آمار
| تعداد نشریات | 5 |
| تعداد شماره‌ها | 111 |
| تعداد مقالات | 1,247 |
| تعداد مشاهده مقاله | 1,199,881 |
| تعداد دریافت فایل اصل مقاله | 1,060,652 |
A modified public key cryptography based on generalized Lucas matrices | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 29 دی 1402 اصل مقاله (429.14 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.28022.1419 | ||
| نویسندگان | ||
| Kalika Prasad1، 2؛ Munesh Kumari1، 2؛ Hrishikesh Mahato* 1 | ||
| 1Department of Mathematics, Central University of Jharkhand, India | ||
| 2Department of Mathematics, Government Engineering College, Bhojpur, Bihar, India | ||
| چکیده | ||
| In this paper, we propose a generalized Lucas matrix (a recursive matrix of higher order) obtained from the generalized Fibonacci sequences. We obtain their algebraic properties such as direct inverse calculation, recursive nature, etc. Then, we propose a modified public key cryptography using the generalized Lucas matrices as a key element that optimizes the keyspace construction complexity. Furthermore, we establish a key agreement for encryption-decryption with a combination of the terms of generalized Lucas sequences under the residue operation. | ||
| کلیدواژه‌ها | ||
| Affine-Hill cipher؛ Cryptography؛ Fibonacci sequence؛ Lucas sequence؛ Lucas matrix | ||
| مراجع | ||
|
[1] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245 (2014), 526–538. https://doi.org/10.1016/j.amc.2014.07.111
[2] G. Cerda-Morales, On generalized Fibonacci and Lucas numbers by matrix methods, Hacet. J. Math. Stat. 42 (2013), no. 2, 173–179.
[3] A. Demir, N. Omur, and Y.T. Ulutas, Optimization by $k$-Lucas numbers, Appl. Math. Comput. 197 (2008), no. 1, 366–371. https://doi.org/10.1016/j.amc.2007.07.045
[4] D.S. Dummit and R.M. Foote, Abstract Algebra, Wiley Hoboken, 2004.
[5] T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inf. Theory 31 (1985), no. 4, 469–472. https://doi.org/10.1109/TIT.1985.1057074
[6] I. Gupta, J. Singh, and R. Chaudhary, Cryptanalysis of an extension of the hill cipher, Cryptologia 31 (2007), no. 3, 246–253. https://doi.org/10.1080/01611190701202465
[7] S. Halici and Ö. Deveci, On Fibonacci quaternion matrix, Notes Number Theory Discrete Math. 27 (2021), no. 4, 236–244. https://doi.org/10.7546/nntdm.2021.27.4.236-244
[8] H.E. Tianxiao, H.C. Jeff and J.S. Peter, Matrix representation of recursive sequences of order 3 and its applications, J. Math. Res. Appl. 38 (2018), no. 3, 221–235. https://doi.org/10.3770/j.issn:2095-2651.2018.03.001
[9] C.H. King, Some further properties of the Fibonacci numbers, Master’s thesis, San Jose State, San Jose, CA, 1960.
[10] T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 2, John Wiley & Sons, 2019.
[11] M. Kumari, K. Prasad, B. Kuloğlu, and E. Özkan, The $k$-Fibonacci group and periods of the $k$-step Fibonacci sequences, WSEAS Trans. Math. 21 (2022), 838–843. http://doi.org/10.37394/23206.2022.21.95
[12] M. Kumari, K. Prasad, and J. Tanti, A note on linear codes with generalized Fibonacci matrices, Jñānābha 52 (2022), no. 2, 77–81. https://doi.org/10.58250/jnanabha.2022.52209
[13] M. Kumari and J. Tanti, On the role of the Fibonacci matrix as key in modified ECC, arXiv preprint arXiv:2112.11013 (2021).
[14] M. Kumari and J. Tanti, Cryptography using multinacci block matrices, International Journal of Nonlinear Analysis and Applications 14 (2023), no. 10, 57–65. https://doi.org/10.22075/ijnaa.2023.29918.4295
[15] E.P. Miles, Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly 67 (1960), no. 8, 745–752. https://doi.org/10.1080/00029890.1960.11989593
[16] E. Özkan and ˙I. Altun, Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials, Commun. Algebra 47 (2019), no. 10, 4020–4030. https://doi.org/10.1080/00927872.2019.1576186
[17] K. Prasad and H. Mahato, Cryptography using generalized Fibonacci matrices with Affine-Hill cipher, J. Discrete Math. Sci. Cryptogr. 25 (2022), no. 8, 2341–2352. https://doi.org/10.1080/09720529.2020.1838744
[18] K. Prasad and H. Mahato, On some new identities of Lucas numbers and generalization of Fibonacci trace sequences, Palest. J. Math. 12 (2023), no. 2, 329–340.
[19] K. Prasad, H. Mahato, and M. Kumari, Some properties of $r$-circulant matrices with $k$-balancing and $k$-Lucas balancing numbers, Bol. Soc. Mat. Mex. 29 (2023), no. 2, Artical ID: 44. https://doi.org/10.1007/s40590-023-00510-6
[20] W. Stallings, Cryptography and network security - principles and practice, 7th edition, Pearson Education India, 2017.
[21] P. Stanimirović, J. Nikolov, and I. Stanimirović, A generalization of Fibonacci and Lucas matrices, Discrete Appl. Math. 156 (2008), no. 14, 2606–2619. https://doi.org/10.1016/j.dam.2007.09.028
[22] D.R. Stinson, Cryptography: Theory and Practice, Chapman and Hall/CRC., New York, 2005.
[23] P. Sundarayya and G.V. Prasad, A public key cryptosystem using affine Hill cipher under modulation of prime number, J. Inf. Optim. Sci. 40 (2019), no. 4, 919–930. https://doi.org/10.1080/02522667.2018.1470751
[24] D. Tasci and E. Kilic, On the order-$k$ generalized Lucas numbers, Appl. Math. Comput. 155 (2004), no. 3, 637–641. https://doi.org/10.1016/S0096-3003(03)00804-X
[25] B. Thilaka and K. Rajalakshmi, An extension of Hill cipher using generalised inverses and mth residue modulo n, Cryptologia 29 (2005), no. 4, 367–376. https://doi.org/10.1080/0161-110591893933 | ||
|
آمار تعداد مشاهده مقاله: 295 تعداد دریافت فایل اصل مقاله: 877 |
||