U={u ∈ GF(2^2k):u^2k+1=1}, it is shown that the property of being vectorial bent implies that Im(F)=GF(2^k)∪{0}, if F is evaluated on U, that is, F(u) takes all possible values of GF(2^k)* exactly once and the zero value is taken twice when u ranges over U. This condition is then reformulated in terms of the evaluation of certain elementary symmetric polynomials related to F, which in turn gives some necessary conditions on the coefficients ai (for binomial trace functions) that can be stated explicitly. Finally, we show that a bent trace monomial of Dillon's type Trk^2k(λx^r(2k-1)) is never a vectorial bent function.