Optimum secret sharing scheme secure against cheating

Abstract

Tompa and Woll introduced a problem of cheating in (k, n) threshold secret sharing schemes. In this problem k - 1 malicious participants aim to cheat an honest one by opening forged shares and causing the honest participant to reconstruct the wrong secret. We first derive a tight lower bound on the size of shares |Vi| for secret sharing schemes that protect against this type of attack: |Vi| ≥ (|S| - 1)/6 + 1, where Vi denotes the set of shares of participant Pi, S denotes the set of secrets, and δ denotes the cheating probability. We next present an optimum scheme, which meets the equality of our bound, by using "difference sets." A partial converse and some extensions are also shown. © 2006 Society for Industrial and Applied Mathematics.