Evolving Secret Sharing with Essential Participants
Komargodski et al. introduced Evolving Secret Sharing which allows an impartial participant, called dealer, to share a secret among unbounded number of participants over any given access structure. In their construction for evolving secret sharing over ge
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Abstract Komargodski et al. introduced Evolving Secret Sharing which allows an impartial participant, called dealer, to share a secret among unbounded number of participants over any given access structure. In their construction for evolving secret sharing over general access structure, the size of share of the ith participant happens to be exponential (O(2i−1 )). They also provided constructions for (k, ∞) threshold secret sharing. We consider the problem of evolving secret sharing with t essential participants, namely, over t-(k, ∞) access structure, a generalization of (k, ∞) secret sharing (t = 0). We further generalize this access structure to a possible case of unbounded number of essential participants and provide a construction for secret sharing on it. Both the constructions are information theoretically secure and reduce the share size of the construction due to Komargodski et al. over general access structure, exponentially. Moreover, the essential participants receive ideal (and hence, optimal) shares in the first construction. Keywords Evolving access structure · Secret sharing · Essential participants · Information theoretic
1 Introduction In secret sharing, one can so share an information (usually a field element) among n (fixed and pre-decided) participants that certain subsets are able to reconstruct it back, while others are not [18]. Given any access structure on a set of participants, J. Pramanik (B) Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, India e-mail: [email protected] A. Adhikari Department of Mathematics, Presidency University, 86/1, College Street Rd, Kolkata 700073, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 D. Bhattacharjee et al. (eds.), Proceedings of International Conference on Frontiers in Computing and Systems, Advances in Intelligent Systems and Computing 1255, https://doi.org/10.1007/978-981-15-7834-2_64
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there exists a secret sharing scheme realizing it. Evolving secret sharing generalizes the notion of usual secret sharing where the participants’ set was to be known beforehand. It allows participants to join one by one, and the dealer hands them their shares without refreshing shares already distributed. Komargodski et al. introduced evolving secret sharing in [8]. We discuss few of these notions in detail in Sect. 2. In Sect. 3, we introduce t-(k, ∞) and (t, ∞, k, ∞) secret sharing and provide two constructions. In Sect. 4, we summarize our results and suggest further research directions. Our Contribution: In this paper, we provide a construction for secret sharing realizing t-(k, ∞) access structure where fixed t participants are essential. Essential participants in this scheme receive a share of size O(1), whereas ith of the other participants receives a share of the size (k − 1) · log i + poly(k, ) · O(log i) for an -bit secret being shared. We further generalize this access structure to (t, ∞, k, ∞) access structure and provi
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