Cryptography using multinacci block matrices

Document Type : Research Paper


1 Department of Mathematics, Central University of Jharkhand, Ranchi, India

2 Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India


In this paper, we propose public key cryptography using recursive block matrices involving generalized Fibonacci numbers over a finite field $\mathbb{Z}_{p}$. For this, we define multinacci block matrices, a kind of upper triangular matrix involving multinacci matrices at diagonal places and give some of its algebraic properties. Moreover, we set up a method for key element agreement at end users, which makes cryptography more efficient. The proposed cryptography comes with a large key space and its security relies on the Discrete Logarithm Problem (DLP).


[1] R. Alvarez, F.-M. Martinez, J.-F. Vicent, and A. Zamora, A new public key cryptosystem based on matrices, 6th WSEAS Int. Conf. Inf. Secur. Privacy Tenerife, Spain, December 14-16, 2007, pp. 36–39.
[2] W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inf. Theory 22 (1976), no. 6, 644–654.
[3] O. Diskaya, E. Avaroglu, and H. Menken, The classical AES-like cryptology via the Fibonacci polynomial matrix, Turk. J. Eng. 4 (2020), no. 3, 123–128.
[4] J. Hoffstein, J. Pipher, J.H. Silverman, and J.H. Silverman, An introduction to mathematical cryptography, vol. 1, Springer, 2008.
[5] J. Kannan, M. Somanath, M. Mahalakshmi, and K. Raja, Encryption decryption algorithm using solutions of Pell equation, Int. J. Math. Appl. 10 (2022), no. 1, 1–8.
[6] M. Kumari, K. Prasad, and J. Tanti, A note on linear codes with generalized Fibonacci matrices, Jnanabha 52 (2022), no. 2, 77–81.
[7] P. Kuppuswamy and S.Q.Y. Al-Khalidi, Hybrid encryption/decryption technique using new public key and symmetric key algorithm, MIS Review: Int. J. 19 (2014), no. 2, 1–13.
[8] M. Mohan, M.K. Kavithadevi, and V.J. Prakash, Improved classical cipher for healthcare applications, Procedia Comput. Sci. 93 (2016), 742–750.
[9] 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.
[10] K. Prasad, H. Mahato, and M. Kumari, A novel public key cryptography based on generalized Lucas matrices, arXiv preprint arXiv:2202.08156 (2022).
[11] R.L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM 21 (1978), no. 2, 120–126.
[12] J. Shtayat and A. Al-Kateeb, The Perrin r-matrix and more properties with an application, J. Discrete Math. Sci. Cryptogr. 25 (2022), no. 1, 41–52.
[13] Y. Soykan, E. Ta,sdemir, and ˙I. Vedat, On matrix sequence of modified Tribonacci-Lucas numbers, MANAS J. Eng. 10 (2022), no. 2, 211–221. [14] D.R. Stinson, Cryptography: Theory and Practice, Chapman and Hall/CRC, 2005.
[15] 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.
[16] M.K. Viswanath and M.R. Kumar, A public key cryptosystem using Hiil’s cipher, J. Discrete Math. Sci. Cryptogr. 18 (2015), no. 1-2, 129–138.
[17] M. Zeriouh, A. Chillali, and A. Boua, Cryptography based on the matrices, Bol. Soc. Parana. Mat. 37 (2019), no. 3, 75–83.
Volume 14, Issue 10
October 2023
Pages 57-65
  • Receive Date: 14 February 2023
  • Revise Date: 13 March 2023
  • Accept Date: 11 April 2023