Almost Optimum Secret Sharing Schemes Secure Against Cheating for Arbitrary Secret Distribution

We consider the problem of cheating in secret sharing schemes, cheating in which individuals submit forged shares in the secret reconstruction phase in an effort to make another participant reconstruct an invalid secret. We introduce a novel technique which uses universal hash functions to detect such cheating and propose two efficient secret sharing schemes that employ the functions. The first scheme is nearly optimum with respect to the size of shares; that is, the size of shares is only one bit longer than its existing lower bound. The second scheme possesses a particular merit in that the parameter for the probability of successful cheating can be chosen without regard to the size of the secret. Further, the proposed schemes are proven to be secure regardless of the probability distribution of the secret.