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Idempotent ideals in integral group rings
Lachman, Dominik ; Příhoda, Pavel (advisor) ; Šaroch, Jan (referee)
This thesis concerns following hyphotesis: whenever I is two-sided idem- potent ideal in group ring ZSn, such that IQ is non-trivial ideal of QSn, IQ has to be so called augmentation ideal. The vylidity of this hypothesis would give us weak version of the fact that in the case of solvable group G, there are no two-sided non-trivial idempotent ideals in ZG. At first I desctibe methodt how to calculate idempotent ideals in ZSn and then show that hy- pothesis holds in the case of S5, but fail in the case of ZS5. In theoretic part, I firstly switch to local point of view and describe two-sided idempo- tent ideals in Z(p)Sn, for primes p dividing order of group Sn, as trace ide- als of finitely generated projective Z(p)Sn-modules. Next, I describe functor −⊗Z(p) Q : Proj(Z(p)Sn) → Mod(QSn) using the language of Grothendiecks groups by matrix E. Matrix E shows to be transposition of decomposition matrix, which we can calculate using Braeur's character. 1
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Diffie and Hellman are exchanging matrices over group rings
Linkeová, Romana ; Příhoda, Pavel (advisor) ; El Bashir, Robert (referee)
Title: Diffie and Hellman are exchanging matrices over group rings Author: Romana Linkeová Department: Department of Algebra Supervisor: Mgr. Pavel Příhoda, Ph.D., Department of Algebra Abstract: The Diffie-Hellman key exchange protocol is not suitable for devices with limited computational power while computing over group Z∗ p (where p is at least a 300-digit number). This fact led to the research of other algebraic structures, which may help in reducing the computational and storage cost of the protocol. D. Kahrobaei et al. posted in 2013 a proposal for working over a structure of small matrices and claimed that this modification will not affect the security of the protocol. We will attempt to attack this modification of the Diffie- Hellman protocol with the help of the theory of symmetric group representations. Firstly, we mention the basics of the theory of representations together with both the classical and the modified Diffie-Hellman protocol. Next, we elaborate the attack step by step and complement some of the steps with examples. Then, we probed security of the modified protocol against the baby-step giant-step attack. Keywords: public key cryptography, symmetric group representations, Diffie-Hellman protocol 1
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