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17 pages, 726 KiB

Open AccessArticle

A Post-Quantum Public-Key Signcryption Scheme over Scalar Integers Based on a Modified LWE Structure

by Mostefa Kara, Mohammad Hammoudeh, Abdullah Alamri and Sultan Alamri

Sensors 2025, 25(15), 4728; https://doi.org/10.3390/s25154728 (registering DOI) - 31 Jul 2025

To ensure confidentiality and integrity in the era of quantum computing, most post-quantum cryptographic schemes are designed to achieve either encryption or digital signature functionalities separately. Although a few signcryption schemes exist that combine these operations into a single, more efficient process, they [...] Read more.

To ensure confidentiality and integrity in the era of quantum computing, most post-quantum cryptographic schemes are designed to achieve either encryption or digital signature functionalities separately. Although a few signcryption schemes exist that combine these operations into a single, more efficient process, they typically rely on complex vector, matrix, or polynomial-based structures. In this work, a novel post-quantum public-key encryption and signature (PQES) scheme based entirely on scalar integer operations is presented. The proposed scheme employs a simplified structure where the ciphertext, keys, and core cryptographic operations are defined over scalar integers modulo n, significantly reducing computational and memory overhead. By avoiding high-dimensional lattices or ring-based constructions, the PQES approach enhances implementability on constrained devices while maintaining strong security properties. The design is inspired by modified learning-with-errors (LWE) assumptions, adapted to scalar settings, making it suitable for post-quantum applications. Security and performance evaluations, achieving a signcryption time of 0.0007 s and an unsigncryption time of 0.0011 s, demonstrate that the scheme achieves a practical balance between efficiency and resistance to quantum attacks. Full article

(This article belongs to the Section Intelligent Sensors)

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13 pages, 295 KiB

Open AccessArticle

On D_α-Spectrum of the Weakly Zero-Divisor Graph of ℤ_n

by Amal S. Alali, Mohd Rashid, Asif Imtiyaz Ahmad Khan and Muzibur Rahman Mozumder

Mathematics 2025, 13(15), 2385; https://doi.org/10.3390/math13152385 - 24 Jul 2025

Viewed by 149

Abstract

Let us consider the finite commutative ring R, whose unity is

1 \neq 0 .

Its weakly zero-divisor graph, represented as

W Γ (R)

, is a basic undirected graph with two distinct vertices,

c_{1}

and

c_{2}

, [...] Read more.

Let us consider the finite commutative ring R, whose unity is

1 \neq 0 .

Its weakly zero-divisor graph, represented as

W Γ (R)

, is a basic undirected graph with two distinct vertices,

c_{1}

and

c_{2}

, that are adjacent if and only if there exist

r \in

ann (c_{1})

and

s \in

ann (c_{2})

that satisfy the condition

r s = 0

. Let

D (G)

be the distance matrix and

T r (G)

be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The

D_{α}

matrix of graph G is defined as

D_{α} (G) = α T r (G) + (1 - α) D (G)

for

α \in [0, 1]

. This article finds the

D_{α}

spectrum for the graph

W Γ (Z_{n})

for various values of n and also shows that

W Γ (Z_{n})

for

n = ϑ_{1} ϑ_{2} ϑ_{3} \dots ϑ_{t} η_{1}^{d_{1}} η_{2}^{d_{2}} \dots η_{s}^{d_{s}} (d_{i} \geq 2, t \geq 1, s \geq 0)

, where

ϑ_{i}

’s and

η_{i}

’s are the distinct primes, is

D_{α}

integral. Full article

(This article belongs to the Section E: Applied Mathematics)

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28 pages, 403 KiB

Open AccessArticle

Domination Parameters of Unit Graphs of Rings

by Ting Du and Aiping Gan

Axioms 2025, 14(6), 399; https://doi.org/10.3390/axioms14060399 - 23 May 2025

Viewed by 261

Abstract

Let R be a finite commutative ring with nonzero identity 1. In this paper, domination parameters, the domination number and the total dominating number, of the unit graph

G (R)

or the closed unit graph

\bar{G} (R)

of [...] Read more.

Let R be a finite commutative ring with nonzero identity 1. In this paper, domination parameters, the domination number and the total dominating number, of the unit graph

G (R)

or the closed unit graph

\bar{G} (R)

of R are investigated. To study the domination number of

G (R)

, we prove that it suffices to consider the case when R is a direct product of fields. Furthermore, we discuss the domination number and total dominating number of the unit graph of the ring of integers modulo n. Full article

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25 pages, 463 KiB

Open AccessFeature PaperArticle

Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes

by Perathorn Pooksombat and Wittawat Kositwattanarerk

Mathematics 2025, 13(10), 1562; https://doi.org/10.3390/math13101562 - 9 May 2025

Viewed by 276

Abstract

Lattice codes play an important role in wireless communication and are closely related to linear codes. Multi-level constructions of complex lattices from codes are known to produce lattice codes with desirable parameters and efficient encoding and decoding of information bits. However, their constellation [...] Read more.

Lattice codes play an important role in wireless communication and are closely related to linear codes. Multi-level constructions of complex lattices from codes are known to produce lattice codes with desirable parameters and efficient encoding and decoding of information bits. However, their constellation usually involves superfluous elements that need to be mapped to a representative within the same coset to reduce average transmission power. One such elegant shaping function is a componentwise modulo, which is known to produce a cubic shaping for Barnes–Wall lattices. In this paper, we generalize this result to lattices over quadratic rings of integers, thus encompassing constructions from p-ary codes, where p is a prime number. We identify all bases that permit cubic modulo shaping. This provides useful insights into practical encoding and decoding of lattice codes from multi-level constructions. Full article

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23 pages, 920 KiB

Open AccessArticle

On Algebraic Properties of Primitive Eisenstein Integers with Applications in Coding Theory

by Abdul Hadi, Uha Isnaini, Indah Emilia Wijayanti and Martianus Frederic Ezerman

Entropy 2025, 27(4), 337; https://doi.org/10.3390/e27040337 - 24 Mar 2025

Cited by 1 | Viewed by 1460

Abstract

An even Eisenstein integer is a multiple of an Eisenstein prime of the least norm. Otherwise, an Eisenstein integer is called odd. An Eisenstein integer that is not an integer multiple of another one is said to be primitive. Such integers can be [...] Read more.

An even Eisenstein integer is a multiple of an Eisenstein prime of the least norm. Otherwise, an Eisenstein integer is called odd. An Eisenstein integer that is not an integer multiple of another one is said to be primitive. Such integers can be used to construct signal constellations and complex-valued codes over Eisenstein integers via a carefully designed modulo function. In this work, we establish algebraic properties of even, odd, and primitive Eisenstein integers. We investigate conditions for the set of all units in a given quotient ring of Eisenstein integers to form a cyclic group. We perform set partitioning based on the multiplicative group of the set. This generalizes the known partitioning of size a prime number congruent to 1 modulo 3 based on the multiplicative group of the Eisenstein field in the literature. Full article

(This article belongs to the Special Issue Discrete Math in Coding Theory)

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22 pages, 1715 KiB

Open AccessArticle

Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)

by Amal Alsaluli, Wafaa Fakieh and Hanaa Alashwali

Axioms 2024, 13(12), 873; https://doi.org/10.3390/axioms13120873 - 15 Dec 2024

Cited by 1 | Viewed by 858

Abstract

In this paper, we examine the interplay between the structural and spectral properties of the unit graph

G (Z_{n})

for

n = p_{1}^{r_{1}} p_{2}^{r_{2}} \dots p_{k}^{r_{k}}

, where [...] Read more.

In this paper, we examine the interplay between the structural and spectral properties of the unit graph

G (Z_{n})

for

n = p_{1}^{r_{1}} p_{2}^{r_{2}} \dots p_{k}^{r_{k}}

, where

p_{1}, p_{2}, \dots, p_{k}

are distinct primes and

k, r_{1}, r_{2}, \dots, r_{k}

are positive integers such that at least one of the

r_{i}

must be greater than 1. We first analyze the structure of the unit graph of

Z_{n}

, treating it as what we will define as a ‘generalized join graph’ under these conditions. We then determine the Laplacian spectrum of

G (Z_{n})

and deduce that it is integral for all n. Consequently, we obtain the Laplacian spectral radius and algebraic connectivity of

G (Z_{n})

. We also prove that the vertex connectivity of

G (Z_{p q})

is

(p - 2) q

, where

2 \neq p < q

. We deduce the vertex connectivity of

G (Z_{n})

when

n = p^{r} q^{s}

, where

2 \neq p < q

are primes and

r, s

are positive integers. Finally, we present conjectures regarding the vertex connectivity of

G (Z_{n})

when

n = p_{1} p_{2} \dots p_{k}

and

n = p_{1}^{r_{1}} p_{2}^{r_{2}} \dots p_{k}^{r_{k}}

, where

p_{i}

are distinct primes,

r_{i}

are positive integers, and

1 \leq i \leq k

. Full article

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17 pages, 512 KiB

Open AccessArticle

Decoding of

Z_{2^{S}}

Linear Generalized Kerdock Codes

by Aleksandar Minja and Vojin Šenk

Mathematics 2024, 12(3), 443; https://doi.org/10.3390/math12030443 - 30 Jan 2024

Cited by 1 | Viewed by 1037

Abstract

Many families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the

Z_{4}

ring (ring of integers modulo 4), by applying the Gray map to the quaternary symbols. Generalized Kerdock codes represent an [...] Read more.

Many families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the

Z_{4}

ring (ring of integers modulo 4), by applying the Gray map to the quaternary symbols. Generalized Kerdock codes represent an extension of classical Kerdock codes to the

Z_{2^{S}}

ring. In this paper, we develop two novel soft-input decoders, designed to exploit the unique structure of these codes. We introduce a novel soft-input ML decoding algorithm and a soft-input soft-output MAP decoding algorithm of generalized Kerdock codes, with a complexity of

O (N^{S} {log}_{2} N)

, where N is the length of the

Z_{2^{S}}

code, that is, the number of

Z_{2^{S}}

symbols in a codeword. Simulations show that our novel decoders outperform the classical lifting decoder in terms of error rate by some 5 dB. Full article

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14 pages, 402 KiB

Open AccessArticle

On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ_n

by Nazim, Nadeem Ur Rehman and Ahmad Alghamdi

Mathematics 2023, 11(20), 4310; https://doi.org/10.3390/math11204310 - 16 Oct 2023

Cited by 5 | Viewed by 1671

Abstract

For a finite commutative ring

R

with identity

1 \neq 0

, the weakly zero-divisor graph of

R

denoted as

W Γ (R)

is a simple undirected graph having vertex set as a set of non-zero zero-divisors of

R

and two [...] Read more.

For a finite commutative ring

R

with identity

1 \neq 0

, the weakly zero-divisor graph of

R

denoted as

W Γ (R)

is a simple undirected graph having vertex set as a set of non-zero zero-divisors of

R

and two distinct vertices

a

and

b

are adjacent if and only if there exist elements

r \in ann (a)

and

s \in ann (b)

satisfying the condition

r s = 0

. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph

W Γ (R)

. Specifically, the investigation is carried out on the weakly zero-divisor graph

W Γ (Z_{n})

for various values of

n

. Full article

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25 pages, 1062 KiB

Open AccessArticle

On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs

by Amal S. Alali, Shahbaz Ali, Muhammad Adnan and Delfim F. M. Torres

Symmetry 2023, 15(10), 1911; https://doi.org/10.3390/sym15101911 - 12 Oct 2023

Cited by 2 | Viewed by 1446

Abstract

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications [...] Read more.

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph

G (Z_{n})

is called a zero-divisor graph over the zero divisors of a commutative ring

Z_{n}

, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded. Full article

(This article belongs to the Special Issue Symmetry in Differential Geometry and Geometric Analysis)

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12 pages, 312 KiB

Open AccessArticle

Spectrum of the Cozero-Divisor Graph Associated to Ring

Z_{n}

by Mohd Rashid, Amal S. Alali, Wasim Ahmed and Muzibur Rahman Mozumder

Axioms 2023, 12(10), 957; https://doi.org/10.3390/axioms12100957 - 11 Oct 2023

Cited by 2 | Viewed by 1709

Abstract

Let R be a commutative ring with identity

1 \neq 0

and let

Z {(R)}^{'}

be the set of all non-unit and non-zero elements of ring R.

Γ^{'} (R)

denotes the cozero-divisor graph of R and [...] Read more.

Let R be a commutative ring with identity

1 \neq 0

and let

Z {(R)}^{'}

be the set of all non-unit and non-zero elements of ring R.

Γ^{'} (R)

denotes the cozero-divisor graph of R and is an undirected graph with vertex set

Z {(R)}^{'}

,

w \notin z R

, and

z \notin w R

if and only if two distinct vertices w and z are adjacent, where

q R

is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs

Γ^{'} (Z_{n})

. We also show that the cozero-divisor graph

Γ^{'} (Z_{p_{1} p_{2}})

is a signless Laplacian integral. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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8 pages, 240 KiB

Open AccessFeature PaperArticle

A General Construction of Integer Codes Correcting Specific Errors in Binary Communication Channels

by Hristo Kostadinov and Nikolai Manev

Mathematics 2023, 11(11), 2521; https://doi.org/10.3390/math11112521 - 31 May 2023

Cited by 1 | Viewed by 1376

Abstract

Integer codes have been successfully applied to various areas of communication and computer technology. They demonstrate good performance in correcting specific kinds of errors. In many cases, the used integer codes are constructed by computer search. This paper presents an algebraic construction of [...] Read more.

Integer codes have been successfully applied to various areas of communication and computer technology. They demonstrate good performance in correcting specific kinds of errors. In many cases, the used integer codes are constructed by computer search. This paper presents an algebraic construction of integer codes over the ring of integers modulo

A = 2^{n} + 1

capable of correcting at least up to two bit errors in a single b-byte. Moreover, the codes can correct some configurations of three or more erroneous bits, but not all possible ones. The construction is based on the use of cyclotomic cosets of 2 modulo A. Full article

(This article belongs to the Section A: Algebra and Logic)

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8 pages, 267 KiB

Open AccessArticle

On Monochromatic Clean Condition on Certain Finite Rings

by Kai An Sim, Wan Muhammad Afif Wan Ruzali, Kok Bin Wong and Chee Kit Ho

Mathematics 2023, 11(5), 1107; https://doi.org/10.3390/math11051107 - 22 Feb 2023

Viewed by 1260

Abstract

For a finite commutative ring R, let

a, b, c \in R

be fixed elements. Consider the equation

a x + b y = c z

where x, y, and z are idempotents, units, and any element in [...] Read more.

For a finite commutative ring R, let

a, b, c \in R

be fixed elements. Consider the equation

a x + b y = c z

where x, y, and z are idempotents, units, and any element in the ring R, respectively. We say that R satisfies the r-monochromatic clean condition if, for any r-colouring

χ

of the elements of the ring R, there exist

x, y, z \in R

with

χ (x) = χ (y) = χ (z)

such that the equation holds. We define

m_{(a, b, c)} (R)

to be the least positive integer r such that R does not satisfy the r-monochromatic clean condition. This means that there exists

χ (i) = χ (j)

for some

i, j \in {x, y, z}

where

i \neq j

. In this paper, we prove some results on

m_{(a, b, c)} (R)

and then formulate various conditions on the ring R for when

m_{(1, 1, 1)} (R) = 2

or 3, among other results concerning the ring

Z_{n}

of integers modulo n. Full article

8 pages, 303 KiB

Open AccessArticle

Polyadic Rings of p-Adic Integers

by Steven Duplij

Symmetry 2022, 14(12), 2591; https://doi.org/10.3390/sym14122591 - 7 Dec 2022

Viewed by 1600

Abstract

In this note, we first recall that the sets of all representatives of some special ordinary residue classes become

(m, n)

-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find [...] Read more.

In this note, we first recall that the sets of all representatives of some special ordinary residue classes become

(m, n)

-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine when the representatives form a

(m, n)

-ring. At very short spacetime scales, such rings could lead to new symmetries of modern particle models. Full article

(This article belongs to the Special Issue Symmetry in Cosmology and Gravity: Topic and Advance)

16 pages, 305 KiB

Open AccessArticle

Bounds for Coding Theory over Rings

by Niklas Gassner, Marcus Greferath, Joachim Rosenthal and Violetta Weger

Entropy 2022, 24(10), 1473; https://doi.org/10.3390/e24101473 - 16 Oct 2022

Cited by 5 | Viewed by 2458

Abstract

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there [...] Read more.

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov’s weight over the integers modulo 2^s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight. Full article

(This article belongs to the Special Issue Information Theoretic Methods for Future Communication Systems)

12 pages, 304 KiB

Open AccessArticle

A_γ Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings

by Bilal Ahmad Rather, Fawad Ali, Asad Ullah, Nahid Fatima and Rahim Dad

Symmetry 2022, 14(8), 1710; https://doi.org/10.3390/sym14081710 - 17 Aug 2022

Cited by 10 | Viewed by 2161

Abstract

The

A_{γ}

matrix of a graph G is determined by

A_{γ} (G) = (1 - γ) A (G) + γ D (G),

where

0 \leq γ \leq 1

, [...] Read more.

The

A_{γ}

matrix of a graph G is determined by

A_{γ} (G) = (1 - γ) A (G) + γ D (G),

where

0 \leq γ \leq 1

,

A (G)

and

D (G)

are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the

A_{γ}

matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more

γ

adjacency-type matrices. In this paper, we obtain the

A_{γ}

eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. Full article

(This article belongs to the Special Issue Topological Graph Theory and Discrete Geometry)

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