arXiv subjects
Publications and source records attributed to V. A. Roman'kov.
We study verbally closed subgroups of free solvable groups. A number of results is proved that give sufficient conditions under whose a verbally closed subgroup is turned to be a retract and so algebraically closed of the full group.
We prove that every verbally closed two-generated subgroup of a free solvable group G of a finite rank is a retract of G.
We show that many known schemes of the public key exchange protocols in the algebraic cryptography, that use two-sided multiplications, are the specific cases of the general scheme of such type. In most cases, such schemes are built on platforms that are subsets of the linear spaces. They have been repeatedly compromised by the linear decomposition method introduced by the first author. The method allows to compute the exchanged keys without computing the private data and therefore without solving the algorithmic problems on which the assumptions are based. We demonstrate that this method can be successfully applied to the general scheme, thus it is in some sense universal.