Geodesic
A fundamental domain in the special linear group $SL_2(\mathbb{f}_p[x])$ and secret sharing on its basis
Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 7-16

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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.
Keywords: a special linear group, a congruence subgroup, a fundamental domain, modular secret sharing, a threshold access structure. 
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     author = {G. V. Matveev and A. A. Osinovskaya and V. I. Yanchevskiǐ},
     title = {A fundamental domain in the special linear group  $SL_2(\mathbb{f}_p[x])$ and secret sharing on its basis},
     journal = {Trudy Instituta matematiki},
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     publisher = {mathdoc},
     volume = {32},
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     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a0/}
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G. V. Matveev; A. A. Osinovskaya; V. I. Yanchevskiǐ. A fundamental domain in the special linear group  $SL_2(\mathbb{f}_p[x])$ and secret sharing on its basis. Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 7-16. http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a0/