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A Publicly Verifiable Secret Sharing (PVSS) scheme, as introduced by Stadler, has a feature where anyone, besides the participants, can verify the validity of the shares distributed by the dealer. Schoenmakers added a new feature, by providing a proof of correctness of the shares released by the players in the reconstruction process. This protocol is claimed to be an improvement on Stadler's and Fujisaki-Okamoto's, both in efficiency and in the type of intractability assumptions. However, Young-Yung improved Schoenmakers' PVSS, using a Discrete-Log instead of a Decision Diffie-Hellman. In this paper, a new PVSS is presented, having an intrinsic difference with its predecessors, that is, the participants can prove the validity of their given shares, implicitly, proving their membership by a zero-knowledge protocol. This feature prevents cheaters from participating in the reconstruction process to gain valid shares. Hence, the new proposed PVSS is more secure than previous ones. Besides, the dealer only sends the amount of commitments limited to the threshold value, regardless of the number of shareholders; this leads to a more dynamic protocol.

Secret Sharing (SS) schemes allow the Sharing of a Secret among a set of trustees in such a way that only some qualified subsets of them can recover the Secret. Ordinary SS schemes assume that the trust to each trustee is fixed over time. However, this is not the case in many real scenarios. Social Secret Sharing (SSS) is a recently introduced type of SS that addresses this issue. It allows the Sharing of a Secret among a set of trustees such that the amount of trust to each participant could be changed over time. There exist only a few SSS schemes in the literature; most of them can share only one Secret during each execution. Hence, these schemes lack the required efficiency in situations where multiple Secrets need to be shared. According to the literature, there exists only one Social multi-Secret Sharing (SMSS) scheme in which, all the Secrets are reconstructed at one stage. However, in many applications, the Secrets should be recovered in multiple stages and even according to some specified order. To address these problems, this paper employs Birkhoff interpolation method and Chinese remainder theorem and proposes a new SMSS scheme. In the proposed scheme, the shareholders can recover the Secrets in different stages and according to the specified order by the dealer. The security analysis of the proposed scheme shows that it provides all the needed security requirements. In addition, the performance analysis of the proposed scheme indicates its overall superiority over the related schemes.

The purpose of this paper is to study the information ratio of perfect Secret Sharing of product of some special families of graphs. We seek to prove that the information ratio of prism graphs Yn are equal to 7/4 for any n³5, and we will gave a partial answer to a question of Csirmaz [10]. We will also study the information ratio of two other families Cm×Cn and Pm×Cn and obtain the exact value of information ratio of these graphs.