Shamir's Secret Sharing Scheme is well established and widely used. It allows a so-called Dealer to split and share a secret k among n Participants such that at least t shares are needed to reconstruct k, where 0 <p > < t ≤ n. Nothing about the secret can belearned from less than t shares. To split secret k, the Dealer generates a polynomial f,whose independent term is k and the coefficients are randomly selected using a uniform distribution. A share is a pair (x,f(x)) where x is also chosen randomly using a uniform distribution. This scheme is useful, for example, to distribute cryptographic keys among different cloud providers and to create multi-factor authentication. The security of Shamir's Secret Sharing Scheme is usually analyzed using a threat model where the Dealer is trusted to split and share secrets as described above. In this paper, we demonstrate that there exists a different threat model where a malicious Dealer can compute shares such that a subset of less than t shares is allowed to reconstruct the secret. We refer to such subsets as hidden sets. We formally define hidden sets and prove lower boundson the number of possible hidden sets for polynomials of degree t-1. Yet, we show how to detect hidden sets given a set of n shares and describe how to create hidden sets while sharing a secret using a modification of Shamir's scheme.
2018 IEEE Symposium on Computers and Communications, ISCC 2018
School of Mathematics and Statistics

De Souza, R.L. (Rick Lopes), Vigil, M. (Martin), Custodio, R. (Ricardo), Caullery, F. (Florian), Moura, L. (Lucia), & Panario, D. (2018). Secret Sharing Schemes with Hidden Sets. In Proceedings - IEEE Symposium on Computers and Communications (pp. 713–718). doi:10.1109/ISCC.2018.8538542