Omsk Algebraic Seminar

Meeting #968

Alexey Nikitin (Deeplay, Omsk)

"Computations in universal classes with applications for graph and order models."

Abstract. In the talk I will present results concerning algebraic geometry over partially ordered sets:
1. Complexity of algorithms for solving systems of equations over partial orders;
2. Noethericity by equations for partial orders;
3. Decidability of theories over partial orders;
4. Construction of radicals and coordinate partial orders for systems of equations over partial orders and strict partially ordered sets.

Meeting #967

Vitaly Roman'kov (Sobolev Institute of Mathematics)

"On the embedding of free nilpotent groups in partially commutative nilpotent groups."

Abstract. An algorithm is presented that allows one to calculate the maximum rank of a free nilpotent group of class k into a given partially commutative nilpotent group of class k.

Meeting #966

Marina Rasskazova (Omsk State Technical University)

"Binary Lie algebras and its application to the theory of Binary Lie superalgebras."

Abstract. The tropical structures min-plus and max-plus are well-known and have many interesting applications.
By definition an algebra \(B\) is binary-Lie algebra iff any two elements \(a,b\in B\) generate a Lie subalgebra. A \({\bf Z}_2-\)graded algebra \(B=B_0\oplus B_1\) is a binary-Lie superalgebra iff \(B\times \Gamma=B_0\otimes \Gamma_0\oplus B_1\otimes \Gamma_1\) is a binary Lie algebra, where \(\Gamma=\Gamma_0\oplus \Gamma_1\) is a Grassman algebra with natural gradation. We apply the theory of binary-Lie algebras for proving the following result.
Theorem. Let \(B=B_0\oplus B_1\) be a simple binary-Lie superalgebra finite dimensional over the field \({\bf C}\) of complex numbers. Then \(B_0\) is a solvable algebra or \(B\) is a Lie superalgebra.
This theorem reduced the problem (open yet!) of classification of a simple binary-Lie superalgebra finite dimensional over the field \({\bf C}\) to the case where subalgebra \(B_0\) is solvable.
This talk is based on the joint paper with A.Grishkov and I.Shestakov.

Meeting #965

Matvei Kotov (Sobolev Institute of Mathematics)

"Tropical approach in cryptography."

Abstract. The tropical structures min-plus and max-plus are well-known and have many interesting applications.
In two papers “Tropical Cryptography” (2014) and “Tropical Cryptography II: extensions by homomorphisms” (2019), V. Shpilrain and D. Grigoriev suggested using tropical algebras as platforms for several cryptographic schemes in order to avoid linear algebra attacks. Another reason for their popularity is efficiency because multiplication is replaced by addition. Also, in “An application of different dioids in public key cryptography” (2014), M. Durcheva offered other cryptographic protocols based on tropical algebras and similar algebras. In recent years, other authors also developed some schemes in this area.
Many of these protocols were analyzed and attacks were developed by M. Kotov and A. Ushakov (2018), A. Muanalifah, and S. Sergeev (2020), D. Rudy and C. Monico (2020), S. Isaac and D. Kahrobaei (2021), K. Ahmed, S. Pal, and R. Mohan (2022).
Recently, we analyzed one of Durcheva’s protocols and implemented a successful attack (joint work with Ivan Buchinskiy and Alexander Treier).
In this talk, I will review the current status of tropical cryptography (protocols, methods to analyze them, and attacks) and discuss our new attack.