Paper 2026/2086

Dynamic Participant Updates to Access and Adversary Structures for Ideal Secret Sharing Schemes Based on Linear Codes

Abstract

The paper investigates secret sharing schemes based on linear codes, focusing primarily on the access structures and adversary structures arising from dynamic participant updates, as well as on whether such access structures can be realized by ideal linear secret sharing schemes. First, we characterize the maximal adversary structure corresponding to the access structure obtained by adding new participants to every authorized subsets of a given access structure.Second, we prove that the original access structure can be realized by a linear code $C(n+1)[n+1,k;q]$ if and only if the access structure obtained after adding $\ell$ new participants can be realized by a linear code $C(n+\ell+1)[n+\ell+1,k+\ell;q]$. In addition, we establish the correspondence between the codewords of these two linear codes. Furthermore, We prove that the original access structure can be realized by an ideal secret sharing scheme if and only if the expanded access structure can be realized by one as well. Finally, we demonstrate that every linear secret sharing scheme realizing a disjoint access structure is ideal. We also prove that access structures whose any two minimal authorized subsets share all but one participant can be realized by an ideal linear secret sharing scheme. In addition, we determine the adversary structure corresponding to this class of access structures. More generally, we obtain the maximal adversary structure when new participants are added to merely part of the minimal authorized subsets.