Document

# Linear Threshold Secret-Sharing with Binary Reconstruction

## File

LIPIcs.ITC.2021.12.pdf
• Filesize: 0.77 MB
• 22 pages

## Acknowledgements

We are grateful to Hoeteck Wee who proposed the problem to us, and provided many useful comments and suggestions towards solving it. We also thank anonymous referees for helpful comments.

## Cite As

Marshall Ball, Alper Çakan, and Tal Malkin. Linear Threshold Secret-Sharing with Binary Reconstruction. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITC.2021.12

## Abstract

Motivated in part by applications in lattice-based cryptography, we initiate the study of the size of linear threshold (`t-out-of-n') secret-sharing where the linear reconstruction function is restricted to coefficients in {0,1}. We also study the complexity of such schemes with the additional requirement that the joint distribution of the shares of any unauthorized set of parties is not only independent of the secret, but also uniformly distributed. We prove upper and lower bounds on the share size of such schemes, where the size is measured by the total number of field elements distributed to the parties. We prove our results by defining and investigating an equivalent variant of Karchmer and Wigderson’s Monotone Span Programs [CCC, 1993]. One ramification of our results is that a natural variant of Shamir’s classic scheme [Comm. of ACM, 1979], where bit-decomposition is applied to each share, is optimal for when the underlying field has characteristic 2. Another ramification is that schemes obtained from monotone formulae are optimal for certain threshold values when the field’s characteristic is any constant. For schemes with the uniform distribution requirement, we show that they must use Ω(nlog n) field elements, for all thresholds 2 < t < n and regardless of the field. Moreover, this is tight up to constant factors for the special cases where any t = n-1 parties can reconstruct, as well as for any threshold when the field characteristic is 2.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational complexity and cryptography
##### Keywords
• Secret sharing
• Span programs
• Lattice-based cryptography

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Shweta Agrawal, Xavier Boyen, Vinod Vaikuntanathan, Panagiotis Voulgaris, and Hoeteck Wee. Functional encryption for threshold functions (or fuzzy IBE) from lattices. In Public Key Cryptography, volume 7293 of Lecture Notes in Computer Science, pages 280-297. Springer, 2012.
2. Rudolf Ahlswede and Levon H. Khachatrian. The complete intersection theorem for systems of finite sets. European Journal of Combinatorics, 18(2):125-136, 1997. URL: http://www.sciencedirect.com/science/article/pii/S0195669885700923.
3. Miklós Ajtai, János Komlós, and Endre Szemerédi. Sorting in c log n parallel sets. Comb., 3(1):1-19, 1983.
4. Amos Beimel. Secure schemes for secret sharing and key distribution, 1996.
5. Amos Beimel. Secret-sharing schemes: A survey. In IWCC, volume 6639 of Lecture Notes in Computer Science, pages 11-46. Springer, 2011.
6. Josh Cohen Benaloh and Jerry Leichter. Generalized secret sharing and monotone functions. In CRYPTO, volume 403 of Lecture Notes in Computer Science, pages 27-35. Springer, 1988.
7. George Robert Blakley. Safeguarding cryptographic keys. In 1979 International Workshop on Managing Requirements Knowledge (MARK), pages 313-318. IEEE, 1979.
8. Andrej Bogdanov, Siyao Guo, and Ilan Komargodski. Threshold secret sharing requires a linear size alphabet. In Theory of Cryptography Conference, pages 471-484. Springer, 2016.
9. Dan Boneh, Rosario Gennaro, Steven Goldfeder, Aayush Jain, Sam Kim, Peter M. R. Rasmussen, and Amit Sahai. Threshold cryptosystems from threshold fully homomorphic encryption. In CRYPTO (1), volume 10991 of Lecture Notes in Computer Science, pages 565-596. Springer, 2018.
10. Ravi B. Boppana. Amplification of probabilistic boolean formulas. Adv. Comput. Res., 5:27-45, 1989.
11. Xavier Boyen. Attribute-based functional encryption on lattices. In TCC, volume 7785 of Lecture Notes in Computer Science, pages 122-142. Springer, 2013.
12. Ronald Cramer and Serge Fehr. Optimal black-box secret sharing over arbitrary abelian groups. In Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology, CRYPTO ’02, page 272–287, Berlin, Heidelberg, 2002. Springer-Verlag.
13. J. Friedman. Constructing o(n log n) size monotone formulae for the k-th elementary symmetric polynomial of n boolean variables. In 25th Annual Symposium onFoundations of Computer Science, 1984., pages 506-515, 1984.
14. Anna Gal. A characterization of span program size and improved lower bounds for monotone span programs. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, page 429–437, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276855.
15. Christopher Godsil and Karen Meagher. Erdős–Ko–Rado Theorems: Algebraic Approaches. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2015.
16. Sergey Gorbunov, Vinod Vaikuntanathan, and Hoeteck Wee. Attribute-based encryption for circuits. In STOC, pages 545-554. ACM, 2013.
17. M. M. Halldorsson, J. Radhakrishnan, and K. V. Subrahmanyam. Directed vs. undirected monotone contact networks for threshold functions. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 604-613, 1993.
18. Kenneth Hoffman and Ray A. Kunze. Linear Algebra. PHI Learning, second edition, 2004. URL: http://www.worldcat.org/isbn/8120302702.
19. Shlomo Hoory, Avner Magen, and Toniann Pitassi. Monotone circuits for the majority function. In APPROX-RANDOM, volume 4110 of Lecture Notes in Computer Science, pages 410-425. Springer, 2006.
20. M. Karchmer and A. Wigderson. On span programs. In [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference, pages 102-111, 1993.
21. Gyula O.H. Katona. Around the complete intersection theorem. Discrete Applied Mathematics, 216:618-621, 2017. Levon Khachatrian’s Legacy in Extremal Combinatorics. URL: http://www.sciencedirect.com/science/article/pii/S0166218X1600010X.
22. Mike Paterson. Improved sorting networks with o(log N) depth. Algorithmica, 5(1):65-92, 1990.
23. Adi Shamir. How to share a secret. Commun. ACM, 22(11):612-613, 1979.
24. Leslie G. Valiant. Short monotone formulae for the majority function. J. Algorithms, 5(3):363-366, 1984.
X

Feedback for Dagstuhl Publishing