Optimum secret sharing scheme secure against cheating

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 \V-i\ for secret sharing schemes that protect against this type of attack: \V-i\ >= (\S\ - 1)/delta + 1, where V-i denotes the set of shares of participant P-i, S denotes the set of secrets, and delta 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.