Perfect uniformly robust secret sharing scheme for a vector access structure

Authors

  • Anton Alekseychuk Institute of Special Communication and Information Protection of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine https://orcid.org/0000-0003-4385-4631
  • Mykhailo Pokydko Administration of the State Service of Special Communications and Information Protection of Ukraine, Kyiv, Ukraine https://orcid.org/0009-0002-9225-5694

DOI:

https://doi.org/10.20535/2411-1031.2026.14.1.365483

Keywords:

cybersecurity, cryptographic information protection, access control, secret sharing scheme, unconditional security, vector access structure, robustness

Abstract

A secret sharing scheme is a cryptographic protocol designed to distribute a secret among designated participants in such a way that only certain (authorized) coalitions of participants can reconstruct the value of the secret by combining the components (shares of the secret) they receive. The set of all authorized coalitions of participants is called an access structure. Vector secret sharing schemes are constructed using linear block codes over finite fields so that the sets of participants’ shares are random codewords whose first coordinate equals the value of the secret. Any access structure for which there exists a vector scheme implementing it is called a vector access structure. One of the requirements imposed on modern secret sharing schemes is unconditional resistance to attacks by “dishonest” participants who may substitute their obtained shares in order to distort the value of the secret during its reconstruction. Such schemes are called robust. A secret sharing scheme is said to be uniformly robust if it remains robust regardless of the method used to select the set of secret keys from the set of all elements that can potentially be distributed. At present, uniformly robust secret sharing schemes are known only for a narrow class of threshold access structures. This paper proposes a construction of a uniformly robust secret sharing scheme for an arbitrary vector access structure. The construction is based on the use of random block codes that are equivalent to a predetermined code defining the underlying vector secret sharing scheme. Informally, the robustness of the resulting scheme relies on the participants’ lack of complete information about the selected code. The scientific basis of the proposed method for constructing uniformly robust secret sharing schemes is an obtained analytical relation that provides an upper bound on the probability of successful substitution of shares by participants of unauthorized coalitions with the aim of distorting the secret. The practical significance of the obtained results lies in the possibility of applying the proposed construction to the development of cryptographic protocols for distributed storage and processing of confidential information with increased resistance to actions of dishonest participants. The proposed approach can be used in access control systems, distributed information systems, and collaborative data storage services where it is necessary to guarantee the integrity and correctness of secret reconstruction.

Author Biographies

Anton Alekseychuk, Institute of Special Communication and Information Protection of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv

doctor of technical science, professor, professor at the state information resources security academic department

Mykhailo Pokydko, Administration of the State Service of Special Communications and Information Protection of Ukraine, Kyiv

Director of the Department of Digital Transformation

References

A. Shamir, “How to share a secret”, Communications of the ACM, vol. 22, no. 11, pp. 612-613, 1979.

E.F. Brickell, “Some ideal secret sharing schemes”, Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 6, pp. 105–113, 1989.

A. Beimel, “Secret-sharing schemes: A survey”, in Proc. Int. Conf. Coding and Cryptology, Berlin, Heidelberg: Springer, 2011, pp. 11-46. [Online]. Available: https://link.springer.com/chapter/10.1007/978-3-642-20901-7_2. Accessed on: Feb. 11, 2026.

R. Cramer, I.B. Damgård, N. Döttling, S. Fehr, and G. Spini, “Linear secret sharing schemes from error correcting codes and universal hash functions”, in Advances in Cryptology – EUROCRYPT 2015, E. Oswald and M. Fischlin, Eds., Berlin, Heidelberg: Springer, 2015, pp. 313-336, Lecture Notes in Computer Science, vol. 9057. https://doi.org/10.1007/978-3-662-46803-6.

O. Farràs, and C. Padró, “Ideal Hierarchical Secret Sharing Schemes” in Theory of Cryptography. TCC 2010. Lecture Notes in Computer Science, vol 5978, Berlin, Heidelberg: Springer, 2010, pp. 219-236. https://doi.org/10.1007/978-3-642-11799-2_14.

A.K. Chattopadhyay et al., “Secret sharing: A comprehensive survey, taxonomy and applications”, Computer Science Review, vol. 51, Art. no. 100608, 2024. doi: https://doi.org/10.1016/j.cosrev.2023.100608.

M. Tompa, and H. Woll, “How to share a secret with cheaters”, Journal of Cryptology, vol. 1, no. 3, pp. 133-138, 1989. [Online]. Available: https://link.springer.com/article/10.1007/BF02252871. Accessed on: Feb. 11, 2026.

J. Pieprzyk, and X.M. Zhang, “Cheating Prevention in Linear Secret Sharing” in Information Security and Privacy. ACISP 2002. Lecture Notes in Computer Science, vol 2384, Berlin, Heidelberg: Springer, 2002, pp. 121-135. doi: https://doi.org/10.1007/3-540-45450-0_9.

C. Padró, G. Sáez, and J.L. Villar, “Detection of cheaters in vector space secret sharing schemes”, Designs, Codes and Cryptography, vol. 16, no. 1, pp. 75-85, 1999. doi: https://doi.org/10.1023/A:1008378426278

S. Cabello, C. Padró, and G. Sáez, “Secret sharing schemes with detection of cheaters for a general access structure”, Designs, Codes and Cryptography, vol. 25, no. 2, pp. 175-188, 2002. doi: https://doi.org/10.1023/A:1013856431727.

W. Ogata, K. Kurosawa, and D.R. Stinson, “Optimum secret sharing scheme secure against cheating”, SIAM Journal on Discrete Mathematics, vol. 20, no. 1, pp. 79-95, 2006. doi: https://doi.org/10.1137/S0895480100378689.

R. Cramer, Y. Dodis, S. Fehr, C. Padró, and D. Wichs, “Detection of algebraic manipulation with applications to robust secret sharing and fuzzy extractors”, in Advances in Cryptology – EUROCRYPT 2008, Lecture Notes in Computer Science, vol. 4965, Berlin, Heidelberg: Springer, 2008, pp. 471-488. doi: https://doi.org/10.1007/978-3-540-78967-3_27.

D.J. Bernstein et al., Classic McEliece: Conservative Code-Based Cryptography, 2019. [Online]. Available: https://cryptojedi.org/papers/mceliecenistr2-20190331.pdf. Accessed on: Feb. 11, 2026.

S. Lang, Algebra. New York, NY, USA: Springer-Verlag, 2002.

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Published

2026-06-26

How to Cite

Alekseychuk, A., & Pokydko, M. (2026). Perfect uniformly robust secret sharing scheme for a vector access structure. Collection "Information Technology and Security", 14(1), 139–150. https://doi.org/10.20535/2411-1031.2026.14.1.365483

Issue

Vol. 14 No. 1 (2026)

Section

MATHEMATICAL AND COMPUTER MODELING

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