Cryptography utilizing the affine-Hill cipher and extended generalized Fibonacci matrices. (English) Zbl 07883997
Summary: We are aware that a major cryptosystem element plays a crucial part in maintaining the security and robustness of cryptography. Various researchers are focusing on creating new forms of cryptography and improving those that already exist using the principles of number theory and linear algebra. In this article, we have proposed an extended generalized Fibonacci matrix (recursive matrix of higher order) having a relation with extended generalized Fibonacci sequences and established some properties in addition to that usual matrix algebra. Further, we proposed a modified public key cryptography using these matrices as keys in affine-Hill cipher and key agreement for encryption-decryption with the combination of terms of extended generalized Fibonacci sequences under prime modulo. This system has a large key space and reduces the time complexity as well as space complexity of the key transmission by only requiring the exchange of pair of numbers (parameters) as opposed to the entire key matrix.
MSC:
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 11B37 | Recurrences |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11C20 | Matrices, determinants in number theory |
| 94A60 | Cryptography |
| 14G50 | Applications to coding theory and cryptography of arithmetic geometry |
| 68P30 | Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science) |
Keywords:
affine Hill cipher; cryptography; Fibonacci sequence and matrix; extended generalized Fibonacci sequence and matrixReferences:
| [1] | Asci,, M., Aydinyuz S., k-Order Fibonacci Polynomials on AES-Like Cryptology. CMES-Computer Modeling in Engineering & Sciences, 131(1),(2022), 277-293. |
| [2] | DUMMIT, D. S., AND FOOTE, R. M.,Abstract algebra, vol. 3. Wiley Hoboken, 2004. · Zbl 1037.00003 |
| [3] | ElGamal, T., A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on information theory 31, 4 (1985), 469-472. · Zbl 0571.94014 |
| [4] | GOULD, H. W., A history of the Fibonacci q-matrix and a higher-dimensional problem. Fibonacci Quart 19, 3 (1981), 250-257. · Zbl 0467.10009 |
| [5] | GUPTA, I., SINGH, J., AND CHAUDHARY, R., Cryptanalysis of an extension of the hill cipher. Cryptologia 31, 3 (2007), 246-253. · Zbl 1134.94005 |
| [6] | Kalman, D., AND Mena, R., The Fibonacci Number-Exposed. The Mathematical Magazine, 2, (2002). |
| [7] | KOSHY, T., Fibonacci and Lucas numbers with applications. John Wiley & Sons, 2019. · Zbl 1409.11001 |
| [8] | KUMARI, M., AND TANTI, J., A model of public key cryptography using multinacci matrices. arXiv preprint arXiv:2003.08634 (2020). |
| [9] | KUMARI, M., AND TANTI, J., On the role of the Fibonacci matrix as key in modified ecc. arXiv preprint arXiv:2112.11013 (2021). |
| [10] | MAO, W., Modern cryptography: theory and practice. Pearson Education India,2003. |
| [11] | PAAR, C., AND PELZL, J., Understanding cryptography: a textbook for students and practitioners. Springer Science & Business Media, 2009. |
| [12] | PRASAD, K., AND MAHATO, H., Cryptography using generalized Fibonacci matrices with an affine-hill cipher. Journal of Discrete Mathematical Sciences and Cryptography (2021), 1-12. |
| [13] | PRASAD, K., MAHATO, H. AND KUMARI, M., A novel public key cryptography based on generalized Lucas matrices.arXiv preprint arXiv:2202.08156v1 (2022). |
| [14] | STALLINGS, W., Cryptography and network security: principles and practice, 7th Ed. Pearson Education Limited, 2017. |
| [15] | STINSON, D. R., Cryptography: theory and practice, 3rd Ed. Chapman and Hall/CRC, Taylor & Francis Group, 2006. · Zbl 1085.94001 |
| [16] | STOTHERS, A. J., On the complexity of matrix multiplication. |
| [17] | SUNDARAYYA, P. AND VARA PRASAD, G., A public key cryptosystem using affine hill cipher under modulation of prime number. Journal of Information and Optimization Sciences 40, 4 (2019), 919-930. |
| [18] | Suleyman Aydinyuz, Mustafa Asci, Error detection and correction for coding theory on k-order Gaussian Fibonacci matrices. Mathematical Biosciences and Engineering, 2023, 20(2), 1993-2010. doi: 10.3934/mbe.2023092. · doi:10.3934/mbe.2023092 |
| [19] | TASCI, D., AND KILIC, E., On the order-k generalized lucas numbers. Applied mathematics and computation 155, 3 (2004), 637-641. · Zbl 1115.11012 |
| [20] | THILAKA, B., AND RAJALAKSHMI, K., An extension of hill cipher using generalized inverses and mth residue modulo n. Cryptologia 29, 4 (2005), 367-376. · Zbl 1325.94142 |
| [21] | VISWANATH, M., AND KUMAR, M. R., A public key cryptosystem using Hill’s cipher. Journal of Discrete Mathematical Sciences and Cryptography 18, 1-2 (2015), 129-138. · Zbl 1495.94073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
