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Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the well-known virtual braid groups. For each Coxeter graph $\Gamma$, they defined the virtual Artin group $VA[\Gamma]$, which is generated by the corresponding Artin group $A[\Gamma]$ and the Coxeter group $W[\Gamma]$, subject to certain mixed relations inspired by the action of $W[\Gamma]$ on its root system $\Phi[\Gamma]$. There is a natural surjection $ \mathrm{VA}[\Gamma] \rightarrow W[\Gamma]$, with the kernel $PVA[\Gamma]$ representing the pure virtual Artin group. In this paper, we explore linearity, crystallographic quotients, and automorphisms of certain classes of virtual Artin groups. Inspired from the work of Cohen, Wales, and Krammer, we construct a linear representation of the virtual Artin group $VA[\Gamma]$. As a consequence of this representation, we deduce that if $W[\Gamma]$ is a spherical Coxeter group, then $VA[\Gamma]/PVA[\Gamma]'$ is a crystallographic group of dimension $ |\Phi[\Gamma]|$ with the holonomy group $W[\Gamma]$. Further, extending an idea of Davis and Januszkiewicz, we prove that all right-angled virtual Artin groups admit a faithful linear representation. The remainder of the paper focuses on conjugacy classes and automorphisms of a subclass of right-angled virtual Artin groups, $VAT_n$, associated with planar braid groups called twin groups. We determine the automorphism group of $VAT_n$ for each $n\geq 5$, and give a precise description of a generic automorphism. As an application of this description, we prove that $VAT_n$ has the $R_\infty$-property for each $n \ge 2$., Comment: 29 pages

In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace $(A, \cdot, \circ)$ induces an action $\lambda^{\op}: (A, \circ) \to \Aut (A, \cdot)$, which gives rise to the group $\Lambda_{A^{\op}} = (A, \cdot) \rtimes_{\lambda^{\op}} (A, \circ)$. We prove that if $A$ and $B$ are isoclinic skew left braces, then $\Lambda_{A^{\op}}$ and $\Lambda_{B^{\op}}$ are also isoclinic under some mild restrictions on the centers of the respective groups. Our key observation is that there is a one-to-one correspondence between the set of equivalence classes of irreducible representations of $(A, \cdot, \circ)$ and that of the group $\Lambda_{A^{\op}}$. We obtain a decomposition of the induced representation of the additive group $(A, \cdot)$ and of the multiplicative group $(A, \circ)$ corresponding to the regular representation of the group $\Lambda_{A^{\op}}$. As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders., Comment: 17 pages, added Proposition 3.3: Proof that if $A$ is any skew left brace, then $\Lambda_{A}$ and $\Lambda_{A^{\op}}$ are isomorphic groups

We give an alternate proof of the left-orderability of the mapping class group of a connected oriented infinite-type surface with a non-empty boundary. Our main strategy involves the inductive construction of a countable stable Alexander system for the surface using a carefully chosen exhaustion by finite-type subsurfaces. In fact, we prove that a generalised ideal arc system for the surface also induces a left-ordering on the big mapping class group. We then prove that two generalised ideal arc systems determine the same left-ordering if and only if they are loosely isotopic. Finally, we prove that the topology on the big mapping class group is the same as the order topology induced by a left-ordering corresponding to an inductively constructed ideal arc system., Comment: 22 pages, 19 figures

It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles, we derive a four-term exact sequence that relates 1-cocycles, second cohomology, and a specific group of automorphisms associated with the extensions of symmetric quandles. This exact sequence shows that the obstruction to lifting and extending automorphisms is found in the second symmetric quandle cohomology. Additionally, some general aspects of dynamical cocycles and extensions are discussed., Comment: 18 pages

In this paper, we investigate residual finiteness and subquandle separability of quandles. The existence of these finiteness properties implies the solvability of the word problem and the generalised word problem for quandles. We prove that the fundamental $n$-quandle of any link in the 3-sphere is residually finite for each $n \ge 2$. This supplements the recent result on residual finiteness of link quandles and the classification of links whose fundamental $n$-quandles are finite for some $n$. We also establish several general results on these finiteness properties and give many families of quandles admitting them., Comment: 16 pages, comments are welcomed

A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this paper, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, this establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the symmetric quandle (co)homology developed by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)]. Our constructions also apply to symmetric racks. We conclude by establishing an explicit isomorphism between the second cohomology of a symmetric quandle and the first cohomology of its associated group., Comment: 30 pages

Twin groups are planar analogues of Artin braid groups and play a crucial role in the Alexander-Markov correspondence for the isotopy classes of immersed circles on the 2-sphere without triple and higher intersections. These groups admit diagrammatic representations, leading to maps obtained by the addition and deletion of strands. This paper explores Brunnian twin groups, which are subgroups of twin groups composed of twins that become trivial when any of their strands are deleted. We establish that Brunnian twin groups consisting of more than two strands are free groups. Furthermore, we provide a necessary and sufficient condition for a Brunnian doodle on the 2-sphere to be the closure of a Brunnian twin. Additionally, we delve into two generalizations of Brunnian twins, namely, $k$-decomposable twins and Cohen twins, and prove some structural results about these groups. We also investigate a simplicial structure on pure twin groups that admits a simplicial homomorphism from Milnor's construction of the simplicial 2-sphere. This gives a possibility to provide a combinatorial description of homotopy groups of the 2-sphere in terms of pure twins., Comment: 23 pages

In this paper, generalising the idea of the Rokhlin property, we explore the concept of the twisted Rokhlin property of topological groups. A topological group is said to exhibit the twisted Rokhlin property if, for each automorphism $\phi$ of the group, there exists a $\phi$-twisted conjugacy class that is dense in the group. We provide a complete classification of connected orientable infinite-type surfaces without boundaries whose mapping class groups possess the twisted Rokhlin property. Additionally, we prove that the mapping class groups of the remaining surfaces do not admit any dense $\phi$-twisted conjugacy class for any automorphism $\phi$. This supplements the recent work of Lanier and Vlamis on the Rokhlin property of big mapping class groups. We also prove that the mapping class group of each connected orientable infinite-type surface without boundary possesses the $R_\infty$-property., Comment: 20 pages, 6 figures

Relative Rota-Baxter groups are generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota-Baxter groups. We prove an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier $M_{RRB}(\mathcal{A})$ of a relative Rota-Baxter group $\mathcal{A} =(A,B,\beta,T)$, and prove that the exponent of $M_{RRB}(\mathcal{A})$ divides $|A||B|$ when $\mathcal{A}$ is finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces., Comment: 29 pages. arXiv admin note: text overlap with arXiv:2309.00692

Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang-Baxter equation. In this paper, we develop an extension theory of relative Rota-Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota-Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota-Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension two., Comment: 30 pages

Elliptic curve cryptography (ECC) is a remarkable mathematical tool that offers the same level of security as traditional public-key cryptography (PKC) with a significantly smaller key size and lower computational requirements. The use of pairing on elliptic curves has emerged as a vibrant field of research that provides enhanced security measures for the next generation of cryptographic systems. This thesis explores using ECC and Pairing-Based Cryptosystems (PBC) as effective mathematical tools for achieving high-security levels with minimal key size and computation cost. Specifically, the research aims to analyze Pairing-Friendly Elliptic Curves (PF-EC) and their practicality in resource-constrained environments. It proposes solutions to some of the limitations of existing applications of pairing-based cryptography. The thesis begins by presenting a comprehensive framework for constructing PF-EC and evaluating the practical security of several families of pairing-friendly curves. The study then explores the limitations of Identity-Based Encryption (IBE), a recognized application of pairing-based cryptography. It proposes mechanisms to address issues such as key escrow, secure key issuing, user slandering, and key abusing problems. The proposed solutions include an Escrow-Free Identity-Based Encryption (EF-IBE) scheme secured against confidentiality and an Escrow-Free Identity-Based Signature (EF-IBS) scheme that is forgeable and secure.

Key escrow refers to storing a copy of a cryptographic key with a trusted third party, typically a government agency or some other organization. Key escrow aims to ensure that law enforcement agencies can access encrypted data when necessary, for example, in criminal investigations or national security matters. However, key escrow also raises concerns about privacy and security. If the trusted third party is compromised, the stored keys could be exposed, and unauthorized parties could access sensitive information. This could result in a significant breach of privacy and potentially harm national security. In identity-based cryptography, the key escrow problem arises because a trusted third party, the Private Key Generator (PKG), generates the private keys for all users. This means that the PKG has complete control over the private keys, which raises concerns about the security and privacy of the users. To balance security and privacy concerns, some approaches have been proposed to address the key escrow problem. We propose an efficient democratic identity-based encryption model that balances the government's and users' rights while ensuring security and privacy. The key objective of the proposed scheme is to provide the government with authority to monitor unlawful messages while ensuring the user's privacy for their lawful messages. To achieve this, the scheme involves two entities: PKG and PKPO. The user's partial key is escrowed at PKG, while the partial key is stored at PKPO. The latter provides a privacy service to the user by confusing their signature, which the user with their personal information can only unlock., Comment: M.Tech Dissertation

Additive manufacturing (AM) offers numerous benefits, such as manufacturing complex and customised designs quickly and cost-effectively, reducing material waste, and enabling on-demand production. However, several security challenges are associated with AM, making it increasingly attractive to attackers ranging from individual hackers to organised criminal gangs and nation-state actors. This paper addresses the cyber risk in AM to attackers by proposing a novel semantic-based threat prioritisation system for identifying, extracting and ranking indicators of compromise (IOC). The system leverages the heterogeneous information networks (HINs) that automatically extract high-level IOCs from multi-source threat text and identifies semantic relations among the IOCs. It models IOCs with a HIN comprising different meta-paths and meta-graphs to depict semantic relations among diverse IOCs. We introduce a domain-specific recogniser that identifies IOCs in three domains: organisation-specific, regional source-specific, and regional target-specific. A threat assessment uses similarity measures based on meta-paths and meta-graphs to assess semantic relations among IOCs. It prioritises IOCs by measuring their severity based on the frequency of attacks, IOC lifetime, and exploited vulnerabilities in each domain.

Ontologies play a critical role in Semantic Web technologies by providing a structured and standardized way to represent knowledge and enabling machines to understand the meaning of data. Several taxonomies and ontologies have been generated, but individuals target one domain, and only some of those have been found expensive in time and manual effort. Also, they need more coverage of unconventional topics representing a more holistic and comprehensive view of the knowledge landscape and interdisciplinary collaborations. Thus, there needs to be an ontology covering Science and Technology and facilitate multidisciplinary research by connecting topics from different fields and domains that may be related or have commonalities. To address these issues, we present an automatic Science and Technology Ontology (S&TO) that covers unconventional topics in different science and technology domains. The proposed S&TO can promote the discovery of new research areas and collaborations across disciplines. The ontology is constructed by applying BERTopic to a dataset of 393,991 scientific articles collected from Semantic Scholar from October 2021 to August 2022, covering four fields of science. Currently, S&TO includes 5,153 topics and 13,155 semantic relations. S&TO model can be updated by running BERTopic on more recent datasets

Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota-Baxter groups with skew left braces, which are well-known to give non-degenerate set-theoretic solutions of the Yang-Baxter equation. We prove that every relative Rota-Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota-Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota-Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota-Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces., Comment: Comments and Suggestions are welcome

The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ are extensions of the symmetric group $S_n$, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group $S_n$ are the pure virtual braid group $VP_n$, the pure virtual twin group $PVT_n$ and the pure virtual triplet group $PVL_n$, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and the commutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete the understanding of these groups, except that of $VB_n^{'}$, for which the existence of a finite presentations is not known for $n \ge 4$. We also prove that $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it., Comment: 29 pages

A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack., Comment: 27 pages, 12 figures, the paper has been revamped with the title changed

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander-Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both $T_n$ and $L_n$ when $n \ge 4$. We also determine subquotients of principal congruence subgroups of $T_n$, and identify the pure twin group $PT_n$ and the pure triplet group $PL_n$ with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that $T_n /PT_n^{'}$, $T_n/T_n^{''}$ and $L_n /PL_n^{'}$ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of $T_n/T_n^{''}$., Comment: 25 pages, to appear in Forum Mathematicum

Major families of pairing-friendly elliptic curves, including BN, BLS12, BLS24, KSS16, and KSS18 have recently been vulnerable to number field sieve (NFS) attacks. Due to the recent attacks on discrete logs in F_(q^k ), selecting such curves became relevant again. This paper revisited the topic of selecting pairing-friendly curves at different security levels. First, we expanded the classification given by Freeman et al. [1] by identifying new families that were not previously mentioned, such as a complete family with variable differentiation and new sparse families of curves. We discussed individual curves and a comprehensive framework for constructing parametric families. We estimated the security and assessed families of the pairing-friendly curve to discover families of curves better than BN, KSS, and BLS in terms of the required key size. We also evaluated the complexity of the optimal ate pairing that has never been discussed before, except by Barbulescu et al. [2]. We demonstrated that the recent attack (TNFS) on pairing needs to increase the key size. We compared families of curves in the context of key size and selected a suitable alternative to an elliptic curve.

The Dehn quandle of a closed orientable surface is the set of isotopy classes of non-separating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than two, we determine all values of $n$ for which the $n$-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles. We also compute the size of the smallest non-trivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle and also determine the smallest non-trivial quotient of a braid quandle., Comment: 16 pages and 3 figures

We show that quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space. We give explicit examples to show that the new invariants are also stronger than the $\Hom$ quandle invariant when the coloring quandles are medial. We provide computer assisted computations of idempotents for all quandles of order less than six, and also determine the ones for which the set of all idempotents is itself a quandle. The data supports our conjecture about triviality of idempotents of integral quandle rings of finite latin quandles. We also determine Peirce spectra for complex quandle algebras of some small order quandles., Comment: 18 pages, 2 figures