Secret sharing in a special linear group

Objectives. The problem of developing the mathematical foundations of modular secret sharing in a special linear group over the ring of integers is being solved. The relevance of the problem is reduced to the fact that a large number of requirements are imposed on secret sharing schemes. These include the ideality of the scheme, the possibility of verification, changing the threshold without the participation of the dealer, the implementation of a non-threshold access structure and some others. Every secret sharing scheme developed to date does not fully satisfy all these requirements. It only has a certain configuration of these properties. The development of a scheme on a new mathematical basis is intended to expand the list of these configurations, which creates more opportunities for the user in choosing the optimal option. Methods. Group theory, modular arithmetic and theory of secret sharing schemes are used. Results. A fundamental domain with respect to the action of the main congruence subgroup by right shifts in the special linear group of second-order matrices over the ring of integers is constructed. On this basis, methods for modular secret sharing and its threshold restoration are proposed. Conclusion. A rigorous mathematical justification is given for the correctness of the algorithms for generating partial secrets and restoring the main secret in the special linear group over the ring of integers. These results will be used to study the configuration of secret sharing properties in this group.