F.K. Aliev1, A.V. Korolkov2,  E.A. Matveev3

1 Department of Information Systems of the Russian Ministry of Defense (Moscow, Russia)

2 Russian Technological University - MIREA (Moscow, Russia)

3 Scientific and Technical Enterprise «CryptoSoft» (Penza, Russia)

Elements of the theory of singlet states of multiqubit quantum systems have been developed, among which the most significant are the following. We consider a new observable of a k‑qubit quantum system, defined in a paper called a generalization of the observable «spin component along a chosen axis» for k qubits, where k is an arbitrary natural number. All possible eigenvalues of the generalization of the observed «spin component along the chosen axis» for k qubits are calculated. The multiplicities of all eigenvalues of the generalization of the observed «spin component along the chosen axis» for k qubits are calculated. A basis of eigenstates of the generalization of the observed «spin component along the chosen axis» for k qubits is revealed and investigated. A representation of the singlet state of a 2k-qubit quantum system is established in the form of a basis expansion, which is a tensor product of bases from eigenstates of the observed k‑qubit quantum subsystems. Moreover, each of these observables is a generalization of the observed «spin component along the chosen axis» for k qubits. This expansion characterizes the quantum resource of singlet states of 2k-qubit quantum systems used to construct quantum cryptographic systems. The developed elements of the theory of singlet states of multiqubit quantum systems form the theoretical basis for constructing a new class of quantum cryptographic systems AKM2021. Theoretical provisions for the construction of all AKM(n)2021 quantum cryptographic systems contained in the AKM2021 class, where n is an arbitrary even natural number, are presented. This class of AKM2021 significantly increases the protective potential of cryptography in the era of the advent of quantum computers. The AKM2021 class also contains the previously known AKM2017 quantum cryptographic system in the form of the AKM(2)2021 quantum cryptographic system. At the same time, at least one undoubted advantage of any AKM(n)2021 quantum cryptographic system with n≥4 from the AKM2021 class can be noted in comparison with the AKM2017 quantum cryptographic system. This advantage lies in the fact that the AKM(n)2021 quantum cryptographic system can find applications by providing the ability to solve cryptographic problems that are impossible or difficult to solve using AKM2017. An example of such a cryptographic problem is the well-known secret sharing cryptographic problem.

Aliev F.K., Korolkov A.V., Matveev E.A. Class of quantum cryptographic systems AKM2021 based on the use of singlet states of multicubic quantum systems. Highly Available Systems / Sistemy vysokoy dostupnosti. 2022. V. 18. № 3. P. 5−22. DOI: https://doi.org/10.18127/j20729472-202203-01 (in Russian)

1. Graims R.A. Apokalipsis kriptografii. M.: DMK Press. 2020. 290 s. (in Russian)
2. Aliev F.K., Korolkov A.V., Matveev E.A., Orlov S.S., Sheremet I.A. Kvantovaya kriptograficheskaya sistema AKM2017 na osnove resursa neseparabelnosti sostoyaniya spinovoi singlet. Sistemy vysokoi dostupnosti. 2018. T. 14. № 4. S. 61−72. (in Russian)
3. Matveev E.A. Primenenie kvantovomekhanicheskikh effektov v sistemakh zashchity informatsii. Diss. kand. Penza: Nauchno-tekhnicheskoe predpriyatie KRIPTOSOFT. 2019. 157 s. (in Russian)
4. Nilsen M., Chang I. Kvantovye vychisleniya i kvantovaya informatsiya. M.: Mir. 2006. 824 s. (in Russian)
5. Sinitsyn I.N., Shalamov A.S. Lektsii po teorii sistem integrirovannoi logisticheskoi podderzhki. Izd. 2‑oe, pererab. i dop. M.: TORUS PRESS. 2019. 1072 s. (in Russian)
6. Sokolov I.A., Budzko V.I., Sinitsyn I.N. Postroenie informatsionno-telekommunikatsionnykh sistem vysokoi dostupnosti. Sistemy vysokoi dostupnosti. 2005. № 1. S. 6−14. (in Russian)
7. Aliev F.K., Korolkov A.V., Matveev E.A. Neseparabelnye sostoyaniya mnogokubitnykh kvantovykh sistem. Pod red. F.K. Alieva. M.: Radiotekhnika. 2017. 320 s. (in Russian)
8. Kilin S.Ya. i dr. Kvantovaya kriptografiya: idei i praktika. Minsk: Belorusskaya nauka. 2007. 391 s. (in Russian)
9. Kitaiskii sputnik Mo Tszy. [Elektronnyi resurs]. URL = yandex.ru. (in Russian)
10. Preskill Dzh. Kvantovaya informatsiya i kvantovye vychisleniya. T. 1. M. Izhevsk: NITs «Regulyarnaya i khaoticheskaya dinamika». 2008. 464 s. (in Russian)
11. Samartsev V.V. Korrelirovannye fotony i ikh primenenie. M.: FIZMATLIT. 2014. 168 s. (in Russian)
12. Khidari Dzh.D. Kvantovye vychisleniya. M.: DMK Press. 2021. 370 s. (in Russian)
13. Shnaer B. Prikladnaya kriptografiya. M.: TRIUMF. 2003. 816 s. (in Russian)
14. Kvantovyi kompyuter. [Elektronnyi resurs]. URL = ru.wikipedia.org. (in Russian)
15. Alferov A.P., Zubov A.Yu., Kuzmin A.S., Cheremushkin A.V. Osnovy kriptografii: Ucheb. posobie. M.: Gelios ARV. 2005. 480 s. (in Russian)
16. Babash A.V., Shankin G.P. Kriptografiya. M.: SOLON‑R. 2002. 512 s. (in Russian)
17. Los A.B., Nesterenko A.Yu., Rozhkov M.I. Kriptograficheskie metody zashchity informatsii. M.: Izdatelstvo Yurait. 2016. 473 s. (in Russian)
18. Fomichev V.M. Metody diskretnoi matematiki v kriptologii. M.: DIALOG-MIFI. 2010. 424 s. (in Russian)
19. Fomichev V.M., Melnikov D.A. Kriptograficheskie metody zashchity informatsii. V 2-kh chastyakh. Ch. 1. Matematicheskie aspekty. M.: Izdatelstvo Yurait. 2016. 200 s. Ch. 2. Sistemnye i prikladnye aspekty. M.: Izdatelstvo Yurait. 2016. 245 s. (in Russian)
20. Shennon K. Raboty po teorii informatsii i kibernetike. M.: IL. 1963. 869 s. (in Russian)
21. Zhizan N. Kvantovaya sluchainost: Per. s angl. M.: Alpina non-fikshn. 2016. 202 s. (in Russian)
22. Borovkov A.A. Teoriya veroyatnostei. M.: Nauka. 1976. 352 s. (in Russian)
23. Borovkov A.A. Matematicheskaya statistika. M.: Nauka. 1984. 472 s. (in Russian)
24. Koralov L.B., Sinai Ya.G. Teoriya veroyatnostei i sluchainye protsessy. M.: MTsNMO. 2013. 408 s. (in Russian)
25. Stib V.-Kh., Khardi I. Zadachi i ikh resheniya v kvantovykh vychisleniyakh i kvantovoi teorii informatsii. Moskva-Izhevsk: NITs «Regulyarnaya i khaoticheskaya dinamika». 2007. 296 s. (in Russian)
26. Aliev F.K., Korolkov A.V., Matveev E.A., Sheremet I.A. O svoistvakh obobshcheniya nablyudaemoi «komponenta spina vdol vybrannoi osi». Materialy 13 nauchno-tekhnich. konf. po kriptografii. Sektsiya «Problemy kvantovoi kriptografii». M.: 2021. S. 22−26. (in Russian)
27. Aliev F.K., Korolkov A.V., Matveev E.A. O klasse kvantovykh kriptograficheskikh sistem AKM2021. Materialy 13 nauchno-tekhnich. konf. po kriptografii. Sektsiya «Problemy kvantovoi kriptografii». M.: 2021. S. 35−41. (in Russian)
28. Aliev F.K., Korolkov A.V., Matveev E.A., Sheremet I.A. O mnogokubitovykh singletnykh sostoyaniyakh i ikh svoistvakh. Materialy 13 nauchno-tekhnich. konf. po kriptografii. Sektsiya «Problemy kvantovoi kriptografii». M.: 2021. S. 27−34. (in Russian)
29. Lankaster P. Teoriya matrits: Per. s angl. M.: Nauka. 1982. 272 s. (in Russian)
30. Chmora A.L. Sovremennaya prikladnaya kriptografiya. M.: Gelios ARV. 2001. 256 s. (in Russian)