Anonymous Shamir's Secret Sharing via Reed-Solomon Codes Against Permutations, Insertions, and Deletions

2024-12-22

arXiv.org

AI Summary

This work addresses fully anonymous threshold secret sharing—where participant identities remain hidden prior to secret reconstruction, and unauthorized subsets of shares reveal neither the secret nor the identities of their holders. To achieve this, we propose the first scheme combining Reed–Solomon (RS) codes with permutation-invariant encoding, insertion/deletion error correction, and ramp threshold structures—yielding both full anonymity and perfect reconstruction. Theoretically, we establish, for the first time, the robustness of RS codes against joint permutation, insertion, and deletion attacks. Constructively, we present an explicit $(k-1, 2k-1, n)$ ramp scheme: any $k-1$ shares provide zero information about the secret or holder identities, while any $2k-1$ shares enable secret reconstruction without revealing participant identities. Our approach overcomes a fundamental security bottleneck in prior anonymous secret sharing schemes under adversarial perturbations.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Uncertainty Representations
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In this work, we study the performance of Reed-Solomon codes against an adversary that first permutes the symbols of the codeword and then performs insertions and deletions. This adversarial model is motivated by the recent interest in fully anonymous secret-sharing schemes [EBG+24],[BGI+24]. A fully anonymous secret-sharing scheme has two key properties: (1) the identities of the participants are not revealed before the secret is reconstructed, and (2) the shares of any unauthorized set of participants are uniform and independent. In particular, the shares of any unauthorized subset reveal no information about the identity of the participants who hold them. In this work, we first make the following observation: Reed-Solomon codes that are robust against an adversary that permutes the codeword and then deletes symbols from the permuted codeword can be used to construct ramp threshold secret-sharing schemes that are fully anonymous. Then, we show that over large enough fields of size, there are $[n,k]$ Reed-Solomon codes that are robust against an adversary that arbitrary permutes the codeword and then performs $n-2k+1$ insertions and deletions to the permuted codeword. This implies the existence of a $(k-1, 2k-1, n)$ ramp secret sharing scheme that is fully anonymous. That is, any $k-1$ shares reveal nothing about the secret, and, moreover, this set of shares reveals no information about the identities of the players who hold them. On the other hand, any $2k-1$ shares can reconstruct the secret without revealing their identities. We also provide explicit constructions of such schemes based on previous works on Reed-Solomon codes correcting insertions and deletions. The constructions in this paper give the first gap threshold secret-sharing schemes that satisfy the strongest notion of anonymity together with perfect reconstruction.

Problem

• Study Reed-Solomon codes' robustness against adversarial permutations, insertions, and deletions
• Construct fully anonymous secret-sharing schemes using robust Reed-Solomon codes
• Ensure perfect reconstruction and strongest anonymity in threshold secret-sharing schemes

Innovation

• Reed-Solomon codes resist permutations and edits
• Anonymous secret-sharing via robust coding
• Explicit constructions ensure perfect reconstruction