Algebra and Cryptography Seminar, Spring 2012

Organizers: Robert Gilman, Alexei Myasnikov, and Vladimir Shpilrain


2:30-3:30 pm
Room 8405, CUNY Graduate Center
365 Fifth Avenue at 34th Street


11:00 am-12:00 pm
Room Peirce 220, Stevens Institute of Technology
Hoboken, NJ


February 3, Graduate Center: Alexander A. Mikhalev (Moscow State University), Free Akivis Algebras
Abstract: A vector space A over a field F is called an Akivis algebra if it is endowed with an anticommutative bilinear operation [x,y] (a commutator) and a trilinear operation (x,y,z) (an associator) that satisfy the identity [[x,y],z] + [[y,z],x] + [[z,x],y] = (x,y,z) + (y,z,x) + (z,x,y) - (y,x,z) - (x,z,y) - (z,y,x). These algebras were introduced by M.A.Akivis as tangent algebras of local analytic loops. If B is an algebra over a field and [x,y]=xy-yx, (x,y,z)=(xy)z-x(yz), then the algebra B with these operations is an Akivis algebra (we denote it by Ak(B)). Let Ak(X) be the free Akivis algebra over a field F with the set X of free generators, F(X) the free nonassociative algebra over the field F with the same set X of free generators. Then the algebra Ak(X) is isomorphic to the subalgebra of Ak(F(X)) generated by the set X.
I.P.Shestakov and U.U.Umirbaev proved that subalgebras of free Akivis algebras are free, i.e. the variety of all Akivis algebras over a field F is a Schreier variety. An element u of Ak(X)) is said to be a primitive element (a coordinate polynomial) if it is an element of some set of free generators of the algebra Ak(X).
In this talk we consider the problem of recognizing automorphisms of free Akivis algebras. We prove the Freiheitssatz for free Akivis algebras. We also show that an element u of Ak(X) is a primitive element if and only if the factor algebra of Ak(X) by the ideal generated by the element u is a free Akivis algebra. We also consider some properties of primitive elements. The talk is based on a joint work with I.P.Shestakov.

February 10, Graduate Center: Alexander V. Mikhalev (Moscow State University), Nonassociative cryptography
Abstract: The main goal of the talk is to show how to use nonassociative algebraic structures in cryptography: cryptosystems over quasigroup rings; Moufang loops, Page loops; alternative rings.

February 24, Graduate Center: Alexander Ushakov (Stevens Institute of Technology), Authenticated commutator key establishment protocol
Abstract: In this talk I will present a commutator key establishment protocol with authentication.

March 9, Graduate Center: Svetla Vassileva (McGill University), Uniform polynomial-time conjugacy in free solvable groups
Abstract: We prove that the conjugacy problem in free solvable groups is decidable in polynomial time uniformly over this class. We will review the Magnus embedding which is used to embed a free solvable group in a wreath product. We give a result on the polynomial time complexity of the conjugacy problem in wreath products. We rely on a result of Miasnikov, Roman'kov, Ushakov and Vershik which computes the Magnus embedding in cubic time.

April 20, Graduate Center: Delaram Kahrobaei (New York City College of Technology), Non-commutative Digital Signatures
Abstract: I will survey several digital signatures proposed in the last decade using non-commutative groups and rings and propose a digital signature using non-commutative groups and analyze its security. This is joint work with Bobby Koupparis (GC and RBC).

April 27, Graduate Center: Igor Lysenok (Moscow Steklov Institute and Stevens Institute), Quadratic equations in free monoids and a classification of surface train tracks
Abstract: An equation in a group or in a monoid is called quadratic if every variable occurs in it exactly twice. In my talk, I will discuss a bound on the size of minimal solutions of quadratic equations in free monoids and its connection to a certain combinatorial classification of surface train tracks.


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