Properties and constraints of cheating-immune secret sharing schemes

A secret sharing scheme is a cryptographic protocol by means of which a dealer shares a secret among a set of participants in such a way that it can be subsequently reconstructed by certain qualiied subsets. The setting we consider is the following: in a rst phase, the dealer gives in a secure way a piece of information, called a share, to each participant. Then, participants belonging to a qualiied subset send in a secure way their shares to a trusted party, referred to as a combiner, who computes the secret and sends it back to the participants. Cheating-immune secret sharing schemes are secret sharing schemes in the above setting where dishonest participants, during the reconstruction phase, have no advantage in sending incorrect shares to the combiner (i.e., cheating) as compared to honest participants. More precisely, a coalition of dishonest participants, by using their correct shares and the incorrect secret supplied by the combiner, have no better chance in determining the true secret (that would have been reconstructed if they submitted correct shares) than an honest participant. In this paper we study properties and constraints of cheating-immune secret sharing schemes. We show that a perfect secret sharing scheme cannot be cheating-immune. Then, we prove an upper bound on the number of cheaters tolerated in such schemes. We also repair a previously proposed construction to realize cheating-immune secret sharing schemes. Finally, we discuss some open problems.