In this paper we introduce a strong version of the Birkhoff–James orthogonality in Hilbert C*-modules. More precisely, we consider elements x and y of a Hilbert C*-module V over a C*-algebra A which satisfy ||x||£||x+ya|| for all a Î A. We show that this relation can be described as the Birkhoff–James orthogonality of appropriate elements of V, and characterized in terms of states acting on the underlying C*-algebra A. Some analogous relations of this type are considered as well.

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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.