Combinatorial Repairability for Threshold Schemes

TL;DR

This paper explores methods for repairing lost or corrupted shares in threshold schemes without dealer participation, ensuring security and efficiency through two approaches: enrollment protocols and combinatorial share distribution.

Contribution

It introduces two novel repair methods for threshold schemes that maintain security and improve efficiency, expanding the practical applicability of such schemes.

Findings

The enrollment protocol allows share repair without dealer involvement.

Distributing multiple shares reduces communication complexity.

The methods preserve the unconditional security of the scheme.

Abstract

In this paper, we consider methods whereby a subset of players in a $(k,n)$-threshold scheme can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme, and it should be efficient, where efficiency is measured in terms of the amount of information exchanged during the repairing process. We study two approaches to repairing. The first method is based on the "enrollment protocol" from \cite{NSG} which was originally developed to add a new player to a threshold scheme (without the participation of the dealer) after the scheme was set up. The second method distributes "multiple shares" to each player, as defined by a suitable combinatorial design. This method results in larger shares, but lower communication complexity, as compared to the first method.