Group Theory Applications in Cryptology

PROGRAM GROUP ALGEBRAS, REPRESENTATIONS AND COMPUTATION ORGANIZERS: Gurmeet Kaur Bakshi, Manoj Kumar and Pooja Singla DATE: 14 October 2019 to 23 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Determining explicit algebraic structures of semisimple group algebras is a fundamental problem, which has played a central role in the development of representation theory of finite groups. The tools of representation theory of finite groups extend in various ways to profinite groups such as compact linear groups over ring of integers of a local field (for example GL_n(Z_p)). However the continuous representations or even representation growth of profinite groups is not well understood and is one of the current exciting areas of research. The importance of computational methods in all pursuits of pure mathematics is no more obscure, and the subject has established itself as a powerful tool, aiding quick maturing of intuition about concrete mathematical structures. The focus of this program is on theoretical aspects of group algebras and representation theory of finite and profinite groups complemented by computational techniques using discrete algebra system GAP. The first part of this program, a 5-day instructional workshop during October 14 - 18, 2019, is planned to touch upon various aspects of group algebras and representation theory, including Wedderburn decomposition as well as idempotents of group algebra and their applications, representations of finite simple groups of Lie type and p-adic analytic groups, representation growth and computational methods. It will consist of three mini-courses: Representations of finite and profinite groups Structure of group algebras Computations with finite groups The plan is to have five hours of teaching and five hours tutorials and guided practical sessions in each mini-course. The second part of the program, a 4-day discussion meeting during October 20-23, 2019, is aimed at exploring some of the recent developments and problems of current mathematical interest in the above mentioned and related areas. Eminent speakers from India as well as abroad will present their work on a variety of topics. To encourage discussions among the participants, plenty of time will be provided between the talks. It is planned to have poster presentations by young researchers to manifest their mathematical thoughts and to foster interactions with the senior mathematicians. We'll dedicate October 21, 2019 to group rings, "Group Ring Day", in honour of the 80th birthday of Professor I.B.S. Passi. The participants, PhD students, post doctoral fellows and young faculty members, are expected to have exposure to advanced group theory and to have a good grasp on basic algebra and representation theory. Eligibility criteria: The program is mainly aimed at Ph.D. students, post doctoral fellows and young faculty members. However, a couple of highly motivated final year master students shall also be given a chance to participate in the first part of the program. CONTACT US: garc2019@icts.res.in PROGRAM LINK: https://www.icts.res.in/program/garc2019 Table of Contents (powered by https://videoken.com) 0:00:00 Group Theory for Cryptology 0:03:22 Block ciphers consist of: A text space which is also the ciphertext space 0:04:23 A set of keys 0:05:12 An (injective) map: 0:06:55 The homomorphism 0:11:11 Differential attack 0:12:07 Substitution-Permutation Networks 0:13:54 Confusion and diffusion 0:14:17 There are many regular subgroups of Sym(V), and many of them are isomorphic to V 0:14:45 Lemma (John D. Dixon, Maximal abelian subgroups of the symmetric groups, can J. Math. XXIII, 3 (1971), 426-438) 0:20:50 Differential attack revisited 0:21:28 A successful example on a toy SPN* 0:23:22 Weak-key subspace 0:26:12 Is the case d=n-2 special? 0:29:52 Theorem (A,C,G,S) 0:32:00 Our next result is certainly the most technical of the whole paper. 0:33:16 Theorem (A, C, G,S) and Corollary 0:35:38 The fact that the Sylow 2-subgroups of Sym(V) are self-normalising was already known to P. 0:36:25 Next? 0:37:46 The Big Problem 0:38:37 Thank you!