Adaptation of the modular number system in threshold secret sharing schemes

Objectives. The purpose of the research is to test the applicability of the adaptation of the modular number system using a masking transformation with a pseudo-random integer value to the original secret S in a modification of Adi Shamir’s (k, n)-threshold secret sharing scheme to reduce the complexity of calculating the basic integral characteristic to a theoretical minimum.

Methods. A modification of Adi Shamir's secret sharing scheme in a threshold cryptosystem based on modular arithmetic (MA cryptosystem) with the generation of shares of secret sharing participants in two stages is considered. Shamir’s scheme was chosen as optimal in terms of complexity, resource intensity, perfection and ideality; in addition, it is scalable – the number of participants can be increased to the order of the field p, without changing the ability to recover the secret. A masking transformation using a term with a pseudo-random integer value C for the shared secret S, the range of change of the pseudo-random parameter C agreed upon the range of changes in the values of the original signal is applied. The interval-modular form of the number of the secret value is applied too.

Results. It is shown that the use of the interval-modular form of the number S̅ – a masking transformation with a pseudo-random parameter of the number S of the original secret – reduces the complexity of calculating basic interval-index characteristics when solving threshold coding problems almost to a theoretical minimum. Adaptive coordination of changes in the pseudo-random parameter of the masking function with the domain of its results makes it possible to implement a minimally redundant modular decomposition of the masking function for any admissible basis of the based scheme.

Conclusion. The results of the presented work allow to conclude for modular threshold cryptosystems of secret sharing in distributed data processing systems that the use of a linear masking function and narrowing the range of changes in the masking analogue S̃ of the original secret S, allowing for minimally redundant coding for the selected p₁, p₂, …, p_n, causes a significant reduction in the computational complexity of the calculated minimal-redundant modular arithmetic relations of integral characteristics within the framework of the model under study. Due to which a higher level of performance is achieved at the stage of decoding the original secret compared to other solutions.

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