Finite Rings Research Articles

Sperner’s theorem for non-free modules over finite chain rings

Sperner’s theorem for non-free modules over finite chain rings

Linear complementary pairs of constacyclic n-D codes over a finite commutative ring

Linear complementary pairs of constacyclic n-D codes over a finite commutative ring

Universal adjacency spectrum of the cozero-divisor graph and its complement on a finite commutative ring with unity

For a finite simple undirected graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree diagonal matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. The cozero-divisor graph [Formula: see text] of a finite commutative ring [Formula: see text] with unity is a simple undirected graph with the set of all nonzero nonunits of [Formula: see text] as vertices and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, we study structural properties of [Formula: see text] by defining an equivalence relation on its vertex set in terms of principal ideals of the ring [Formula: see text]. Then we obtain the universal adjacency eigenpairs of [Formula: see text] and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of [Formula: see text] and [Formula: see text] in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo [Formula: see text] in a simpler form.

Hulls of constacyclic codes over finite non-chain rings and their applications in quantum codes construction

Hulls of constacyclic codes over finite non-chain rings and their applications in quantum codes construction

On the rank decoding problem over finite principal ideal rings

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings.In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

Some results on the finite rings with maximal size of pairwise non-commuting elements is 5

Let R be a finite ring and let X be a non-empty subset of R. If ab=ba for any two distinct a,b∈X, then X is called a set of pairwise non-commuting elements of R. Moreover, X is said to be a set of pairwise non-commuting elements of R with maximal size if its cardinality is the largest one among all suchsets. In this paper, we study the structures for some finite rings with maximal size of pairwise non-commuting elements is 5.

Integrated Shared Random Key Agreement Protocol for Wireless Sensor Network

The secured data transmission in Wireless Sensor Network (WSN) relies on effective key generation and secured sharing. The generated key must be random to enhance data confidentiality. The processes associated with the security in WSN must be designed at reduced computing time and communication cost. Our research work aims to design a novel lightweight key-sharing protocol that is needed for ensuring data confidentiality. The protocol must meet the constraints of WSN by being lightweight and consuming less energy. Security breaches in WSNs occur due to insecure keys. This can be overcome by generating shared keys which are generated once using the dynamic features of Sensor Nodes (SNs) when the Cluster Heads (CHs) are selected. In this research work, we have generated the Master Shared Key (MSK) at the transmitter node by forming a Galois Ring (GR) using WSN parameters and derived the Shared Random Key (SRK) using matched positions of exchanged Random Sequences (RSs). It is protected using a Physically Unclonable Function (PUF). The novelty lies in the SRK generation from MSK which is chosen at random from the polynomials generated during the formation of GR. The MSK is securely shared with the receiver node by encrypting using a Preloaded Key (PK). After this exchange, the key for encryption and decryption is derived by the transmitter and the receiver by exchanging RSs. The SRK is then encrypted using a key which is a unique fingerprint of the SN generated using PUF and stored in the SNs and the CHs to prevent node capture attack that occurs in WSN. Our proposed Shared Random Key Agreement Protocol (SRKAP) is comparable to the Localized Encryption and Authentication Protocol (LEAP) and performs better compared to the Elliptic Curve Diffie Hellman (ECDH) algorithm

XOR count and block circulant MDS matrices over finite commutative rings

<p>Block circulant MDS matrices are used in the design of linear diffusion layers for lightweight cryptographic applications. Most of the work on construction of block circulant MDS matrices focused either on finite fields or $ GL(m, \mathbb{F}_2) $. The main objective of this paper is to extend the above study of block circulant MDS matrices to finite commutative rings. Additionally, we examine the behavior of the XOR count distribution under different reducible polynomials of equal degree over $ \mathbb{F}_2 $. We show that the determinant of a block circulant matrix over a ring can be expressed in a simple form. We construct $ 4 \times 4 $ and $ 8 \times 8 $ block circulant matrices over a ring. Furthermore, for non-negative integer $ l $, we identify the conditions under which a ring $ \mathfrak{R}_l = \frac{\mathbb{F}_2[x]}{\langle (f(x))^{2^l} \rangle} $, contains a finite field of order $ 2^m $, where $ f(x) $ is an irreducible polynomial of degree $ m $. To facilitate efficient implementation, we analyze XOR distributions within specific rings, such as $ R_1 = \frac{\mathbb{F}_2[x]}{\langle (1+x^2+x^6) \rangle} $ and $ R_2 = \frac{\mathbb{F}_2[x]}{\langle (1+x^4+x^6) \rangle} $. Our calculations reveal distinct XOR distributions when utilizing two reducible polynomials of equal degree, with XOR count distributions 776 and 764, respectively. However, when using irreducible polynomials of the same degree, the XOR count distributions remain the same. This difference is advantageous for applications in lightweight cryptography.</p>

Common zero-divisor graph over a commutative ring

Let [Formula: see text] be a finite commutative ring with non-zero identity and [Formula: see text] be the set of all non-zero zero-divisors of [Formula: see text]. In this paper, for a non-empty subset [Formula: see text] of [Formula: see text], we introduce a new graph, denoted by [Formula: see text], with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there is [Formula: see text] such that [Formula: see text]. Then, we investigate some properties of the graph [Formula: see text].

On the hardness of the Lee syndrome decoding problem

<p style='text-indent:20px;'>In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric. We first prove that the decisional version of the problem is NP-complete, by a reduction from the <inline-formula><tex-math id="M1">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-dimensional matching problem. Then, we study the complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite fields to the Lee metric over finite rings, as well as proposing some novel solutions. For the analyzed algorithms, we assess the computational complexity in the asymptotic regime and compare it to the corresponding algorithms in the Hamming metric.</p>

Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

<p style='text-indent:20px;'>In this work, we define a modification of a bordered construction for self-dual codes which utilises <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.</p>

Number of cliques of Paley-type graphs over finite commutative local rings

Number of cliques of Paley-type graphs over finite commutative local rings

Classification of Certain Cyclic LCD Codes

We show that a necessary and sufficient condition for a cyclic code C of length N over a finite chain ring R (whose maximal ideal has nilpotence 2) to be an LCD code is that C = (f(x)), where f(X) is a self-reciprocal monic divisor of XN − 1 in R[X] and x = X + (XN − 1) in R[X]/(XN − 1). A similar, but slightly different, theorem was proved in 2019 by Z. Liu and J. Wang for general finite chain rings (Theorem 25 in [5]). We provide two proofs, both completely different than the proof of Liu and Wang.

On permutation polynomials modulo 7n

A polynomial that permutes the elements of a finite ring R is known as a permutation polynomial. The conditions for the coefficients of a polynomial of degree d to be a permutation polynomial modulo pn, are known for p = 2, 3, 5. In this paper, we obtain the necessary and sufficient conditions on the coefficients of a polynomial modulo 7n to be a permutation polynomial.

Group of units of a certain class of finite rings of characteristic p2

Group of units of a certain class of finite rings of characteristic p2

A new public key cryptosystem based on the non-commutative ring R

In this paper, we present a new public cryptosystem based on one of the most problems (word problem, solving a non-linear system problem, integer factorization problem and discrete logarithm problem,...) is the conjugal classical problem (CCP) over a non commutative ring. In generally this cryptosystem is very difficult to solve for decipher. Firstly, by using the elliptic curve over the finite ring Fq[e], where e4 = e3, with q = pd and p is prime number greater than or equal to 5, d ϵ ℕ* [12] we define the non-commutative ring R. Secondly, by using the protocol of Diffie-Hellman [4] we have proposed a novel encryption scheme on R and we study the problem CCP over it. Precisely, we will make a new fully homomorphic encryption scheme over the ring R based on the two difficult problems; conjugal classical problem and discrete logarithm problem.

Szeged Index and Padmakar-Ivan Index of Nilpotent Graph of Integer Modulo Ring with Prime Power Order

Recently, graphs have started to be used to represent a finite ring. Nikmehr and Khojasteh in the article defined the nilpotent graph of a ring . Denoted , is a graph with the set of vertices being all the elements in the ring Two vertices and are adjacent if and only if is nilpotent elements in the ring . Topological index is a field that discusses graph structure based on the degree of each vertex of a graph and the distance between vertices. In this study, the author will gives the general formula of the Szeged index and Padmakar-Ivan index of the nilpotent graph graph of the modulo ring with prime power order. The result of this research is a general formula for the topological indices of nilpotent graphs of the integer modulo ring, called the Szeged index and the Padmakar-Ivan index.

On the ℓ-DLIPs of codes over finite commutative rings

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of ℓ-dimension linear intersection pairs (ℓ-DLIPs) of codes over a finite commutative ring (R), for some positive integer ℓ. In this paper, we study ℓ-DLIP of codes over R in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) ℓ-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring which helps us to get an important characterization of ℓ-DLIP of constacyclic codes. Finally, the ℓ-DLIP of constacyclic codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.

Additive conjucyclic codes over a class of Galois rings

As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except Fq2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_{q^2}$$\\end{document}) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document}, where r≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 2$$\\end{document} is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document} and the family of linear cyclic codes of length 2n over Z2r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2^r}$$\\end{document}. This correspondence helps to obtain additive conjucyclic codes over GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document} via known linear cyclic codes over Z2r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2^r}$$\\end{document}. We prove that the trace dual CTr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {C}}^{Tr}$$\\end{document} of an additive conjucyclic code C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {C}}$$\\end{document} is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z2r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2^r}$$\\end{document} contained in additive conjucyclic codes over GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document}. Last but not least, we explicitly derive the generator matrices for these codes.

Elliptic curves over a finite ring

Let Fq be a finite field, where q is a power of a prime p such that p>5. Let \alpha be a root of a monic polynomial of a minimal degree over Fq. In this paper, we will study elliptic curves over (Fq[\alpha],+,*), where + is the usual addition and * represent a non-standard product law over Fq[\alpha]. Elliptic curves over this ring can be used to create a new type of cryptography. The method is fast, simple, and secure.