Cryptosystem and Protocol for Transmition of Confidential Data without Any Preliminary Distribution of Secret and Public Keys, Based on the Use of a Commutative Encryption Procedure

Commutative encryption, previously proposed by A. Shamir and published about 30 years ago in a monograph by B. Schneier “Applied Cryptography”, has not found practical application due to the lack of known strong ciphers possessing the commutativity property. This paper confirms that such well-known ciphers as AES, GOST-2015, El-Gamal and Mc-Eliece ciphers, indeed, do not possess this property. However, the authors managed to construct a modification of the RSA cipher using a new version of the protocol, which allows the exchange of confidential information without any preliminary distribution of both public and secret encryption keys between legitimate users. This property is just the relevance of the current paper, because, as a rule, a keys distribution problem is a bottleneck of Cryptosystem creation for their application to real confidential digital telecommunication system. Although such properties are close to the properties of so-called public-key cryptosystems, unlike them, the proposed cryptosystem can use the same public keys for an unlimited number of users. Such keys can be made publicly available, for example, by storing them in the cloud. It is this property of the proposed Cryptosystem that reflects the novelty of the approach, since, as the authors know, no key system has yet been described that does not require preliminary key distribution.

This property can be useful for certain scenarios involving the exchange of confidential data, such as passwords and broadcast information. In the first case we have in mind that it is necessary to authenticate users by some server. If it has in data base user’s passwords stored, then users are authenticated only upon presentation of the corresponding passwords. However, communication channel used for such authentication is vulnerable to adversary’s interception, but our scheme prevents password’s disclosing. Another practical outcome of the proposed Cryptosystem consists in application to the broadband channels, if it is necessary to minimize the number of encryption keys used.

1. Schneier B. Applied Cryptography. Moscow: Triumph Publ.; 2002. (in Russ.)

2. Korzhik V.I., Yakovlev V.A., Starostin V.S., Buinevich M.V. Advance in Applied Cryptography Theory: Survey and Some New Results. Part 2. Keyless Cryptography. Proceedings of Telecommunication Universities. 2024;10(6):79–98. (in Russ.) DOI:10.31854/1813-32X-2024-10-6-79-98. EDN:HPBOWG

3. Korzhik V., Starostin V., Yakovlev V., Kabardov M., Krasov A., Adadurov S. Advance in Keyless Cryptography. In: Ramakrishnan S. (ed.) Lightweight Cryptographic Techniques and Cybersecurity Approaches. 2022. p.97–117. DOI:10.5772/intechopen.104429

4. Menezes A.J., van Oorschot P.C., Vanstone S.A. Handbook of Applied Cryptography. Boca Raton; 1997. DOI:10.1201/ 9780429466335

5. Korzhik V.I., Yakovlev V.A. Fundamentals of Cryptography. St. Petersburg: Intermedia Publ.; 2016. 216 p. (in Russ.) EDN:WEQWMN

6. Shor P.W. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing. 1997;26(5):1484–1509. DOI:10.1137/S0097539795293172

7. Korzhik V.I., Starostin V.S., Kabardov M.M., Gerasimovich A.M., Yakovlev V.A., Zhuvikin A.G. Information-theoretically secure key sharing protocol with constant noiseless public channels. Mathematical Aspects of Cryptography. 2021;12(3):125–141. DOI:https://doi.org/10.4213/mvk378

8. Mincciancio D., Regev O. Lattice-based Cryptography. In: Bernstein D.J., Buchmann J., Dahmen E. (eds) Post-Quantum Cryptography. Berlin, Heidelberg: Springer; 2009. p.147–191. DOI:10.1007/978-3-540-88702-7_5

9. Myasnikov A., Shpilrain V., Ushakov A. Non-Commutative Cryptography an Complexity of Groupe-Theoretical Problems. American Mathematical Society, 2011. 385 p. EDN:GPBUOR

10. Moldovyan A., Moldovyan D., Moldovyan A. Post-Quantum Public-Key Cryptoschemes On Finite Algebras. Informatics and Automation. 2024:3(4):1246–1276. (in Russ.) DOI:10.15622/ia.23.4.12

11. Duong M.T., Moldovyan A.A., Moldovyan D.N., Nguyen M.H., Do B.T. Structure of quaternion-type algebras and post-quantum structure algorithm. International Journal of Electrical and Computer Engineering. 2025;15(3):2965–2976. DOI:10.11591/ijece.v15i3.pp2965-2976