The inherent linearity of Hill cipher renders it vulnerable to security threats, such as known-plaintext attacks. To solve this problem, an enhanced variant was designed by extending the Hill cipher from the integer ring to the finite field. First, a 2D finite field hyper chaotic map (2D-FFHCM) with desirable dynamical properties was constructed to dynamically generate both the invertible key matrices (via the product of invertible triangular matrices) and a strong S-Box. Subsequently, an enhanced variant of Hill cipher based on the S-Box was constructed, achieving strong confusion and diffusion properties. Finally, the proposed algorithm was evaluated through comprehensive tests, including key space, key sensitivity, information entropy, differential attack analysis and correlation analysis. Experimental results demonstrated the excellent performance of the proposed algorithm, confirming it can effectively resist common cryptographic attacks.

Let g ( x ) ∈ Z [ x ] be a monic irreducible polynomial of degree n. We say that g ( x ) is monogenic if, for a root θ of g ( x ) , the set { 1 , θ , … , θ n − 1 } forms an integral basis of the ring of integers Z K of the number field K = Q ( θ ) . Consider f ( x ) = x n + a x 2 + dx + b ∈ Z [ x ] with d 2 = 4 ab , and g ( x ) = x m + c ∈ Z [ x ] , where n ≥ 3 and m ≥ 1 , such that ( f ° g ) ( x ) = ( x m + c ) n + a ( x m + c ) 2 + d ( x m + c ) + b is irreducible over Q . In this study, we establish necessary and sufficient conditions involving a, b, c, d, m, n for the polynomial ( f ° g ) ( x ) to be monogenic. Additionally, we examine the nature of solutions to specific differential equations, and present a class of monogenic polynomials with non-square-free discriminants as an application.

Abstract We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.

Abstract Given a number field F with ring of integers $\mathcal {O}_{F}$ , one can associate to any torsion free subgroup of $\operatorname {SL}(2,\mathcal {O}_{F})$ of finite index a complete Riemannian manifold of finite volume with fibered cusp ends. For natural choices of flat vector bundles on such a manifold, we show that analytic torsion is identified with the Reidemeister torsion of the Borel-Serre compactification. This is used to obtain exponential growth of torsion in the cohomology for sequences of congruence subgroups.

We consider the problem of finding dense rank one module sublattices in module lattices of a given rank, where by module lattice we mean a lattice obtained as R-linear combinations of vectors with coefficients in a number field K, with ring of integers 𝒪 K , and R ⊆ 𝒪 K an order.We provide generic upper bounds on the normalized volume of rank one module sublattices for K totally real or CM, specific bounds for quadratic extensions and some cyclotomic extensions, as well as exact values for some norm Euclidean quadratic extensions.

Let f(x)∈Z[x] be an Nth degree polynomial that is monic and irreducible over Q. We say that f(x) is monogenic if {1,θ,θ2,…,θN−1} is a basis for the ring of integers of Q(θ), where f(θ)=0. We say that f(x) is cyclic if the Galois group of f(x) over Q is the cyclic group of order N. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences.

Abstract An irreducible monic polynomial f ( x ) ∈ Z [ x ] $f(x)\in \mathbb{Z}[x]$ is said to be monogenic if Z [ θ ] $\mathbb{Z}[\theta ]$ is the ring of integers of the number field Q ( θ ) $\mathbb{Q}(\theta )$ for a root θ of f ( x ). Let f ( x ) = x n + ax n −1 + bx n −2 + c , a 2 = 4 b , and g ( x ) ∈ Z [ x ] $g(x)\in \mathbb{Z}[x]$ be such that f ( g ( x )) is irreducible over Q $\mathbb{Q}$ . In this article, we establish a criterion that characterises the primes dividing the index, [ Q ( α ) : Z [ α ] ] $[\mathbb{Q}(\alpha ):\mathbb{Z}[\alpha ]]$ where α is a root of the composition f ◦ g ( x ). In fact, for certain choices of g ( x ), we prove that f ( x ) is monogenic if and only if any prime dividing disc( f ( x )) does not divide the index [ O K : Z [ α ] ] $[{\mathcal{O}}_{K}:\mathbb{Z}[\alpha ]]$ . We utilise this result to construct infinite families of monogenic polynomials.

We defined Krull modules over an integral domain as a completely integrally closed module satisfying ascending chain condition on integral v-submodules. However, if the dimension of the module of fractions over the quotient field is more than 1, then there are many non-integral submodules. Thus, we then defined strongly Krull modules which involves all non-zero submodules (whether it’s integral or non-integral). In this paper, we will prove that the Cartesian product of the ring of integers and the ring of p-integers is an example of Krull module which is not strongly Krull, showing that our definition of strongly Krull modules is indeed properly stronger than Krull modules.

When studying pure number fields, the discriminant and integral basis are cornerstone ideas that shed light on their structure and mathematical characteristics. The algebraic invariants and arithmetic behavior of a number field are affected by the discriminant, which encodes crucial information regarding the field’s ramification and the geometry of its ring of integers. To examine the field’s features, like the structure of its ideal class group and the solutions to Diophantine equations, an integral basis is a set of elements in the ring of integers that forms a basis over the integers. Class number analysis, ideal class group determination, and Hilbert symbol computing are only a few of the many important areas of number theory that benefit greatly from these ideas. Computerising discriminants for fields of high degree and discovering minimal integral bases for fields with complex ramification remain challenging tasks. We still require a deeper understanding of algebraic number theory and more advanced computational approaches to address these challenges, despite the significant progress we’ve made. In mathematical physics, coding theory, and cryptography, where the algebraic properties of number fields influence the efficacy and security of various algorithms, the study of integral bases and discriminants is fundamental for both theoretical and practical reasons.

Classical covering systems, introduced by Erdo’s, consist of congruences ai (mod ni) whose union covers all integers. Despite extensive work on their structural and extremal properties, little is known about analogues in algebraic settings. In this paper, we develop a unified framework for covering systems over algebraic domains, focusing on the ring of integers of a number field and the polynomial ring [x]. We define algebraic covering systems in both environments and establish necessary norm and degree-based conditions for full coverage and demonstrate that restricted families of ideals or polynomial moduli cannot yield coverings unless their reciprocal norm sums exceed explicit thresholds. We further provide structural examples, and counterexamples illustrating how factorization patterns, prime splitting, and residue structure influence covering behavior. Our results show that polynomial rings admit sharper and more uniform obstruction criteria than number fields, while number-field coverings exhibit arithmetic constraints governed by prime ideal decomposition.

In abstract algebra at the undergraduate level, the ring Z[sqrt(5)] is often used as a simple example of an integral domain that does not satisfy the unique factorization domain (UFD) but Z[sqrt(5)] is Halfway Factorial Domain (HFD). Unlike the ring of integers (Z) or the Gaussian integers (Z[i]) . Z[sqrt(5)] contains elements that admit non-unique factorizations, making it an interesting subject of study. A key challenge in analyzing the structure of lies in its limited group of units, consisting only of +-1, as well as the existence of irreducible elements that are not necessarily prime. This phenomenon leads to ambiguity in factorization, necessitating a deeper investigation into its arithmetic properties. This research aims to explore the factorization characteristics in Z[sqrt(5)], analyze irreducible elements and their relation to primality, and examine the implications of non-unique factorization on its algebraic structure. The findings are expected to contribute to a more comprehensive understanding of quadratic rings and their applications in number theory.

The Fundamental Theorem of Arithmetic (FTA), first formalized in Euclids Elements around 300 BCE, established the foundation for classical number theory, composed around 300 BCE. The principle of unique factorization later became central to the rise of modern mathematics. In the mid-19th century, mathematicians such as J.W.R. Dedekind and D. Hilbert extended number-theoretic questions into quadratic fields and rings of algebraic integers, creating the foundations of algebraic number theory. Steinitzs work in the early twentieth century of 1910 further generalized algebraic structures, marking the beginning of abstract algebra as an independent field. The purpose of this essay is to examine the Fundamental Theorem of Algebra's proof. and applies it to several representative problems in elementary number theory. It then extends to related corollaries and conceptual developments of unique factorization, including notable cases in non-unique factorization rings where the property does not hold. Finally, it introduces approaches grounded in properties of the FTA that have been applied to the ongoing study of major open problems, including the Goldbach Conjecture. Overall, through both review and mathematical analysis, the paper shows that the structural foundation provided by the FTA underlies the verification and proof of many of the most difficult results and open conjectures in mathematics, including Fermats Last Theorem and the Goldbach Conjecture.

The article considers modern directions of development and features of the architecture of neurocomputers. This study is devoted to the analysis of issues related to the organization and functioning of modern computers and their main components. The main components are considered and the components of the structure and functioning of modern computers of the von Neumann type are analyzed. The structure of computer memory is analyzed, and the organization and procedure of interaction between its levels are studied, in particular, the features of segmental organization of memory and problems and ways to solve them regarding ensuring protection are considered. The main principles of organization of computer buses, operating devices and other functioning systems are substantiated. The leading trends in the architecture of modern processors have been investigated, the principles of construction and architectural features of parallel computer systems have been established, new approaches to the study of computer architecture and the features of their construction have been proposed. The conceptual apparatus for neurocomputers has been proposed and improved, and their architectural features have been determined. It is substantiated that modern computer architectures are based on the principles of the von Neumann architecture, which allows for the joint storage of commands and data in a single address space of memory. The significant structural components of the research system, which are necessary for the effective functioning and fulfillment of the tasks, are analyzed. The peculiarities of the functioning of components of modern architecture are investigated and the main requirements for it are determined. The necessity and feasibility of the requirements determined by the specifics of the functioning of information systems, which ensure high speed, reliability and universality of use, are substantiated. The requirements for the operation of high-performance technical means, considered according to the criterion of compliance with productivity and resource consumption, can be considered a method of their research and development of new logical systems for organizing activities and productivity, as well as improving existing principles of organizing computer calculations. It has been established that the most effective way to improve functioning, considering this problem through the prism of the above-mentioned features, is parallel processing of actions, mediated by the simultaneous execution of programs or their constituent components, procedures, subroutines, on independent devices. The structure of a modular nanocomputer has been improved, ensuring the execution of specified operations and realizing a number of advantages of multitasking. The ways of enhancing the functional capabilities of a nanocomputer designed to represent and process both real and complex input data, as well as when processing them in a ring of integers, are determined by neural network algorithms.

We verify that each natural number $t>1$ can be expressed as a sum of unitary fractions of different natural numbers such that the least common multiple among them is coprime with a fixed value $v$. Using this fact, we characterize the real quadratic integers which appear in the representation of 1 as a unitary fraction sum over the real quadratic integer ring.

Abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.

Pick a random matrix γ in Γ = SL ( 2 , ℤ ) . Denote by 𝒪 𝕂 the Dedekind ring generated by its eigenvalues, and let Δ 𝕂 , Δ γ and Δ = Tr ( γ ) 2 - 4 be the respective discriminant of the rings 𝒪 𝕂 , the multiplier ring M ( 2 , ℤ ) ∩ ℚ [ γ ] and ℤ [ γ ] . We show that their ratios admit probability limit distributions when ordered by Frobenius norm. In particular, 42% of the elements of Γ have a fundamental discriminant, and ℤ [ γ ] is a ring of integers with probability 32%.

Abstract We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper bounds for the minimal ranks of ‐universal quadratic forms. For , and , we classify, up to equivalence, all classical, additively indecomposable binary quadratic forms.

Abstract The usual division algorithms on ${\mathbb {Z}}$ and ${\mathbb {Z}}[i]$ measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi _R$ . To the author’s knowledge, the integers, ${\mathbb {Z}}$ and the Gaussians, ${\mathbb {Z}}[i]$ are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi _{{\mathbb {Z}}}$ and $\phi _{{\mathbb {Z}}[i]}$ . This article presents the first division algorithm (that the author knows of) for ${\mathbb {Z}}[i]$ relative to $\phi _{{\mathbb {Z}}[i]}$ , empowering readers to perform the Euclidean algorithm on ${\mathbb {Z}}[i]$ using its minimal Euclidean function.

This research introduces a fortified cryptographic paradigm for safeguarding RGB image data, founded upon a substitution–permutation network (SPN) architecture defined over the residue class rings of Eisenstein integers modulo a selected prime element. The framework employs a dual-tier configuration of two independently constructed :8times:8 substitution boxes, each synthesized within the algebraic domain of :Z{left[omega:right]}_{pi:}, where :omega: represents a primitive cubic root of unity. The generation process for these substitution layers leverages affine transformations and their respective inverses, ensuring the simultaneous attainment of strong nonlinearity, bijective mapping, and algebraic robustness. Within the encryption pipeline, the initial substitution box functions exclusively as the confusion layer, whereas the second is designed to integrate both permutation and diffusion properties. To further amplify cryptographic intricacy, an auxiliary substitution box is formulated through the bitwise XOR amalgamation of the two primary layers, thereby intensifying inter-channel diffusion across the RGB spectrum. By harnessing the unique arithmetic and lattice geometry of Eisenstein integer residue classes, including their modular and inherently non-Euclidean characteristics, the scheme achieves superior confusion–diffusion balance. Compared with conventional image encryption techniques that rely solely on chaotic maps, Gaussian integers, or quaternion-based transformations, the proposed system offers enhanced algebraic structure exploitation, triple-layer substitution for richer nonlinear complexity, and explicit multi-channel diffusion. This leads to higher entropy, lower adjacent-pixel correlation, and greater resilience to both differential and linear cryptanalysis. Empirical assessments confirm the effectiveness of the proposed methodology, yielding near-optimal entropy, negligible adjacent-pixel correlation, and robust defense capabilities against an extensive range of established cryptanalytic strategies, thereby positioning the scheme as a compelling solution for secure imagetransmission in modern communication networks.

Abstract Let be a subset of , the ring of all algebraic integers. A polynomial is said to be integral‐valued on if for all . The set of all integral‐valued polynomials on forms a subring of containing . We say that is trivial if , and nontrivial otherwise. We give a collection of necessary and sufficient conditions on in order for to be nontrivial. Our characterizations involve, variously, topological conditions on with respect to fixed extensions of the ‐adic valuations to ; pseudo‐monotone sequences contained in ; ramification indices and residue field degrees; and the polynomial closure of in .