Zoom link: https://videoconf-colibri.zoom.us/j/92261549490
 
Abstract:
This talk will explore the connection between combinatorial group theory and cryptographic protocols, particularly in the context of key exchange mechanisms, where two parties seek to establish a common group element (key) by making use of the difficulty encoded by an algorithmic problem [5]. These protocols not only provide practical cryptographic applications but also drive the theoretical study of algorithmic problems, such as the conjugacy problem or the decomposition problem, and the study of properties of particular groups, such as the probability of two elements being conjugate, motivating the identification of groups that enhance the security of the protocol.

To ensure secure communication, it is crucial that protocols minimize the information revealed about the final key. We will examine the main criteria that define protocol security and investigate the group-theoretic properties that enhance it. In particular, we will introduce the concepts as the degree of commutativity [4, 1] and the conjugacy ratio [3].

This talk will conclude with a discussion of future research, including potential contributions to existing problems in group theory and possible extensions to semigroup theory. By bridging algebraic structures with cryptographic challenges, this work aims to further the understanding of secure communication protocols and their mathematical foundations.

References:
[1] Y. Antolı́n, A. Martino, E. Ventura. Degree of Commutativity of Infinite Groups. Proc. Amer. Math. Soc, 145(2): 479-485, 2017.
[2] J. Araújo, M. Kinyon, J. Konieczny, A. Malheiro. Three Notions of Conjugacy for Abstract Semigroups, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2015.
[3] L. Ciobanu, C. G. Cox, and A. Martino. The conjugacy ratio of groups, Proc. Edinb. Math. Soc., 62:895-911, 2019.
[4] W. H. Gustafson. What is the probability that two group elements commute?. Amer. Math, Monthly, 80:1031-1034, 1973.

[5] A. Myasnikov, V. Shpilrain, and A. Ushakov. Group-based Cryptography, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2008.

More information is available here.

More information is available here.

Statistics and Risk Management Seminar

Department of Mathematics, NOVA MATH/FCT NOVA

Title: Estimation and Testing in Some Mean-Reverting Processes

Speaker: Severien Nkurunziza, University of Windsor, Windsor, Canada

Date | Time: April 23, 2025 | 14h00

 Zoom: https://videoconf-colibri.zoom.us/j/88333359956

Abstract: We consider inference problem concerning the drift parameter in some generalized mean-reverting processes with unknown change-points in the context where the target parameter is suspected to satisfy some restrictions. We generalize some recent findings in five ways. First, the established method incorporates the uncertain prior knowledge. Second, we derive the unrestricted estimator (UE) and the restricted estimator (RE) as well as their asymptotic properties. Third, we propose a test for testing the hypothesized restrictions and we establish its asymptotic power. Fourth, we construct a class of shrinkage estimators (SEs) which includes as special cases the UE, RE, andclassical SEs. Fifth, we study the relative performance of the proposed class of SEs, and we prove that James-Stein type estimators dominate the UE. Beyond such interesting findings, the additional novelty of the derived results consists in the fact that the dimensions of the proposed estimators are random. Because of that, the asymptotic power of the proposed test and the relative efficiencies do not follow
from classical methods. To overcome this problem, we derive an asymptotic result which is useful in its own.

Short Bio: Dr. Sévérien Nkurunziza is a Professor of Statistics in the Department of Mathematics and Statistics at the University of Windsor. He joined the department in July 2005 as an Assistant Professor, after completing his PhD in statistics at the Université du Québec à Montréal. Since 2006, he has been funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and since 2017, he is Elected Member of International Statistical Institute. His research topics include the models with breakpoints, shrinkage-type estimators and asymptotic theory in Statistics. In particular, he focuses on statistical methods that may have applications in survival analysis, in financial mathematics as well as in brain imaging. More details about Dr. S. Nkurunziza’s research contributions can be found at https://www.uwindsor.ca/science/math/711/faculty-dr-s%C3%A9v%C3%A9rien-nkurunziza.

Organizers: Mina Norouzirad & Isabel Natário

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This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications)