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Ilia Ponomarenko

Publications and source records attributed to Ilia Ponomarenko.

51 records · Page 3Linked to original sources
arXiv2017

Recognizing and testing isomorphism of Cayley graphs over an abelian group of order $4p$ in polynomial time

We construct a polynomial-time algorithm that given a graph $X$ with $4p$ vertices ($p$ is prime), finds (if any) a Cayley representation of $X$ over the group $C_2\times C_2\times C_p$. This result, together with the known similar result for circulant graphs, shows that recognising and testing isomorphism of Cayley graphs over an abelian group of order $4p$ can be done in polynomial time.

math.CO↗
arXiv2017

Finding a cycle base of a permutation group

A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved that a cycle base of a permutation group of degree $n$ can be constructed in polynomial time in~$n$.

math.GR↗
arXiv2016

Coherent configurations associated with TI-subgroups

Let X be a coherent configuration associated with a transitive group G. In terms of the intersection numbers of X, a necessary condition for the point stabilizer of G to be a TI-subgroup, is established. Furthermore, under this condition, X is determined up to isomorphism by the intersection numbers. It is also proved that asymptotically, this condition is also sufficient. More precisely, an arbitrary homogeneous coherent configuration satisfying this condition is associated with a transitive group, the point stabilizer of which is a TI-subgroup. As a byproduct of the developed theory, recent results on pseudocyclic and quasi-thin association schemes are generalized and improved. In particular, it is shown that any scheme of prime degree p and valency k is associated with a transitive group, whenever p>1+6k(k-1)^2.

math.CO↗
arXiv2016

Cartan coherent configurations

The Cartan scheme $\cal X$ of a finite group $G$ with a $(B,N)$-pair is defined to be the coherent configuration associated with the action of $G$ on the right cosets of the Cartan subgroup $B\cap N$ by the right multiplications. It is proved that if $G$ is a simple group of Lie type, then asymptotically, the coherent configuration $\cal X$ is 2-separable, i.e., the array of 2-dimensional intersection numbers determines $\cal X$ up to isomorphism. It is also proved that in this case, the base number of $\cal X$ equals 2. This enables us to construct a polynomial-time algorithm for recognizing the Cartan schemes when the rank of $G$ and order of the underlying field are sufficiently large. One of the key points in the proof of the main results is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.

math.CO↗
arXiv2007

A modular absolute bound condition for primitive association schemes

The well-known absolute bound condition for a primitive symmetric association scheme (X,S) gives an upper bound for |X| in terms of |S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for |X| for an arbitrary primitive scheme (X,S). They do not depend on |S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field.

math.CO↗
arXiv2007

The basis digraphs of p-schemes

It is proved that association schemes with bipartite basis graphs are exactly 2-schemes. This result follows from a characterization of p-schemes for an arbitrary prime p in terms of basis digraphs.

math.CO↗
arXiv2006

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme C over a finite near-field. It is proved that if C is nontrivial, then Aut(C)<AGL(V) where V is the linear space associated with the near-field. In many cases we are able to get more specific information about Aut(C).

math.CO↗
arXiv2006

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 $Aut(C)$ is a subgroup of the one-dimensional affine semilinear group over $R$. 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 $R^\times$ defining $C$, is given. This condition is proved to be sufficient for a class of local rings including the Galois rings of odd characteristic.

math.CO↗
arXiv2006

On amorphic C-algebras

An amorphic association scheme has the property that any of its fusion is also an association scheme. In this paper we generalize the property to be amorphic to an arbitrary C-algebra and prove that any amorphic C-algebra is determined up to isomorphism by the multiset of its degrees and an additional integer equal 1 or -1. Moreover, we show that any amorphic C-algebra with rational structure constants is the fusion of an amorphic homogeneous C-algebra. Finally, as a special case of our results we obtain the well-known Ivanov's characterization of intersection numbers of amorphic association schemes.

math.CO↗
arXiv2005

Constructions in public-key cryptography over matrix groups

The purpose of the paper is to give new key agreement protocols (a multi-party extension of the protocol due to Anshel-Anshel-Goldfeld and a generalization of the Diffie-Hellman protocol from abelian to solvable groups) and a new homomorphic public-key cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups of a given group. To support these and other known cryptographic schemes we present a general technique to produce a family of instances being matrix groups (over finite commutative rings) which play a role for these schemes similar to the groups $Z\_n^*$ in the existing cryptographic constructions like RSA or discrete logarithm.

math.GR↗
arXiv2003

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 latters we introduce Schur rings over a finite commutative ring and prove an analog of this statement for them. Besides, the finite local commutative rings are characterized in the permutation group terms.

math.GR↗
arXiv2003

Homomorphic public-key cryptosystems over groups and rings

We propose a new homomorphic public-key cryptosystem over arbitrary nonidentity finite group based on the difficulty of the membership problem for groups of integer matrices. Besides, a homomorphic cryptosystem is designed for the first time over finite commutative rings.

cs.CR↗
arXiv2003

Homomorphic public-key cryptosystems and encrypting boolean circuits

In this paper homomorphic cryptosystems are designed for the first time over any finite group. Applying Barrington's construction we produce for any boolean circuit of the logarithmic depth its encrypted simulation of a polynomial size over an appropriate finitely generated group.

cs.CR↗