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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{TIMB_2024_32_2_a0,
     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},
     pages = {7--16},
     year = {2024},
     volume = {32},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a0/}
}
TY  - JOUR
AU  - G. V. Matveev
AU  - A. A. Osinovskaya
AU  - V. I. Yanchevskiǐ
TI  - A fundamental domain in the special linear group $SL_2(\mathbb{f}_p[x])$ and secret sharing on its basis
JO  - Trudy Instituta matematiki
PY  - 2024
SP  - 7
EP  - 16
VL  - 32
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a0/
LA  - ru
ID  - TIMB_2024_32_2_a0
ER  - 
%0 Journal Article
%A G. V. Matveev
%A A. A. Osinovskaya
%A V. I. Yanchevskiǐ
%T A fundamental domain in the special linear group $SL_2(\mathbb{f}_p[x])$ and secret sharing on its basis
%J Trudy Instituta matematiki
%D 2024
%P 7-16
%V 32
%N 2
%U http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a0/
%G ru
%F TIMB_2024_32_2_a0
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/

[1] Cramer R., Damgard I., Nielsen J., “Multiparty computation from threshold homomorphic encryption”, LNCS, 2045, 2001, 280–300 | DOI | MR

[2] Bethencourt J., Sahai A., Waters B., “Ciphertext-policy attribute-based encryption”, 2007 IEEE Symposium on Security and Privacy (SP'07), IEEE, 2007, 321–334 | DOI

[3] Benaloh J., “Secret sharing homomorphisms: keeping shares of a secret sharing”, LNCS, 263, 1987, 251–260 | DOI | MR

[4] Shamir A., “How to share a secret”, Communications of the ACM, 22 (1979), 612–613 | DOI | MR

[5] Asmuth C., Bloom J., “A modular approach to key safeguarding”, IEEE Transactions on Information Theory, 29 (1983), 156–169 | DOI | MR

[6] Mignotte M., “How to share a secret”, LNCS, 149, 1983, 371–375 | DOI | MR

[7] Galibus T., Matveev G., Shenets N., “Some structural and security properties of the modular secret sharing”, Proceedings of SYNASC'08, IEEE, Los Alamitos, 2009, 197–200 | DOI

[8] Galibus T., Matveev G., “Generalized Mignotte's sequences over polynomial rings”, Electronic Notes in Theoretical Computer Science, 186 (2007), 43–48 | DOI | MR

[9] Galibus T., Matveev G., “Finite fields, Gröbner bases and modular secret sharing”, Journal of Discrete Mathematical Sciences and Cryptography, 15 (2012), 339–348 | DOI | MR

[10] Vaskouski M. M., Matveev G. V., “Verification of modular secret sharing”, Journal of the Belarusian State University. Mathematics and Informatics, 2017, no. 2, 17–22 (in Russian) | MR

[11] Matveev G. V., Matulis V. V., “Perfect verification of modular scheme”, Journal of the Belarusian State University. Mathematics and Informatics, 2018, no. 2, 4–9 (in Russian)

[12] Yanchevski\u{i} V. I., Havarushka I. A., Matveev G. V., “Secret sharing in a special linear group”, Informatics, 21:3 (2024), 23–31 (in Russian) | DOI | DOI

[13] Rosen M., Number theory in function fields, Springer-Verlag, New York, 2002, 358 pp. | MR

[14] Taylor D. E., The geometry of the classical groups, Herdelmann Verlag, Berlin, 1992, 229 pp. | MR

[15] Nagao H., “On $GL(2;K[X])$”, Journal of the Institute of Polytechnics, Osaka City University. Series A: Mathematics, 10 (1959), 117–121 | MR

[16] Platonov V. P., Rapinchuk A. S., Algebraic groups and number theory, Nauka, M., 1991, 656 pp. (in Russian) | MR