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Lower Bounds for Secret-Sharing Schemes for k-Hypergraphs

Author Amos Beimel

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Amos Beimel
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

Research supported by ERC grant 742754 (project NTSC) and Israel Science Foundation grant no. 391/21.

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Amos Beimel. Lower Bounds for Secret-Sharing Schemes for k-Hypergraphs. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITC.2023.16

Abstract

A secret-sharing scheme enables a dealer, holding a secret string, to distribute shares to parties such that only pre-defined authorized subsets of parties can reconstruct the secret. The collection of authorized sets is called an access structure. There is a huge gap between the best known upper bounds on the share size of a secret-sharing scheme realizing an arbitrary access structure and the best known lower bounds on the size of these shares. For an arbitrary n-party access structure, the best known upper bound on the share size is 2^{O(n)}. On the other hand, the best known lower bound on the total share size is much smaller, i.e., Ω(n²/log(n)) [Csirmaz, Studia Sci. Math. Hungar.]. This lower bound was proved more than 25 years ago and no major progress has been made since. 
In this paper, we study secret-sharing schemes for k-hypergraphs, i.e., for access structures where all minimal authorized sets are of size exactly k (however, unauthorized sets can be larger). We consider the case where k is small, i.e., constant or at most log(n). The trivial upper bound for these access structures is O(n⋅ binom(n-1,k-1)) and this can be slightly improved. If there were efficient secret-sharing schemes for such k-hypergraphs (e.g., 2-hypergraphs or 3-hypergraphs), then we would be able to construct secret-sharing schemes for arbitrary access structures that are better than the best known schemes. Thus, understanding the share size required for k-hypergraphs is important. Prior to our work, the best known lower bound for these access structures was Ω(n log(n)), which holds already for graphs (i.e., 2-hypergraphs). 
We improve this lower bound, proving a lower bound of Ω(n^{2-1/(k-1)}/k) on the total share size for some explicit k-hypergraphs, where 3 ≤ k ≤ log(n). For example, for 3-hypergraphs we prove a lower bound of Ω(n^{3/2}). For log(n)-hypergraphs, we prove a lower bound of Ω(n²/log(n)), i.e., we show that the lower bound of Csirmaz holds already when all minimal authorized sets are of size log(n). Our proof is simple and shows that the lower bound of Csirmaz holds for a simple variant of the access structure considered by Csirmaz. Using our results, we prove a near quadratic separation between the required share size for realizing an explicit access structure and the monotone circuit size describing the access structure, i.e., the share size in Ω(n²/log(n)) and the monotone circuit size is O(nlog(n)) (where the circuit has depth 3).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Cryptographic primitives
Keywords
  • Secret Sharing
  • Share Size
  • Lower Bounds
  • Monotone Circuits

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