We analyze algorithms for open construction of a key on some noncommutative group. Algorithms of factorization and decomposition for associative algebras (of small dimension) are considered. A survey of applications (in particular, in cryptography) of so-called “hidden matrices” is given.

Cryptographic Algorithms on Groups and Algebras

Matrices, idealisers and integer quaternions

It is known that structural matrix rings pro-vide a natural passage from complete matrix rings to upper and lower triangular matrix rings, and they often explain the peculiarities regarding certain properties of complete matrix rings on the one hand and of triangular matrix rings on the other hand. In this paper the concept of a set of matrix units in a ring associated with a quasi-order relation is introduced and used to provide an internal char-acterisation of structural matrix rings.

An internal characterisation of structural matrix rings

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M2(R), under an additional assumption that both pairs generate M2(R) as an R-algebra.

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Separable algebras over infinite fields are 2-generated and finitely presented

Matrix rings satisfying column sum conditions versus structural matrix rings

This paper provides several new criteria for a ring to be a complete matrix ring. Some applications demonstrate their efficacy; and their relative strengths are indicated by calculating the structures they impose on universal algebras.

J. C. Robson

Papers

Recognition of matrix rings II