On a fundamental domain in a special linear group
EDN: HQZGXB
Abstract
Modular secret sharing in the group $SL_2(\mathbb{Z})$ was recently proposed by Yanchevskiy, Matveev, and Govorushko. In this paper we have constructed in explicit form the entire fundamental domain under the action of left shifts of the principal congruence subgroup on the group $SL_2(\mathbb{Z})$, which presents additional possibilities for constructing schemes, since the domain is the space of stored secrets of the secret sharing scheme.
About the Authors
G. V. Matveev
Belarusian State University
Russian Federation
Russian Federation
Minsk
A. A. Osinovskaya
Institute of Mathematics of the National Academy of Sciences of Belarus
Russian Federation
Russian Federation
Minsk
References
1. Yanchevskiy V. I., Govorushko I. O., Matveev G. V. Secret sharing in the special linear group. Informatics, 2024, vol. 21, no. 3, pp. 23–31 (in Russian). https://doi.org/10.37661/1816-0301-2024-21-3-23-31
2. Harcos G. CEU lecture notes: a set of representatives for Γ0(q) SL2(Z) [Electronic resourse]. – Mode of access: https://users.renyi.hu/ĝHarcos/CEU_Gamma0(q).pdf
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4. Di Matteo G. The action of SL2(Z) on the upper-half complex plane [Electronic resourse]. – Mode of access: https://www.dimatteo.is/Mathematics/Courses/Modular-forms/02-SL2Z.pdf
5. Matveev G. V., Osinovskaya A. A., Yanchevskii V. I. A fundamental domain in the special linear group and secret sharing on its basis. Proceedings of the Institute of Mathathematics of the NAS of Belarus, 2024, vol. 32, no. 2, pp. 23–31 (in Russian).
6. Mignotte M. How to share a secret. Lecture Notes in Computer Science, 1983, vol. 149, pp. 371–375.
Review
For citations:
Matveev G.V., Osinovskaya A.A. On a fundamental domain in a special linear group. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2025;33(2):21-27. (In Russ.) EDN: HQZGXB
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