Generalized Secret Sharing Schemes Using N$^\mu$MDS Codes

Abstract

Mehta et al. [11] recently proposed an ${{\,\mathrm{NMDS}\,}}$ code-based secret sharing scheme having a richer access structure than the traditional (t, n) threshold secret sharing schemes, and is based on two mutually nonmonotonic sets of user groups of sizes t and $t-1$ respectively, where $n \ge t > 1$ corresponds to the total number of users. We give a full generalization of their scheme with complete security proofs. We propose an efficient generalized secret sharing scheme constructed using ${{\,\mathrm{N^{\mu }MDS}\,}}$ codes with time complexity of $O(n^3)$. The scheme accepts an access structure constructed using $\mu +1$ mutually nonmonotonic sets of user groups with sizes, $t, t-1, \dots , t-\mu$, respectively, where $1 \le \mu < t$, and the parameter t defines the threshold such that all user groups of size greater than t can recover the secret. The proposed secret sharing scheme is perfect and ideal and has robust cheating detection and cheater identification features.