Modification of Anshel-Anshel-Goldfeld Postquantum Algorithm / Protocol Based on Algebraic Braid Groups, in Order to the Span–Cyberattack Neutralization

This article deals with relatively young field of cryptography, namely cryptography based on the theory of infinite abstract groups. The research identifies the main problems on which this type of cryptography is based, and the cryptoanalysis conducts of one of the algorithms grounded on the problem of mating (conjugate) element finding, on the base of which the protocol for common key generating is developed. As the algorithm under study, the protocol for generating the common key based on Anshel-Anshel-Goldfeld algorithm, built on the algebraic braid groups, is investigated. During the study of this protocol, one of possible cyberattacks was identified, which allows to get hold of the secret keys of subscribers. To eliminate and to neutralize this span–cyberattack, the new modification of Anshel–Anshel– Goldfeld (AAG) algorithm was developed, which significantly reduces the probability of this cyberattack successful implementation. Analysis of this modification operating time was carried out also.

Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

Tied links in various topological settings

Tied links in [Formula: see text] were introduced by Aicardi and Juyumaya as standard links in [Formula: see text] equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces [Formula: see text], in handlebodies of genus [Formula: see text], and in the complement of the [Formula: see text]-component unlink. We introduce the tied braid monoids [Formula: see text] by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an [Formula: see text]-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

Right-angled Artin groups as normal subgroups of mapping class groups

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

Group Theory: Mathematical Expression of Symmetry in Physics

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

Key agreement protocol in Braid group representation level

In this paper the key agreement protocol is given and the applicationof it in Braid groups is suggested. The one way of protocol is being justified.