This work introduces two algebraic variants of the Massey-Omura cryptosystem based on newly defined generalized (k,t)-Jacobsthal p-numbers and their extensions to finite groups. We first generalize the classical Jacobsthal recurrence and establish structural properties including periodicity, invertibility conditions, and recurrence behavior modulo finite integers. These results are then extended to group-theoretic settings, where we construct the corresponding (k,t)-Jacobsthal sequences in specific finite groups and derive their sequence periods. Leveraging these algebraic foundations, we propose two Massey-Omura-type encryption schemes in which private exponents are selected from the generalized Jacobsthal sequences. We formally prove the correctness of both constructions and analyze the implications of periodicity on exponent invertibility and protocol feasibility. The proposed schemes do not introduce new hardness assumptions beyond those inherent in the underlying platform group. Instead, they provide a mathematically structured alternative to classical exponent selection in three-pass protocols. The results highlight a new connection between recurrence-defined sequences and multiplicative exponentiation in finite groups, offering an algebraically motivated direction for exploring generalized exponent families in symmetric and non-abelian cryptosystems.

Given any integers \(q\ge 2\) and \(p\ge 3\), a multidimensional array \[[m(i_1,i_2, \dots, i_q)\colon 1\le i_1\le p, \dots , 1\le i_q\le p],\] with non-repeated entries from the set \(\{1,2,\dots, p^q\}\) will be called a \(q\) dimensional magic hyper-square of order \(p\) if the sum of the numbers in any of its axes or diagonals is \(p(p^q+1)/2\). In this paper, we study odd-order uniform step magic hyper-squares of the form \[m(i_1, \dots, i_q)=1+\sum\limits_{j=1}^{q} p^{j-1}\left[\left(\sum\limits_{k=1}^qa_{j,k} i_k+d_j\right) \bmod p \right] .\] We derive necessary and sufficient conditions for these coefficients to generate magic hyper-squares and use a specific choice of coefficients to get new formula that generates magic hyper-squares of all odd orders greater than two.

In this paper, we consider two ways of breaking a graph’s symmetry: distinguishing labelings and fixing sets. A distinguishing labeling ϕ of G colors the vertices of G so that the only automorphism of the labeled graph (G, ϕ) is the identity map. The distinguishing number of G, D(G), is the fewest number of colors needed to create a distinguishing labeling of G. A subset S of vertices is a fixing set of G if the only automorphism of G that fixes every element in S is the identity map. The fixing number of G, Fix(G), is the size of a smallest fixing set. A fixing set S of G can be translated into a distinguishing labeling ϕ_S by assigning distinct colors to the vertices in S and assigning another color (e.g., the “null” color) to the vertices not in S. Color refinement is a well-known efficient heuristic for graph isomorphism. A graph G is amenable if, for any graph H, color refinement correctly determines whether G and H are isomorphic or not. Using the characterization of amenable graphs by Arvind et al. as a starting point, we show that both D(G) and Fix(G) can be computed in O((|V(G)|+|E(G)|)log |V(G)|) time when G is an amenable graph.

This paper studies the classification problem of block-transitive \( t \)-designs. Let \(\cal D = (\mathcal{P}, \mathcal{B}) \) be a non-trival \( t\)-\((v,k,\lambda) \) design with \( \lambda \leq 5 \), and let \( G \) be a block-transitive, point-primitive automorphism group of \(\cal D \). We prove that if \( \text{Soc}(G) \) is a sporadic simple group, then up to isomorphism, there are exactly 15 such designs \( \cal D \).

The Mostar invariants are newly introduced bond-additive, distance-related descriptors that compute the degree of peripherality of specific edges as well as the entire graph. These invariants have attracted significant attention in both classical applications of chemical graph theory and studies of complex networks. They have proven to be useful for exploring the topological aspects of these networks. For a graph ℋ, the edge Mostar index Mo_e is defined as the sum of the magnitudes of the differences between m_ℋ(x) and m_ℋ(g) across all edges xg of ℋ. Here, m_ℋ(g) (or m_ℋ(x)) represents the cardinality of the edges in ℋ that are closer to g (or x) than x (or g). In this paper, we determine the trees that maximize and minimize the edge Mostar index for fixed order, diameter, and number of pendent vertices. Sharp upper and lower bounds for this index are established, and the corresponding extremal trees are characterized. Moreover, the correlation of the edge Mostar index with certain physicochemical properties is examined.