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Development of Post-quantum Cryptography and its Applications

Research keywords: · post-quantum cryptography · error-correction codes · cybersecurity
Horizon Europe keywords: · Increased cybersecurity · Next generation computing and data technologies

Institute of Mathematics Department of Algebra

Introduction of the Research Group

Post-quantum cryptographic algorithms can be basically divided into three groups: (1) Code-based algorithms following the classical McEliece scheme, whose security is based on the NP-complete problem of decoding linear error-correcting codes. (2) Lattice-based procedures, whose underlying problem is the determination of the shortest vector of finite dimensional lattices. (3) Isogeny-based procedures, which are based on pseudorandom walks on graphs of supersingular elliptic curves. The basic idea of the first two dates back 3-4 decades, the third is relatively young, only 10 years old. The Department of Algebra at BME has a long tradition of studying parameters and automorphisms of algebraic-geometric codes. Research has also been carried out on the use of sub-field sub-codes of AG codes in code-based post-quantum cryptography, with a special emphasis on sub-field sub-codes of Hermitian codes. It is clear that using this class of codes can significantly reduce the public key size, but further research is needed to investigate what parameter choices can guarantee that known Schur-filtration attack methods are not applicable to this class. A further challenge is to develop and implement efficient error correction methods, extending the methods used so far on high-performance computer algebra systems to embedded IoT devices.
Achievements Publications Journals Projects Conferences

Achievements

-We proved new formulas and estimates for the true dimension of subfield subcodes of Hermitian codes. We improved the bounds on the minimum distance of certain Hermitian codes and computed their automorphism groups. We gave a necessary condition for the parameters of a simple orthogonal array; this enables new bound for the minimum Hamming weight correlation-immune Boolean functions. We constructed a new identification system based on the explicit isomorphism problem.

Publications

[1] S. El Khalfaoui, G.P. Nagy, On the dimension of the subfield subcodes of 1-point Hermitian codes, Adv. Math. Commun. 15 (2021) 219–226. https://doi.org/10.3934/amc.2020054.
[2] R. Kiss, G.P. Nagy, On the nonexistence of certain orthogonal arrays of strength four, Prikl. Diskretn. Mat. (2021) 65–68. https://doi.org/10.17223/20710410/52/3.
[3] S. El Khalfaoui, G.P. Nagy, Estimating the dimension of the subfield subcodes of Hermitian codes, Acta Cybernet. 24 (2020) 625–641. https://doi.org/10.14232/actacyb.285453.
[4] G. Korchmáros, G.P. Nagy, M. Timpanella, Codes and gap sequences of Hermitian curves, IEEE Trans. Inform. Theory. 66 (2020) 3547–3554. https://doi.org/10.1109/tit.2019.2950207.
[5] S. Z. Kiss, P. Kutas, An identification system based on the explicit isomorphism problem. Manuscript, 2022.

Journals

Designs, Codes and Cryptography; IEEE Trans. Inform. Theory, Advances in Mathematics of Communications

Projects

Random Network Coding and Designs over GF(q) COST Action 2012-2015 substitute MC member
Graphs, groups, configurations, geometries NKFIH OTKA 2015-2018 16MFt
Algebras and algorithms NKFIH OTKA K 115288 2016-2021 8MFt
Security Enhancing Technologies for the Internet of Things NKFIH NKP 2018-1.2.1-NKP-2018-00004 2018-2022 300MFt

Conferences

G.P. Nagy (joint work with Sabira El Khalfaoui). Towards the security of McEliece’s cryptosystem based on Hermitian subfield subcodes. 21st Central European Conference on Cryptology. Debrecen, June 25, 2021
G.P. Nagy (joint work with Sabira El Khalfaoui). Towards the security of McEliece’s cryptosystem based on Hermitian subfield subcodes. 20th International Conference on Cryptology "Siberian Scienti#c School-seminar" named after G. P. Agibalov. Novosibirsk, September 6-11, 2021
G.P. Nagy. Old and new classes of abstract unitals, and their embeddings in finite projective planes. Changsha Combinatorics Seminar (China). September 1, 2021 (Talk_GPNagy_Changsha_seminar_2021.pdf)
R. Kiss (joint work with Gábor P. Nagy). Correlation-immune Boolean functions and parameters of orthogonal arrays. Developments in Computer Science, ELTE, Hungary. June 17-19, 2021
G. P. Nagy, Hermitian codes and restricted Hermitian codes: their parameters, automorphisms, implementations and applications in the McEliece Cryptosystem, ISACS’19: International Conference on Intelligent Systems and Advanced Computing Sciences, Taza (Morocco), Dec 26-27, 2019.