Ilia Ponomarenko

Permutation group approach to association schemes

We survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed.

On a New High Dimensional Weisfeiler-Lehman Algorithm

We investigate the following problem: how different can a cellular algebra be from its Schurian closure, i.e., the centralizer algebra of its automorphism group? For this purpose we introduce the notion of a Schurian polynomial approximation scheme measuring this difference. Some natural examples of such schemes arise from high dimensional generalizations of the Weisfeiler-Lehman algorithm which constructs the cellular closure of a set of matrices. We prove that all of these schemes are dominated by a new Schurian polynomial approximation scheme defined by the m-closure operators. A sufficient condition for the m-closure of a cellular algebra to coincide with its Schurian closure is given.

Schur rings

In this paper we survey the recent developments in the theory of Schur rings and its applications to different problems that appear in theory of association schemes, Cayley graphs and other parts of algebraic combinatorics.

On pseudocyclic association schemes

The notion of a pseudocyclic association scheme is generalized to the non-commutative case. It is proved that any pseudocyclic scheme the rank of which is much more than the valency is the scheme of a Frobenius group and is uniquely determined up to isomorphism by its intersection number array. An immediate corollary of this result is that any scheme of prime degree, valency k and rank at least k 4 is schurian.

Homomorphic Public-Key Cryptosystems and Encrypting Boolean Circuits

Given an arbitrary finite nontrivial group, we describe a probabilistic public-key cryptosystem in which the decryption function is chosen to be a suitable epimorphism from the free product of finite Abelian groups onto this finite group. It extends the quadratic residue cryptosystem (based on a homomorphism onto the group of two elements) due to Rabin – Goldwasser – Micali. The security of the cryptosystem relies on the intractability of factoring integers. As an immediate corollary of the main construction, we obtain a more direct proof (based on the Barrington technique) of Sander-Young-Yung result on an encrypted simulation of a boolean circuit of the logarithmic depth.

A NEW LOOK AT THE BURNSIDE–SCHUR THEOREM

The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite commutative ring, and an analogue of this statement is proved for them. Also, the finite local commutative rings are characterized in permutation group terms.

Algorithmic Algebraic Combinatorics and Grbner Bases

Tutorials.- Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics.- Siamese Combinatorial Objects via Computer Algebra Experimentation.- Using Grobner Bases to Investigate Flag Algebras and Association Scheme Fusion.- Enumerating Set Orbits.- The 2-dimensional Jacobian Conjecture: A Computational Approach.- Research Papers.- Some Meeting Points of Grobner Bases and Combinatorics.- A Construction of Isomorphism Classes of Oriented Matroids.- Algorithmic Approach to Non-symmetric 3-class Association Schemes.- Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings.- A Construction of Designs from PSL(2,q) and PGL(2,q), q=1 mod 6, on q+2 Points.- Approaching Some Problems in Finite Geometry Through Algebraic Geometry.- Computer Aided Investigation of Total Graph Coherent Configurations for Two Infinite Families of Classical Strongly Regular Graphs.

Normal cyclotomic schemes over a finite commutative ring

We study cyclotomic association schemes over a finite commutative ring R with identity. The main interest for us is to identify the normal cyclotomic schemes C, i.e. those for which Aut(C) ≤ AL 1(R). The problem is reduced to the case when the ring R is local in which a necessary condition of normality in terms of the subgroup of R × defining C, is given. This condition is proved to be sufficient for a classof local rings including the Galois rings of odd characteristic.

Tensor Rank: Matching Polynomials and Schur Rings

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elementsxa0