Greatest Common Divisor Research Articles

Infinite chains in the tree of numerical semigroups

AbstractOne major problem in the study of numerical semigroups is determining the growth of the semigroup tree. We study infinite chains of numerical semigroups in the semigroup tree, firstly introduced by Bras-Amorós (Semigroup Forum, 71:561–574, 2009). Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus $$g\ge 5$$ g ≥ 5 there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. Bras-Amorós (Semigroup Forum, 71:561–574, 2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multiplicity is 4 or 6 we prove a self-replication behavior in the subtree and provide a formula for the number of semigroups in infinite chains of a given genus and multiplicity 4 and 6, respectively.

On a union of homogeneous sequences

A sequence of positive integers is said to be quasi-primitive if there do not exist three distinct integers such that the greatest common divisor of two elements is not the third. It is clear that the sequence of prime numbers is quasi-primitive. The Erdős quasi-primitive conjecture states that the sum of the reciprocal of the product of an element by its logarithm over any quasi-primitive sequence is maximized by the sum over the sequence of all powers of prime numbers. Note that Erdős, before introduce his conjecture, showed that for any quasi-primitive sequence different from one, this sum is convergent and bounded by a real constant. In this article, by studying the analogous Erdős conjecture, we prove that when the real is sufficiently large, there exists a quasi-primitive sequence such that the sum of the reciprocal of the product of an element by the sum of its logarithm and , over this quasi-primitive sequence is very large than the sum over the sequence of all powers of prime numbers. Obviously, if the case where equals zero is also true, then the conjecture is false.

Generalized NTRU Algorithms on Algebraic Rings

The NTRU (Number Theory Research Unit) is a prominent post-quantum public key cryptography algorithm and a current focus of research. Although many NTRU variants have been proposed, a comprehensive generalization of these variants is still lacking. To address this issue, we propose the G-NTRU (Generalized NTRU), an algorithmic framework for NTRU variants on algebraic rings. This framework generalizes specific ring extensions to more general algebraic structures, allowing the unification of various NTRU variants. We then analyze the key properties of the G-NTRU, including correctness, lattice-based characteristics, and security. The semantic security of the G-NTRU is demonstrated through the introduction of the G-AGCD (Generalized Approximate Greatest Common Divisor). To validate the generality of the G-NTRU framework, we introduce the CNTRU (Complex Number NTRU), a variant of the NTRU over the ring of complex numbers. The CNTRU shares the same properties as the G-NTRU, further confirming the versatility of the G-NTRU in studying NTRU variants over algebraic rings.

Fault Tolerant Metric Dimension of Arithmetic Graphs

For a graph G , two vertices x , y ∈ G are said to be resolved by a vertex s ∈ G , if d ( x | s ) ≠ d ( y | s ) . The minimum cardinality of such a resolving set R in G is called its metric dimension. A resolving set R is said to be fault-tolerant, if for every p ∈ R , R − p preserves the property of being a resolving set. A fault-tolerant metric dimension of G is a minimal possible order fault-tolerant resolving set. A wide variety of situations, in which connection, distance, and connectivity are important aspects, call for the utilisation of metric dimension. The structure and dynamics of complex networks, as well as difficulties connected to robotics network design, navigation, optimisation, and facility positioning, are easier to comprehend as a result of this. As a result of the relevant concept of metric dimension, robots are able to optimise their methods of localization and navigation by making use of a limited number of reference locations. As a consequence of this, numerous applications of robotics, including collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph A l is defined as the graph with its vertex set as the set of all divisors of l , where l is a composite number and l = p γ 1 1 p η 2 2 , … , p n n , where p n ≥ 2 and the p i ’s are distinct primes. Two distinct divisors x , y of l are said to have the same parity if they have the same prime factors (i.e., x = p 1 p 2 and y = p 2 1 p 3 2 have the same parity). Further, two distinct vertices x , y ∈ A l are adjacent if and only if they have different parity and gcd ( x , y ) = p i (greatest common divisor) for some i ∈ { 1 , 2 , … , t } . This article is dedicated to the investigation of the arithmetic graph of a composite number l , which will be referred to throughout the text as A l . In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph A l , where l is a composite number.

Efficient quantum secure multi-party greatest common divisor protocol and its applications in private set operations

Private set intersection (PSI) has important application value, however, current quantum PSI protocols are either unsuitable for multi-party scenarios or inefficient. Recently, Imran (arXiv:2303.17196v3, 2023) proposed two quantum secure multi-party greatest common divisor (GCD) protocols that can be used for PSI, but with the downside of information leakage and resource consumption. In this paper, we propose a novel quantum secure multi-party GCD protocol that has higher security and lower complexity. To hide privacy, each party randomly selects a coefficient within a range determined by his input integer, and with the assistance of a semi-honest third party TP, all parties secretly calculate the linear combination of their inputs under these coefficients. Once enough linear combinations are collected, TP calculates the GCD of these combinations, which is equal to the GCD of all input integers. To verify the honesty of participants, a quantum zero-knowledge proof sub-protocol is designed. Analysis shows that our GCD protocol is correct and has security against malicious attacks. Moreover, its complexity is polynomial level and lower than Imran’s. Furthermore, we demonstrate the scalability of our GCD protocol in private set operations, such as private set intersection, private set intersection cardinality, private multi-set intersection, etc.

Loading conditions for self‐organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability that a particle changes type at each step. In the case where , the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers . It has been proved that, if , and whether or and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of and be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net . This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.

On generalized Star-GCD extensions

– Let ⋆ be a star operation on a ring extension R ⊆ S . The ring extension is said to be a generalized ⋆ -greatest common divisor (G- ⋆ GCD) extension if the intersection of two ⋆ -invertible ideals of R is ⋆ -invertible. We investigate properties of G- ⋆ GCD extensions. We give special attention to polynomial rings and Nagata rings.

The prime-power hypothesis on the codegrees of irreducible characters

For a character χ of a finite group G, the number cod ( χ ) = [ G : Ker χ ] χ ( 1 ) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod ( χ ) ≠ cod ( ϕ ) , the greatest common divisor of cod ( χ ) and cod ( ϕ ) is a prime-power, then either G is solvable or G / Sol ( G ) is isomorphic to one of the groups Alt 5 or PSL 2 ( 8 ) , where Sol ( G ) is the solvable radical of the group G.

Gauged permutation invariant matrix quantum mechanics: partition functions

The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U(N) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.

Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation

To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with multivariate polynomials with a priori errors on their coefficients, using such modular algorithms becomes challenging. This is because any resulting approximate GCD computed in one variable may have perturbations depending on the evaluation point and may not be an image of the same desired multivariate approximate GCD. This necessitates computing it as given multivariate polynomials, and operating with large matrices whose size is exponential in the number of variables. In this paper, we present a new modular algorithm, suitable for dense cases and effective for sparse ones, called “SLRA interpolation”. This algorithm uses the multidimensional fast Fourier transform (FFT) and the structured low-rank approximation (SLRA) of non-square block diagonal matrices. The SLRA interpolation technique may reduce the time-complexity for one iteration in the computation of approximate GCD of several multivariate polynomials, especially for the sparse case.

Finding The Inverse of A Polynomial Modulo in The Ring Z[X] Based on The Method of Undetermined Coefficients

This paper presents the theoretical foundations of finding the inverse of a polynomial modulo in the ring Z[x] based on the method of undetermined coefficients. The use of the latter makes it possible to significantly reduce the time complexity of calculations avoiding the operation of finding the greatest common divisor. An example of calculating the inverse of a polynomial modulo in the ring Z[x] based on the proposed approach is given. Analytical expressions of the time complexities of the developed and classical methods depending on the degrees of polynomials are built. The graphic dependence of the complexity of performing the operation of finding the inverse of a polynomial in the ring Z[x] is presented, which shows the advantages of the method based on undetermined coefficients. It is found that the efficiency of the developed method increases logarithmically with an increase in the degrees of polynomials.

Value Recognition: The International Communication of Shared Human Values and Aesthetics: A Case Study of the Overseas Dissemination and Reception of Chinese Literature

International communication should fundamentally aim at promoting the exchange and mutual learning of civilizations, achieving a transformation from “cultural clashes” to “civilizational exchange and mutual learning.” The manifestation of international communication that facilitates this exchange and mutual learning is that the values contained in the communication texts ultimately become integrated into the culture of the receiving party and even into human culture as a whole due to this exchange and mutual learning. Achieving civilizational exchange and mutual learning in international communication requires “value recognition” efforts between the disseminating and receiving parties, focusing on the universal values of human happiness for all, which is the “greatest common divisor” of value recognition among different receiving subjects. International communication that facilitates the exchange and mutual learning of civilizations must take the inevitable route of such international communication: the path of aesthetic communication.

Characterization of some specific lee distance codes over finite rings

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over [Formula: see text] which have applications in error detection and error correction.

Structured ramp secret sharing schemata over rings of real polynomials

Two new ramp secret sharing schemata based on polynomials are proposed. For both schemata, the secret is considered to be a polynomial created by the dealer. The participants are separated into ℓ⩾2, groups, that are specified by the dealer's levels Li for i=1,2,…,ℓ and each level Li, i⩾2, is separated into subsets. The shares of the secret are given to participants in the form of polynomials. For the first proposed scheme, the dealer creates ℓ polynomials one for each level. Specific participants from every subset of each level have to cooperate all together in order to construct the polynomial of their level. Next all the authorized participants cooperate for computing the greatest common divisor of the polynomials in order to retrieve the secret. In the second scheme, the authorized participants cooperate per two levels using a bottom-up procedure. In both schemata the greatest common divisor can be evaluated by implementing numerical linear algebra methods, and precisely factorization of matrices of special form such as Sylvester matrices. The triangularization of these matrices can be obtained by exploiting their special structure for the reduction of the required floating point operations. The innovative idea of the paper at hand is the use of real polynomials in secret sharing schemata. This is particularly useful since the greatest common divisor can always be computed with efficient accuracy using effective numerical methods. New theoretical results are proved and provided that support the error analysis of our approach.

Enhancing Global Blockchain Privacy via a Digital Mutual Trust Mechanism

Blockchain technology, initially developed as a decentralized and transparent mechanism for recording transactions, faces significant privacy challenges due to its inherent transparency, exposing sensitive transaction data to all network participants. This study proposes a blockchain privacy protection algorithm that employs a digital mutual trust mechanism integrated with advanced cryptographic techniques to enhance privacy and security in blockchain transactions. The contribution includes the development of a new dynamic Byzantine consensus algorithm within the Practical Byzantine Fault Tolerance framework, incorporating an authorization mechanism from the reputation model and a proof consensus algorithm for robust digital mutual trust. Additionally, the refinement of homomorphic cryptography using the approximate greatest common divisor technique optimizes the encryption process to support complex operations securely. The integration of a smart contract system facilitates automatic and private transaction execution across the blockchain network. Experimental evidence demonstrates the superior performance of the algorithm in handling privacy requests and transaction receipts with reduced delays and increased accuracy, marking a significant improvement over existing methods.

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

A complex case of Vojta’s general abc conjecture and cases of Campana’s orbifold conjecture

We show a truncated second main theorem of level one with explicit exceptional sets for analytic maps into P 2 \mathbb P^2 intersecting the coordinate lines with sufficiently high multiplicities. The proof is based on a greatest common divisor theorem for an analytic map f : C ↦ P n f:\mathbb C\mapsto \mathbb P^n and two homogeneous polynomials in n + 1 n+1 variables with coefficients which are meromorphic functions of the same growth as the analytic map f f . As applications, we study some cases of Campana’s orbifold conjecture for P 2 \mathbb P^2 and finite ramified covers of P 2 \mathbb P^2 with three components admitting sufficiently large multiplicities. In addition, we explicitly determine the exceptional sets. Consequently, it implies the strong Green-Griffiths-Lang conjecture for finite ramified covers of G m 2 \mathbb G_m^2 .

The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps

We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein’s greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities.

Euclid’s Algorithm and its Role in Solving Modular Multiplicative Inverse

In this article, the prime motivation is to demonstrate how calculating modular multiplicative inverse can be simplified computationally with the help of Euclid’s algorithm, which is usually attributed to finding the Greatest Common Divisor.

A criterion for nondensity of integral points

AbstractWe give a general criterion for Zariski degeneration of integral points in the complement of a divisor with components in a variety of dimension defined over or over a quadratic imaginary field. The key condition is that the intersection of the components of is not well approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a greatest common divisor (GCD) bound for algebraic points in varieties, which can be of independent interest.