Ideal Hierarchical (t, n) Secret Sharing Schemes

A secret sharing scheme divides a secret into multiple shares by a dealer and shared among shareholders in such a way that any authorized subset of share-holders can reconstruct the secret; whereas any un-authorized subset of share-holders cannot recover the secret. If the maximal length of shares is equal to the length of the secret in a secret sharing scheme, the scheme is called ideal. If the shares corresponding to each un-authorized subset provide absolutely no information, in the information-theoretic sense, the scheme is called per-fect. Shamir proposed the first (t, n) threshold secret sharing scheme and it is ideal and perfect. In this paper, we propose two modifications of Shamir’s secret sharing scheme. In our first modification, each shareholder keeps both x-coordinate and y-coordinate of a polynomial as private share. In our second modification, dealer uses polynomial with degree larger than the threshold value t to generate shares for a (t, n) threshold scheme. We show that thes