Rekonstruksi Struktur Modul $E_p^((m))$ dan Aplikasinya dalam Kriptanalisis Protokol DHDP
Abstract
The module structure over local rings, particularly the module , offers a novel approach to the analysis of matrix-based cryptographic systems. In the context of the security of the Diffie–Hellman Decomposition Problem (DHDP) protocol, the utilization of an isomorphism between and its submodule enables the reduction of attacks on the protocol to systems of ordinary linear equations. This article systematically discusses the concepts of , , and two key propositions regarding their structure and representation, and then applies these to construct a deterministic attack on DHDP. The findings indicate that this approach not only simplifies the cryptanalysis process but also opens new pathways in the design of cryptographic protocols based on module structures.
Keywords: Cryptanalysis, DHDP, E_p^((m)), F_p^((m)), Module structure
Abstrak
Struktur modul atas ring lokal , khususnya modul , memberikan pendekatan baru dalam analisis sistem kriptografi berbasis matriks. Dalam konteks keamanan protokol Diffie–Hellman Decomposition Problem (DHDP), pemanfaatan sifat Isomorfisma antara dan submodul memungkinkan reduksi serangan terhadap protokol menjadi sistem persamaan linear biasa. Artikel ini membahas secara sistematik konsep , , serta dua proposisi penting mengenai struktur dan representasi , lalu menerapkannya dalam membangun serangan deterministik terhadap DHDP. Hasil menunjukkan bahwa pendekatan ini tidak hanya menyederhanakan proses kriptanalisis, tetapi juga membuka jalur baru dalam desain protokol kriptografi berbasis struktur modul.
Keywords
Full Text:
PDFReferences
Ahn, J., Hussain, R., Kang, K., & Son, J. (2025). Exploring encryption algorithms and network protocols: a comprehensive survey of threats and vulnerabilities. IEEE Communications Surveys & Tutorials, 27(5), 2765-2792.
Atani, R. E., Atani, S. E., & Mirzakuchaki, S. (2009). A novel public key crypto system based on semi-modules over quotient semi-rings. Journal of Applied Mathematics and Computing, 30(2), 237-248.
Dubois, V., & Kammerer, J. (2011). Cryptanalysis of cryptosystems based on non-commutative skew polynomials. Journal of Symbolic Computation, 46(3), 355-371.
Durcheva, M. (2025). Cryptography based on (idempotent) semirings: abandoning tropicality?. Encyclopedia, 5(1), 1-20.
Ghadafi, E. (2017). How low can you go? short structure-preserving signatures for Diffie-Hellman vectors. Proceedings of the IMA International Conference on Cryptography and Coding 2017. Springer, 1-19.
Grover, C., Mendelsohn, A., Ling, C., et al. (2022). Non-commutative ring learning with errors from cyclic algebras. Journal of Cryptology, 35(3), 1-67.
Hanoymak, T., & Küsmüş, Ö. (2019). A new multi-party key exchange protocol and symmetric key encryption scheme over non-commutative group rings. Journal of Advances in Mathematics, 8(1), 11–16.
Hatri, Y., Otmani, A., & Guenda, K. (2018). Cryptanalysis of an identity-based authenticated key exchange protocol. Designs, Codes and Cryptography, 86(11), 2445-2458.
Kamal, A. A., & Youssef, A. M. (2012). Cryptanalysis of a key exchange protocol based on the endomorphisms ring End (Z_p×Z_(p^2 )). Applicable Algebra in Engineering, Communication and Computing, 23(3), 143–149.
Khamalwa, M. S. (2024). Exploring how commutative algebra underpins cryptographic protocols and encryption methods used in secure communications and data protection. Newport International Journal of Scientific and Experimental Sciences, 5(3), 58–62.
Khathuria, K., Micheli, G., & Weger, V. (2021). On the algebraic structure of E_p^((m)) and applications to cryptography. Applicable Algebra in Engineering, Communication and Computing, 32(4), 495–505.
Kumari, M., & Tanti, J. (2022). A public key cryptography using multinacci block matrices. Sādhanā, 47(1), 1-13.
Maze, G., Monico, C., Rosenthal, J., & Climent, J. J. (2004). Public key cryptography based on simple modules over simple rings. Designs, Codes and Cryptography, 32(3), 213-225.
Selikh, B., Chillali, A., Mihoubi, D., & Ghadbane, N. (2024). A new public key cryptosystem based on the non-commutative ring R. Journal of Discrete Mathematical Sciences & Cryptography, 27(1), 75–93.
Takagi, T. (2018). Recent developments in post-quantum cryptography, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E101A(1), 3–11.
Ustimenko, V. A. (2018). On new symbolic key exchange protocols and cryptosystems based on a hidden tame homomorphism. Dopovidi Nacional'noi Akademii Nauk Ukraini, (10), 26–36.
Ustimenko, V., & Pustovit, O. (2021). New cryptosystems of noncommutative cryptography based on eulerian semigroups of multivariate transformations. CEUR Workshop Proceedings, 2923, 18–26.
Ustimenko, V. A., & Woldar, A. N. (2019). Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem. Bulletin of the South Ural State University. Series: Mathematics, Mechanics, Computer Science, 12(1), 66–75.
DOI: https://doi.org/10.17509/jem.v13i2.87588
Refbacks
- There are currently no refbacks.
Copyright (c) 2025 Mathematics Study Program, Universitas Pendidikan Indonesia
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.