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Csirmaz’s Duality Conjecture and Threshold Secret Sharing

Author Andrej Bogdanov

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Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Cryptographic primitives
  • Mathematics of computing → Information theory
  • Security and privacy → Mathematical foundations of cryptography
Keywords
  • Threshold secret sharing
  • Fourier analysis

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Abstract

We conjecture that the smallest possible share size for binary secrets for the t-out-of-n and (n-t+1)-out-of-n access structures is the same for all 1 ≤ t ≤ n. This is a strenghtening of a recent conjecture by Csirmaz (J. Math. Cryptol., 2020). We prove the conjecture for t = 2 and all n. Our proof gives a new (n-1)-out-of-n secret sharing scheme for binary secrets with share alphabet size n.

Cite As Get BibTex

Andrej Bogdanov. Csirmaz’s Duality Conjecture and Threshold Secret Sharing. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 3:1-3:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITC.2023.3

Author Details

Andrej Bogdanov
  • University of Ottawa, Canada

Funding

  • Bogdanov, Andrej: This work was supported by RGC GRF grant CUHK 14301519 and NSERC grant RGPIN-2023-05006.

Acknowledgements

Part of the research was carried out while the author was with the Chinese University of Hong Kong. I thank the anonymous ITC 2023 reviewers for helpful suggestions.

References

  1. G.R. Blakley. Safeguarding cryptographic keys. In Proceedings of the 1979 AFIPS National Computer Conference, pages 313-317, Monval, NJ, USA, 1979. AFIPS Press. Google Scholar
  2. Andrej Bogdanov, Siyao Guo, and Ilan Komargodski. Threshold secret sharing requires a linear-size alphabet. Theory of Computing, 16(2):1-18, 2020. URL: https://doi.org/10.4086/toc.2020.v016a002.
  3. Ronald Cramer, Ivan Damgård, and Jesper Buus Nielsen. Secure Multiparty Computation and Secret Sharing. Cambridge University Press, 2015. URL: http://www.cambridge.org/de/academic/subjects/computer-science/cryptography-cryptology-and-coding/secure-multiparty-computation-and-secret-sharing?format=HB&isbn=9781107043053.
  4. László Csirmaz. Secret sharing and duality. J. Math. Cryptol., 15(1):157-173, 2020. URL: https://doi.org/10.1515/jmc-2019-0045.
  5. Satyanarayana V. Lokam. Complexity Lower Bounds Using Linear Algebra. Now Publishers Inc., Hanover, MA, USA, 2009. Google Scholar
  6. Ron Roth. Introduction to Coding Theory. Cambridge University Press, 2006. URL: https://doi.org/10.1017/CBO9780511808968.
  7. Adi Shamir. How to share a secret. Commun. ACM, 22(11):612-613, November 1979. URL: https://doi.org/10.1145/359168.359176.
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