A method for constructing robust multisecret sharing schemes based on polynomial transformations over finite commutative principal ideal rings

Abstract

This paper is devoted to the development of a method for constructing secret sharing schemes intended for registry management systems of the National Center for Backup of State Information Resources. Such schemes represent cryptographic protocols that enable cryptographic keys to be distributed among multiple participants and stored at different locations or by different participants, while allowing secret reconstruction only for predefined authorized coalitions. Registry management systems of the National Center require unconditionally secure multiple secret sharing schemes characterized by low computational complexity of share generation and secret reconstruction procedures, as well as robustness, i.e., unconditional resistance against attacks by dishonest participants who may substitute their own shares in order to corrupt the reconstructed secrets. Existing constructions of robust secret sharing schemes are mainly based on auxiliary transformations over finite fields and are applicable only to single-secret sharing schemes. The paper proposes a method for constructing robust multisecret sharing schemes based on vector secret sharing schemes for a single secret and polynomial transformations over finite commutative principal ideal rings. The proposed method generalizes a previously known approach for constructing robust secret sharing schemes for vector access structures over finite fields. At the same time, the resulting constructions generalize previously known multiple secret sharing schemes defined over residue rings modulo a natural number. An analytical upper bound on the probability of successful share substitution by participants of an arbitrary forbidden coalition is obtained, and conditions are established under which this probability can be made arbitrarily small. The proposed method can be applied to the construction of cryptographic protocols for distributed storage and processing of confidential information with enhanced resistance against dishonest participants, as well as to access control systems, distributed information systems, and collaborative data storage services requiring guaranteed integrity and correctness of secret reconstruction.