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‎‎Secret Sharing is an important issue in cryptography which has many applications‎. ‎In a Secret Sharing scheme‎, ‎a Secret is shared by a dealer among several participants in such a way that any authorized subset of participants can recover the Secret by pooling their shares‎. ‎Recently‎, ‎several schemes based on elliptic curves and bilinear maps have been presented‎. ‎Some of these schemes need a secure channel‎, ‎there are restrictions on the number of Secrets‎, ‎or the participants or the dealer are unable to verify the validity of the shares‎. ‎In this paper‎, ‎we present a new verifiable (t‎, ‎n)-threshold Multi-Secret Sharing scheme based on elliptic curves and pairings that does not have any of the above restrictions‎. ‎The hardness of a discrete logarithm problem on elliptic curves guarantees the security of the proposed scheme‎.

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.

In a (t; n)-threshold Secret Sharing scheme, a Secret s is distributed among n participants such that any group of t or more participants can reconstruct the Secret together, but no group of fewer than t participants can do. In this paper we propose a veri able (t; n)-threshold Multi-Secret Sharing scheme based on Shao and Cao, and the intractability of the elliptic curve discrete logarithm problem (ECDLP) by using generalized Jacobian of elliptic curves. The proposed scheme has all the bene ts of Shao and Cao, however, our scheme no need to a secure channel. Furthermore, we exploit the techniques via elliptic curves to perform the scheme. This can be very important, because the hardness of discrete logarithm problem on elliptic curves increases security of the proposed scheme.

A Multi-Secret Sharing scheme (MSS) allows a dealer to share Multiple Secrets among a set of participants, in such a way A Multi-Secret Sharing scheme (MSS) allows a dealer to share Multiple Secrets among a set of participants, in such a way that any authorized subset of participants can reconstruct the Secrets. Up to now, existing MSSs either require too long shares for participants to be perfect secure, or do not have a formal security analysis/proof. In 2013, Herranz et al. provided the first formal definition of computational security for Multi-Stage Secret Sharing scheme (MSSS) in the standard model and proposed a practical and secure scheme. As far as we know, their scheme is the only computational secure MSS in the standard model, and there is no formal definition of the computational security for other categories of MSSs. Based on this motivation, in this paper, we de ne the first formal model of in distinguishability against the chosen Secret attacks (CSA) for other types of MSSs in the standard model. Furthermore, we present two practical CSA-secure MSSs, belonging to the different types of MSSs and enjoying the advantage of short shares. They are also provably secure in the standard model. Based on the semantic security of the underlying encryption schemes, we prove the security of our schemes.