Non-isomorphic 2-groups with isomorphic modular group algebras

We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

An identification system based on the explicit isomorphism problem

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

Corrigendum to: V. V. Vasilchikov, “Parallel Algorithm for Solving the Graph Isomorphism Problem”, Modeling and analysis of information systems, vol. 27, no. 1, pp. 86–94, 2020. DOI: https://doi.org/10.18255/1818-1015-2020-1-86-94

In the article by V. V. Vasilchikov “Parallel Algorithm for Solving the Graph Isomorphism Problem” ( Modeling and analysis of information systems, vol. 27, no. 1, pp. 86–94, 2020; DOI: https://doi.org/10.18255/1818-1015-2020-1-86-94) there was a misprint in the layout. In the Table 1, in the last column of the row “Degree of graph” the value should be 3000 (instead of 300). The corrected “Table 1” is shown below. The editors apologise for the inconvenience.

On the modular isomorphism problem for groups of class 3 and obelisks

We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.

Joining Forces for Reconstruction Inverse Problems

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.