Geodesic
Construction of quasi-cyclic alternant codes and~their application in code-based cryptography
Prikladnaâ diskretnaâ matematika, no. 3 (2024), pp. 84-109.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper presents an overview of quasi-cyclic alternant codes and their structural analysis regarding the classification of automorphisms. We also have detailed methods for recovering the structure of a given code. The attractiveness of the family of considered codes lies in their cryptographic applications and, as in theory, in reducing the key length of post-quantum code-based schemes. In addition, this method of constructing codes is universal and can be used to obtain subfield subcodes of quasi-cyclic algebraic-geometric codes associated with an arbitrary curve with a known group of automorphisms. However, as a result of constructing quasi-cyclic alternant codes, it becomes possible to reduce the key security of the source code to a code with smaller parameters, which may not be resistant to a structural attack.
Mots-clés : quasi-cyclic codes, alternant codes, invariant codes, automorphism group of a code.
Keywords: algebraic-geometric code, function fields 
@article{PDM_2024_3_a4,
     author = {A. A. Kuninets and E. S. Malygina},
     title = {Construction of quasi-cyclic alternant codes and~their application in code-based cryptography},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {84--109},
     publisher = {mathdoc},
     number = {3},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2024_3_a4/}
}
TY  - JOUR
AU  - A. A. Kuninets
AU  - E. S. Malygina
TI  - Construction of quasi-cyclic alternant codes and~their application in code-based cryptography
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2024
SP  - 84
EP  - 109
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2024_3_a4/
LA  - ru
ID  - PDM_2024_3_a4
ER  - 
%0 Journal Article
%A A. A. Kuninets
%A E. S. Malygina
%T Construction of quasi-cyclic alternant codes and~their application in code-based cryptography
%J Prikladnaâ diskretnaâ matematika
%D 2024
%P 84-109
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2024_3_a4/
%G ru
%F PDM_2024_3_a4
A. A. Kuninets; E. S. Malygina. Construction of quasi-cyclic alternant codes and~their application in code-based cryptography. Prikladnaâ diskretnaâ matematika, no. 3 (2024), pp. 84-109. http://geodesic.mathdoc.fr/item/PDM_2024_3_a4/

[1] Barelli E., On the Security of Some Compact Keys for McEliece Scheme, 2018, arXiv: 1803.05289

[2] Kuninets A. A. and Malygina E. S., “Calculation of error-correcting pairs for an algebraic-geometric code”, Prikladnaya Diskretnaya Matematika, 2024, no. 63, 65–90 (in Russian) | MR | Zbl

[3] Malygina E. S., Kuninets A. A., Ratochka V. L., et al., “Algebraic-geometry codes and decoding by error-correcting pairs”, Prikladnaya Diskretnaya Matematika, 2023, no. 62, 83–105 (in Russian) | MR

[4] Stichtenoth H., Algebraic Function Fields and Codes, Springer Verlag, 1991 | MR

[5] Stichtenoth H., “On automorphisms of geometric Goppa codes”, J. Algebra, 130:1 (1990), 113–121 | DOI | MR | Zbl

[6] Conrad K., The Minimal Polynomial and some Applications, , 2008 https://kconrad.math.uconn.edu/blurbs/linmultialg/minpolyandappns.pdf

[7] Clark P. L., Linear Algebra: Invariant Subspaces, 2013 http://alpha.math.uga.edu/p̃ete/invariant_subspaces.pdf

[8] Faugére J.-C., Otmani A., Perret L., et al., “Folding alternant and Goppa codes with non-trivial automorphism groups”, IEEE Trans. Inform. Theory, 62:1 (2016), 184–121 | DOI | MR