Strong (n, t, n) verifiable secret sharing scheme

A (t, n) secret sharing divides a secret into n shares in such a way that any t or more than t shares can reconstruct the secret; but fewer than t shares cannot reconstruct the secret. In this paper, we extend the idea of a (t, n) secret sharing scheme and give a formal definition on the (n, t, n) secret sharing scheme based on Pedersen’s (t, n) secret sharing scheme. We will show that the (t, n) verifiable secret sharing (VSS) scheme proposed by Benaloh can only ensure that all shares are t-consistent (i.e. any subset of t shares defines the same secret); but shares may not satisfy the security requirements of a (t, n) secret sharing scheme. Then, we introduce new notions of strong t-consistency and strong VSS. A strong VSS can ensure that (a) all shares are t-consistent, and (b) all shares satisfy the security requirements of a secret sharing scheme. We propose a strong (n, t, n) VSS based on Benaloh’s VSS. We also prove that our proposed (n, t, n) VSS satisfies the definition of a strong VSS.