ALGORITHMS FOR REDUCING THE TIME REQUIRED TO PERFORM OPERATIONS OF THE DOMINGO-FERRER CRYPTOSYSTEM
Abstract
An analysis of the literature on the topic of fully homomorphic encryption is carried out. A brief description of the completely homomorphic Domingo-Ferrer cryptographic system based on the number factorization problem is presented, and the time characteristics of the stages of an attack with a known plaintext on this cryptosystem are given. The time characteristics of cryptosystem operations are analyzed, methods and means of their practical implementation are described. New algorithms for implementing the operations of the Domingo-Ferrer cryptosystem are proposed to reduce their execution time. The justification of estimates of the time costs of cryptosystem operations is formed on the basis of theoretical calculations, as well as the results of experimental studies. The aim of the study is to reduce the execution time of the Domingo-Ferrer cryptosystem by developing algorithms for their modification, taking into account the specifics of practical implementation. The main result of this work is a reduction in the execution time of the following operations of the Domingo-Ferrer cryptosystem: encryption by 10-15%, decryption by 2 times, homomorphic multiplication by 64 times for a chain of 200 multiplications using the degree of polynomials of the ciphertext representation d=100 and a slight increase in the time spent on key generation. The conducted research represents a significant contribution to the development of a fully homomorphic Domingo-Ferrer cryptosystem based on the integer factorization problem. This work has practical significance because it significantly improves the performance of homomorphic calculations of this cryptosystem. The results obtained can become the basis for the development of efficient (in terms of required computing costs and the level of security provided) cloud computing software and hardware systems using a fully homomorphic Domingo-Ferrer cryptosystem to ensure the confidentiality of processed information
References
1. Munjal K., Bhatia R. A systematic review of homomorphic encryption and its contributions in healthcare industry, Complex & Intelligent Systems, 2023, Vol. 9, No. 4, pp. 3759-3786. DOI: 10.1007/s40747-022-00756-z.
2. Yang W. et al. A review of homomorphic encryption for privacy-preserving biometrics, Sensors, 2023, Vol. 23, No. 7, pp. 3566. DOI: 10.3390/s23073566.
3. Sadeeq M.M. et al. IoT and Cloud computing issues, challenges and opportunities: A review, Qubahan Academic Journal, 2021, Vol. 1, No. 2, pp. 1-7. DOI: 10.48161/qaj.v1n2a36.
4. Marcolla C. et al. Survey on fully homomorphic encryption, theory, and applications, Proceedings of the IEEE, 2022, Vol. 110, No. 10, pp. 1572-1609. DOI: 10.36227/techrxiv.19315202.v5.
5. Rivest R.L. et al. On data banks and privacy homomorphisms, Foundations of secure computation, 1978, Vol. 4, No. 11, pp. 169-180.
6. Rivest R.L., Shamir A., Adleman L. A method for obtaining digital signatures and public-key cryptosys-tems, Communications of the ACM, 1978, Vol. 21, No. 2, pp. 120-126.
7. Gentry C. A fully homomorphic encryption scheme. – Stanford university, 2009.
8. Al Badawi A. et al. Openfhe: Open-source fully homomorphic encryption library, Proceedings of the 10th workshop on encrypted computing & applied homomorphic cryptography, 2022, pp. 53-63. DOI: 10.1145/3560827.3563379.
9. Xie Q. et al. Efficiency optimization techniques in privacy-preserving federated learning with homomor-phic encryption: A brief survey, IEEE Internet of Things Journal, 2024, Vol. 11, No. 14,
pp. 24569-24580. DOI: 10.1109/JIOT.2024.3382875.
10. Cheon J.H. et al. Homomorphic encryption for arithmetic of approximate numbers, International confer-ence on the theory and application of cryptology and information security. Cham: Springer International Publishing, 2017, pp. 409-437. DOI: 10.1007/978-3-319-70694-8_15.
11. Chillotti I., Gama N., Georgieva M., and Izabachène M. TFHE: Fast fully homomorphic encryption over the torus, J. Cryptol, 2019, Vol. 33, No. 1, pp. 1-58. DOI: 10.1007/s00145-019-09319-x.
12. El Mestari S.Z., Lenzini G., Demirci H. Preserving data privacy in machine learning systems, Comput-ers & Security, 2024, Vol. 137, pp. 103605. DOI: 10.1016/j.cose.2023.103605.
13. Doan T.V.T. et al. A survey on implementations of homomorphic encryption schemes: DT Van Thao et al., The Journal of Supercomputing, 2023, Vol. 79, No. 13, pp. 15098-15139. DOI: 10.1007/s11227-023-05233-z.
14. Domingo-Ferrer J. A provably secure additive and multiplicative privacy homomorphism, International Conference on Information Security. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002,
pp. 471-483. DOI: 10.1007/3-540-45811-5_37.
15. Wagner D. Cryptanalysis of an algebraic privacy homomorphism, Information Security: 6th Interna-tional Conference, ISC 2003, Bristol, UK, October 1-3, 2003. Proceedings 6. Springer Berlin Heidel-berg, 2003, pp. 234-239. DOI: 10.1007/10958513_18.
16. Babenko L.K., Starodubtsev V.S., El'chaninova N.B. Otsenka vremeni vypolneniya poiska sostavlyay-ushchikh klyucha v atake s izvestnym otkrytym tekstom na kriptosistemu Domingo-Ferrera [Estimation of the search time for key components in a known plaintext attack on the Domingo-Ferrer cryptosystem], Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2025, No. 3 (245), pp. 110-118. DOI: 10.18522/2311-3103-2025-3-110-118.
17. Babenko L.K., Starodubtsev V.S. Otsenka vremeni vypolneniya ataki s izvestnym otkrytym tekstom na kriptosistemu Domingo-Ferrera [Estimating the execution time of a known plaintext attack on the Do-mingo-Ferrer cryptosystem], Voprosy kiberbezopasnosti [Cybersecurity Issues], 2026, No. 1 (71), pp. 139-147. DOI: 10.21681/2311-3456-2026-1-139-147.
18. Babenko L.K., Starodubtsev V.S. Otsenka vremeni vypolneniya operatsiy shifrovaniya, rasshifrovaniya, gomomorfnykh vychisleniy s ispol'zovaniem kriptosistemy Domingo-Ferrera [Estimation of the execu-tion time of encryption, decryption, homomorphic calculations using the Domingo-Ferrer cryptosystem], Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2024, No. 5 (241), pp. 6-15. DOI: 10.18522/2311-3103-2024-5-6-15.
19. Alhassan E.A. et al. On some algebraic properties of the Chinese remainder theorem with applications to real life, Journal of Applied Mathematics and Computation, 2021, Vol. 5, No. 3. DOI: 10.26855/jamc.2021.09.008.
20. Babenko L.K., Starodubtsev V.S. Osobennosti realizatsii sistem kriptoanaliza gomomorfnykh shifrov, osnovannykh na zadache faktorizatsii chisel, na primere kriptosistemy MORE [Features of the imple-mentation of the cryptanalysis systems of homomorphic ciphers based on the problem of factorization of numbers, using the example of the cryptosystem MORE], Voprosy kiberbezopasnosti [Cybersecurity Is-sues], 2024, No. 3 (61), pp. 141-145. DOI: 10.21681/2311-3456-2024-3-141-145.
21. Rajaby E., Sayedi S.M. A structured review of sparse fast Fourier transform algorithms, Digital Signal Processing, 2022, Vol. 123, pp. 103403. DOI: 10.1016/j.dsp.2022.103403.
22. Pornin T. Efficient and complete formulas for binary curves, Cryptology ePrint Archive, 2022.
23. Gao X. et al. A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems, Numerische Mathematik, 2026, Vol. 158, No. 2, pp. 533-585. DOI: 10.48550/arXiv.2412.04595.
24. Mateer T. Fast Fourier algorithms with applications: Ph. D. Thesis. Clemson University, 2008.
