Non-abelian Group Research Articles

Retracted: A New Key Exchange Protocol Based on Infinite Non-Abelian Groups

Retracted: A New Key Exchange Protocol Based on Infinite Non-Abelian Groups

Secure Image Encryption using E-Fractal-Based Non-Commutative Group and Hash Function

Due to the importance of data security at present, an encryption algorithm based on the principle of non-commutative group, hash function, and E-fractal has been proposed. Key generation depends on the array values generated by the braid group and using them as input to the SHA-3 (256) which increases the strength of the key because it generates values in one direction only. The results of the hash function are determined by the Lorenz hyper-chaotic system initial values, and four image-size arrays are generated as the final image-encryption keys using the RC4 algorithm. Then the use of the E-fractal diffusion method. To bolster the security of the encryption, it is employed to disperse the pixel values., adopting the control word that increases the randomness of the data. The security analysis of the proposed encryption algorithm demonstrates that it is difficult to decipher due to the presence of the hash function in addition to its easy implementation within seconds, and the efficiency scale was calculated that showed its strength in effectively resisting attacks by others.

Генерическая NP-полнота проблем разрешимости систем уравнений над конечными группами, полугруппами и полями

In this paper we prove that problems of solvability of systems of equations over finite fields, non-Abelian finite groups and non-commutative finite monoids are complete with respect to generic polynomial reducibility in generic analogue of the NP class.

The security analysis of the key exchange protocol based on the matrix power function defined over a family of non-commuting groups

<p>In this paper, we revisited the previously proposed key exchange protocol based on the matrix power function. We prove that the entries of the public key matrices of both parties of the protocol are uniform. Using this result we defined a security game for our protocol and show that the malicious attacker cannot gain any significant advantage in winning this game by applying faithful representation or the linearization approaches. Moreover, we showed that the shared key is computationally indistinguishable from the imitation key if the security parameters are properly chosen.</p>

CONJUGATE S3-MAGIC GRAPHS

In this paper, we introduce conjugate A-magic labeling of graphs where A is a finite non-abelian group and investigate the graphs that are conjugate S3magic.

QSym2: A Quantum Symbolic Symmetry Analysis Program for Electronic Structure.

Symmetry provides a powerful machinery to classify, interpret, and understand quantum-mechanical theories and results. However, most contemporary quantum chemistry packages lack the ability to handle degeneracy and symmetry breaking effects, especially in non-Abelian groups, and they are not able to characterize symmetry in the presence of external magnetic or electric fields. In this article, a program written in Rust entitled QSym2 that makes use of group and representation theories to provide symmetry analysis for a wide range of quantum-chemical calculations is introduced. With its ability to generate character tables symbolically on-the-fly and by making use of a generic symmetry-orbit-based representation analysis method formulated in this work, QSym2 is able to address all of these shortcomings. To illustrate these capabilities of QSym2, four sets of case studies are examined in detail in this article: (i) high-symmetry C84H64, C60, and B9- to demonstrate the analysis of degenerate molecular orbitals (MOs); (ii) octahedral Fe(CN)63- to demonstrate the analysis of symmetry-broken determinants and MOs; (iii) linear hydrogen fluoride in a magnetic field to demonstrate the analysis of magnetic symmetry; and (iv) equilateral H3+ to demonstrate the analysis of density symmetries.

Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

This article presents a new implementation of encryption based on MST, focused on generalized Suzuki 2-groups. The well-known MST cryptosystem, based on Suzuki groups, is constructed using a logarithmic signature at the center of the group, leading to a large array of logarithmic signatures. The proposed encryption is based on multi-parameter noncommutative groups, with a focus on the generalized multi-parameter Suzuki 2-group. This approach responds to the progress in the development of quantum computers, which may pose a threat to the security of many open cryptosystems, especially those based on factorization problems and discrete logarithms, such as RSA or ECC. The use of noncommutative groups to create quantum-resistant cryptosystems has been a known approach for the last two decades.
 The unsolvable word problem, proposed by Wagner and Magyarik, is used in the field of permutation groups and is key to the development of cryptosystems. Logarithmic signatures, introduced by Magliveras, represent a unique type of factorization suitable for finite groups. The latest version of such an implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras introduced the LS transitivity limit for the MST3 cryptosystem, and later Swaba proposed an improved version of the cryptosystem, eMST3. In 2018, T. van Trung suggested applying the MST3 approach using strong aperiodic logarithmic signatures for abelian p-groups. Kong and his colleagues conducted a deep analysis of MST3 and noted that due to the absence of publications on the quantum vulnerability of this algorithm, it can be considered a potential candidate for use in the post-quantum era.
 The main distinction of the new system is the use of homomorphic encryption to construct logarithmic signature coverings for all group parameters, which improves the secrecy of the cryptosystem, particularly against brute-force attacks.

Weak commutativity, virtually nilpotent groups, and Dehn functions

The group \mathfrak{X}(G) is obtained from G\ast G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. We make significant additions to the list of properties that the functor \mathfrak{X} is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrak{X}(G) . Subtle features of \mathfrak{X}(G) are encoded in a normal abelian subgroup W<\mathfrak{X}(G) that is a module over \mathbb{Z} Q , where Q= H_1(G,\mathbb{Z}) . We establish a structural result for this module and illustrate its utility by proving that \mathfrak{X} preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrak{X}(G) when G lies in a class that includes Thompson's group F and all non-fibred Kähler groups. The word problem is soluble in \mathfrak{X}(G) if and only if it is soluble in G . The Dehn function of \mathfrak{X}(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

Finite groups with isomorphic non-commuting graphs have the same nilpotency property

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph ΓG associated to G is the graph whose vertex set is G∖Z(G) and two distinct elements x,y are adjacent if and only if xy≠yx. We prove that if G and H are non-abelian groups with isomorphic non-commuting graphs, such that G is nilpotent, then H is nilpotent, provided |Z(G)|≥|Z(H)|. Actually we prove conjecture 3 proposed in V. Grazian and C. Monetta (2023) [5].

Neutrino Flavor Model Building and the Origins of Flavor and CP Violation

The neutrino sector offers one of the most sensitive probes of new physics beyond the Standard Model of Particle Physics (SM). The mechanism of neutrino mass generation is still unknown. The observed suppression of neutrino masses hints at a large scale, conceivably of the order of the scale of a rand unified theory (GUT), which is a unique feature of neutrinos that is not shared by the charged fermions. The origin of neutrino masses and mixing is part of the outstanding puzzle of fermion masses and mixings, which is not explained ab initio in the SM. Flavor model building for both quark and lepton sectors is important in order to gain a better understanding of the origin of the structure of mass hierarchy and flavor mixing, which constitute the dominant fraction of the SM parameters. Recent activities in neutrino flavor model building based on non-Abelian discrete flavor symmetries and modular flavor symmetries have been shown to be a promising direction to explore. The emerging models provide a framework that has a significantly reduced number of undetermined parameters in the flavor sector. In addition, such a framework affords a novel origin of CP violation from group theory due to the intimate connection between physical CP transformation and group theoretical properties of non-Abelian discrete groups. Model building based on non-Abelian discrete flavor symmetries and their modular variants enables the particle physics community to interpret the current and anticipated upcoming data from neutrino experiments. Non-Abelian discrete flavor symmetries and their modular variants can result from compactification of a higher-dimensional theory. Pursuit of flavor model building based on such frameworks thus also provides the connection to possible UV completions: in particular, to string theory. We emphasize the importance of constructing models in which the uncertainties of theoretical predictions are smaller than, or at most compatible with, the error bars of measurements in neutrino experiments. While there exist proof-of-principle versions of bottom-up models in which the theoretical uncertainties are under control, it is remarkable that the key ingredients of such constructions were discovered first in top-down model building. We outline how a successful unification of bottom-up and top-down ideas and techniques may guide us towards a new era of precision flavor model building in which future experimental results can give us crucial insights into the UV completion of the SM.

Universal adjacency spectrum of the commuting and non-commuting graphs on AC-groups

For a simple undirected graph [Formula: see text] and [Formula: see text], [Formula: see text], the universal adjacency matrix [Formula: see text], where [Formula: see text] is the identity matrix, [Formula: see text] is the all-ones matrix, and [Formula: see text] and [Formula: see text] are the adjacency and degree diagonal matrices of [Formula: see text], respectively. For a finite non-abelian group [Formula: see text] and [Formula: see text], the commuting graph [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and the adjacency rule is commuting. The non-commuting graph of [Formula: see text] is the complement of [Formula: see text]. An AC-group is a non-abelian group such that the centralizer of every non-central element is abelian. In this paper, we obtain the universal adjacency eigenpairs of the commuting and non-commuting graphs for an arbitrary AC-group [Formula: see text] with [Formula: see text] or [Formula: see text]. Moreover, we determine the eigenpairs of [Formula: see text] when [Formula: see text] is a specific kind of group.

Orders on free metabelian groups

Abstract A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group ( G , ⩽ ) (G,\leqslant) is convex if, for all f ⩽ g f\leqslant g in 𝑆, every element h ∈ G h\in G satisfying f ⩽ h ⩽ g f\leqslant h\leqslant g belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.

Mixed moments and the joint distribution of random groups

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random p-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random p-adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups.

Degree and distance based topological descriptors of power graphs of finite non-abelian groups

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order pq, dihedral, and generalized quaternion groups, where p,q(q≥p) are distinct primes.

Diagrammatic structures of the Nielsen identity

The Γ-function, or the effective potential of a gauge field theory should comply with the Nielsen identity, which implies how the effective potential evolves as we shift the gauge-fixing term. In this paper, relying on an abelian toy model, we aim at proving this identity in a diagrammatic form with the R‾ξ gauge. The basic idea is to find out the ghost chain after partially differentiating the diagram by the ξ parameter, and shrink the waists of the diagram into points to separate the bulk-part and C-part of the diagrams. The calculations can be generalized to the models implemented with non-abelian groups, multiple Higgs and fermion multiplets, and to the finite temperature cases.

Provably secure zero knowledge undeniable signature scheme over non-abelian group

Provably secure zero knowledge undeniable signature scheme over non-abelian group

Method of Security Improvement for MST3 Cryptosystem Based on Automorphism Group of Ree Function Field


 
 
 This article is a part of a research endeavor focused on creating a quantum-resistant cryptosystem for secure encryption and decryption. Our approach employs a challenging word problem while emphasizing cost-effective implementation. Previous research has involved the development of encryption schemes based on high-order groups, offering potential security enhancements. The choice of the non-abelian group is a critical factor in shaping the encryption algorithms, feasibility of implementation, and system parameters. Our central objective is to design a cryptosystem that effectively thwarts quantum cryptanalysis. To achieve this, we employ a logarithmic signature along with a random cover across an entire finite non-abelian group. Our unique contribution lies in optimizing finite group selection, parameters, and circuit solutions for the logarithmic signature to meet specific security and implementation criteria. Within this paper, we introduce an encryption scheme utilizing automorphisms of the Ree functional field and propose a method for enhancing resistance to cryptanalysis through the binding of session keys.
 
 

Spin-1 thermal targets for dark matter searches at beam dump and fixed target experiments

The current framework for dark matter (DM) searches at beam dump and fixed target experiments primarily relies on four benchmark models, the so-called complex scalar, inelastic scalar, pseudo-Dirac and finally, Majorana DM models. While this approach has so far been successful in the interpretation of the available data, it a priori excludes the possibility that DM is made of spin-1 particles — a restriction which is neither theoretically nor experimentally justified. In this work we extend the current landscape of sub-GeV DM models to a set of models for spin-1 DM, including a family of simplified models (involving one DM candidate and one mediator — the dark photon) and an ultraviolet complete model based on a non-abelian gauge group where DM is a spin-1 Strongly Interacting Massive Particle (SIMP). For each of these models, we calculate the DM relic density, the expected number of signal events at beam dump experiments such as LSND and MiniBooNE, the rate of energy injection in the early universe thermal bath and in the Intergalactic Medium (IGM), as well as the helicity amplitudes for forward processes subject to the unitary bound. We then compare these predictions with experimental results from Planck, CMB surveys, IGM temperature observations, LSND, MiniBooNE, NA64, and BaBar and with available projections from LDMX and Belle II. Through this comparison, we identify the regions in the parameter space of the models considered in this work where DM is simultaneously thermally produced, compatible with present observations, and within reach at Belle II and, in particular, at LDMX. We find that the simplified models considered here are strongly constrained by current beam dump experiments and the unitarity bound, and will thus be conclusively probed (i.e. discovered or ruled out) in the first stages of LDMX data taking. We also find that the vector SIMP model explored in this work predicts the observed DM relic abundance, is compatible with current observations and within reach at LDMX in a wide region of the parameter space of the theory.

On the characterization of some non-abelian simple groups using codegree set

Let G be a finite group and χ ∈ Irr ( G ) . The codegree of χ is defined as cod ( χ ) = | G : ker ( χ ) | / χ ( 1 ) and cod ( G ) = { cod ( χ ) | χ ∈ Irr ( G ) } is called the set of codegrees of G. In this paper, we show that the set of codegrees of U 4 ( 2 ) , S 4 ( q ) ( q = 2 f ≥ 4 ) , U 4 ( 3 ) , 2 F 4 ( 2 ) ′ , J 3 , G 2 ( 3 ) , A 9 , J 2 , PSL ( 4 , 3 ) , McL , S 4 ( 5 ) , G 2 ( 4 ) , HS , O ′ N and M 24 determines the group up to isomorphism.

A new characterization of simple K3-groups using same-order type

Let G be a group, define an equivalence relation ∼ as below: ∀ g, h ∈ G, g ∼ h ⇐⇒|g| = |h| the set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denoted by α(G). And G is said a αn-group if |α(G)| = n. Let π(G) be the set of prime divisors of the order of G. A simple group of G is called a simple Kn-group if |π(G)| = n. We give a new characterization of simple K3-groups using same-order type. Indeed we prove that a nonabelian simple group G has same-order type {r, m, n, k, l} if and only if G ≅ PSL(2,q), with q = 7, 8 or 9. This result generalizes the main results in (4), (6) and (8). Moreover based on the main result in (8) we have the natural question: Let S be a nonabelian simple αn-group and G a αn-group such that |S| = |G|. Then S ≅ G. In this paper with a counterexample we give a negative answer to this question.