Assessing the security of cryptographic primitives for infinite groups

This paper considers the application of group theory to cryptography using a nonabelian infinite group (the braid group). The practical application of cryptographic protocols are determined by their security and feasibility. Both research papers and experiments will be used to measure feasibility and security of the protocol, with the intention of ultimately deeming the protocol either effective or ineffective. Having secure cryptography is vital to providing anonymity, confidentiality and integrity to data and as the quantum threat creeps towards us, the ever greater importance of new secure cryptography is becoming clear

Product set growth in mapping class groups

We study product set growth in groups with acylindrical actions on quasi-trees and hyperbolic spaces. As a consequence, we show that for every surface $S$ of finite type, there exist $\alpha,\beta>0$ such that for any finite symmetric subset $U$ of the mapping class group $MCG(S)$ we have $|U|\geqslant (\alpha|U|)^{\beta n}$, so long as $\langle U\rangle$ is not contained non-trivially in a direct product with a virtually free abelian factor. This result for mapping class groups also applies to all of their subgroups, including right-angled Artin groups. We separately prove that we can quickly generate loxodromic elements in right-angled Artin groups, which by a result of Fujiwara shows that the set of growth rates for many of their subgroups are well-ordered.Comment: 48 pages, 1 figure. This is a significant rewrite of an earlier paper on right-angled Artin groups, which now includes new results about mapping class group

Geometric Structures in Group Theory (hybrid meeting)

The conference focused on the use of geometric methods to study infinite groups and the interplay of group theory with other areas. One of the central techniques in geometric group theory is to study infinite discrete groups by their actions on nice, suitable spaces. These spaces often carry an interesting large-scale geometry, such as non-positive curvature or hyperbolicity in the sense of Gromov, or are equipped with rich geometric or combinatorial structure. From these actions one can investigate structural properties of the groups. This connection has become very prominent during the last years. In this context non-discrete topological groups, such as profinite groups or locally compact groups appear quite naturally. Likewise, analytic methods and operator theory play an increasing role in the area

A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus

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Modelling and Structural Studies of a Gelling Polysaccharide: Agarose.

This thesis details work carried out over a period of three years on the two gelling carbohydrates agarose and carrageenan. The major part of the work deals with agarose. Two approaches have been used which yield information from different angles; these are the experimental (laboratory) and the simulation (computational) approaches. There is a large field of interest in gelling carbohydrates from the point of view of the food industry. Their extraordinary ability to form stable gels and emulsions incorporating other food ingredients makes them important in many deserts and dairy products. In the present work, models for agarose and carrageenan carbohydrates were developed using structural x-ray data and related carbohydrate literature. The models were treated with two different solvent simulation methods. It was found that the inclusion of individual solvent molecules (the closest approximation to a real solution) was extremely uneconomical when the demands on computing time were taken into account, and in fact the long term outcome of the simulation was the same for both methods. Inclusion of solvent simply reduces diffusion rates and the time constant for chain flexing. Gel permeation chromatography and differential scanning calorimetry were used to prepare samples of agarose molecules of known size, and to probe temperature dependent phase transitions. This work was done at the UNILEVER laboratories at Colworth House, Sharnbrook in Bedfordshire. It was found that only molecules longer than fifteen residues displayed the molecular ordering transition typical of agarose polymer, and a value for the enthalpy of the transition of -1.5kcal per mole of residues was measured. It was predicted that in agarose itself, helical regions of a size of approximately 40 residues should exist. Simulations were then done on several agarose molecules of different sizes in order to parallel the experimental work. The differences in energy between molecules in various conformations were compared. These results were also related to helix-coil transition theory. The modelling predicts an enthalpy per mole of residues for the agarose coil to helix transition of approximately -2kcal, and indicates that single agarose coils may be of some importance in agarose gel structure. The work illustrates the difficulty in modelling such complex systems, and in fact it remains impossible to observe agarose molecules undergoing the transition between a random coil and a helical conformation

On a conjecture for ℓ\ell-torsion in class groups of number fields: from the perspective of moments

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments.