A fundamental domain in the special linear group $SL_2(\mathbb{F}_p[x])$ and secret sharing on its basis
Abstract
The problem of developing the mathematical foundations of modular secret sharing in the special linear group over the ring of polynomials in one variable over the finite Galois field with $p$ elements is being solved. Secret sharing schemes should meet a large number of requirements: perfectness and ideality of a scheme, possibility of verification, changing a threshold without participation of a dealer, implementation of a non-threshold access structure and some others. Every secret sharing scheme developed to date does not fully satisfy all these requirements. The development of a scheme on a new mathematical basis is intended to expand the list of these configurations, thereby creating more possibilities for a user to choose the optimal option. A fundamental domain with respect to the action of the main congruence subgroup by right shifts in the special linear group of dimension 2 over the ring of polynomials is constructed. On this basis, methods for modular threshold secret sharing and its reconstruction are proposed.
About the Authors
G. V. Matveev
Belarusian State University
Belarus
Belarus
Minsk
A. A. Osinovskaya
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Minsk
V. I. Yanchevskii
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Minsk
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Review
For citations:
Matveev G.V., Osinovskaya A.A., Yanchevskii V.I. A fundamental domain in the special linear group $SL_2(\mathbb{F}_p[x])$ and secret sharing on its basis. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2024;32(2):7-16. (In Russ.)
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