Cheolhoon Choi  Minsoo Ryu

Vol. 13,  No. 2, pp. 34-40, Feb.  2024
https://doi.org/10.3745/TKIPS.2024.13.2.34

Hierarchical Secret Sharing Lagrange interpolation polynomial

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Secret sharing is a cryptographic technique that involves dividing a secret or a piece of sensitive information into multiple shares or parts, which can significantly increase the confidentiality of a secret. There has been a lot of research on secret sharing for different contexts or situations. Tassa's conjunctive secret sharing method employs polynomial derivatives to facilitate hierarchical secret sharing. However, the use of derivatives introduces several limitations in hierarchical secret sharing. Firstly, only a single group of participants can be created at each level due to the shares being generated from a sole derivative. Secondly, the method can only reconstruct a secret through conjunction, thereby restricting the specification of arbitrary secret reconstruction conditions. Thirdly, Birkhoff interpolation is required, adding complexity compared to the more accessible Lagrange interpolation used in polynomial-based secret sharing. This paper introduces the multi-compartment secret sharing method as a generalization of the conjunctive hierarchical secret sharing. Our proposed method first encrypts a secret using external groups’ shares and then generates internal shares for each group by embedding the encrypted secret value in a polynomial. While the polynomial can be reconstructed with the internal shares, the polynomial just provides the encrypted secret, requiring external shares for decryption. This approach enables the creation of multiple participant groups at a single level. It supports the implementation of arbitrary secret reconstruction conditions, as well as conjunction. Furthermore, the use of polynomials allows the application of Lagrange interpolation.