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.