A New Application to Cryptology Using Generalized Fibonacci Matrices, Inner Product and Self Adjoint Operator

Abstract

In this paper, we develop a public key cryptosystem using key exchange based on the relationship between inner product and orthogonality. Then we created encoding and decoding algorithms using this key exchange, self adjoint operator, and generalized m-step Fibonacci sequence.

Keywords

• Fibonacci matrix
• Inner product
• Orthogonality
• Self adjoint
• Key exchange
• Coding and decoding algorithm

References

1. Basu, M., Prasad, M., Coding theory on the m–extension of the Fibonacci p–numbers, Chaos, Solitons and Fractals, 42(2009), 2522–2530.
2. Basu, M.,Prasad, M., The generalized relations among the code elements for Fibonacci coding theory, Chaos, Solitons and Fractals, 41(2009), 2517–2525.
3. Basu, M., Das, M., Tribonacci matrices and a new coding theory, Discrete Mathematics, Algorithm and Applications, 6(1)(2014).
4. Basu, M., Das, M., Coding theory on Fibonacci n-step numbers, Discrete Mathematics Algorithms and Applications, 6(2)(2014).
5. Basu, M., Das, M., Coding theory on constant coefficient Fibonacci n-step numbers, communicated.
6. Baumslag, G., Fine, B., Kreuzer, M., Rosenberger, G., A Course in Mathematical Cryptography, Walter de Gruyter, Berlin, 2015.
7. Buchmann, J., Introduction to Cryptography, Springer, 2004.
8. Elgamal, T., A., Public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Transactions on Information Theory, 31(4)(1985), 469–472.

9. Esmaili, M.,Esmaeili, M., A Fibonacci–polynomial based coding method with error detection and correction, Computers and Mathematics with Applications, 60(2010), 2738–2752.
10. Esmaili, M., Moosavi, M., Gulliver, T.A., A new class of Fibonacci sequence based error correcting codes, Cryptography and Communications, 9(2017), 379–396.
11. Gupta, I., Singh, J., Chaudhary, R., Cryptanalysis of an extension of the hill cipher, Cryptologia, 31(3)(2007), 246–253.
12. Hankerson, D., Menezes, A., Vanstone, S., Guide to Elliptic Curve Cryptography, Springer-Verlag, New York, 2004.
13. Hirata-Kohno, N., Petho, A., A Key Exchange protocol based on Diophantine equation and S-integers, JSIAM Letters, 6(0)(2014), 85–88.
14. İrge, V., Soykan, Y., A new application to cryptology using generalized Jacobsthal matrices, inner product and self-adjoint operator, Discrete Mathematics, Algorithms and Applications, 16(08)(2024), 2350113.
15. İrge, V., Soykan, Y., A new coding algorithm using Fibonacci numbers, linear mappings, and change of basis, Discrete Mathematics, Algorithms and Applications, (2024), 2450081.
16. Kameswari, P.A., Kumar, L.P., LU-A method for recovering a key in the key exchange cryptosystem by Diophantine equations, Third Edition, Addison-Wesley, 1998.
17. Kameswari, P.A., Sriniasarao, S.S., Belay, A., An applications of linear diophantine equations to cryptography, Advances in Mathematics: Scientific Journal, 10(6)(2021), 2799–2806.
18. Koblitz, N., A Course in Number Theory and Cryptography, Springer Verlag, Berlin, 1984.
19. Koblitz, N., Elliptic curve cryptosystems, Math. Comp., 48(177)(1987), 203–209.
20. Ling, S., Wang, H., Xing, C., Algebraic Curves in Cryptography, Taylor & Francis Group, 2013.
21. Mao, W., Modern Cryptography: Theory and Practice. Pearson Education India, 2003.
22. Myasnikov, A.G., Shpilrain, V., Ushakov, A., Group-Based Cryptography, Adv. Courses in Math. – CRM Barcelona, Birkh¨auser, Basel, 2008.
23. Özgür, N., Uçar, S., Taş, N., Kaymak, Ö.Ö., A new encoding/decoding algorithm via Fibonacci–Lucas tree diagrams, Discrete Mathematics, Algorithms & Applications, 17(4)(2025).
24. Prasad, K., Mahato, H., Cryptography using generalized Fibonacci matrices with Affine-Hill cipher, Journal of Discrete Mathematical Sciences and Cryptography.
25. Prasad, B., Coding theory on (h(x), g(y))–extension of Fibonacci p–numbers polynomials, Universal Journal of Computational Mathematics, 2(2014), 6–10.
26. Prasad, B., High rates of Fibonacci polynomials coding theory, Discrete Mathematics, Algorithms and Applications, 6(2014).
27. Prasad, B., Coding theory on Lucas p numbers, Discrete Mathematics, Algorithms and Applications, 8(2016).
28. Rynne, B.P., Youngson, M.A., Linear Functional Analysis, Springer-Verlag, 2000.
29. Stakhov, A.P., Fibonacci matrices, a generalization of the Cassini formula, and a new coding theory, Chaos, Solitons and Fractals, 30(2006), 56–66.
30. Stakhov, A., Massingue, V., Sluchenkov, A., Introduction into Fibonacci Coding and Cryptography, Osnova, Kharkov, 1999.
31. Stallings, W., Cryptography and Network Security: Principles and Practice, 7th Ed. Pearson Education Limited, 2017.
32. Stinson, D.R., Cryptography: Theory and Practice, 3rd Ed. Chapman and Hall/CRC, Taylor & Francis Group, 2006.
33. Sundarayya, P., 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.
34. Taş, N., Uçar, S., Özgür, N.Y., Kaymak, Ö ., A new coding/decoding algorithm using Fibonacci numbers, Discrete Math. Algorithms Appl., 10(02)(2018), 1850028.
35. Thilaka, B., Rajalakshmi, K., An extension of hill cipher using generalized inverses and mth residue modulo n, Cryptologia, 29(4)(2005), 367–376.
36. Uçar, S., Taş, N., Özgür, N.Y., A new application to coding theory via Fibonacci and Lucas numbers, Mathematical Sciences and Applications E-notes, 7(1)(2019), 62–70.
37. Viswanath, M., Kumar, M.R., A public key cryptosystem using hill’s cipher, Journal of Discrete Mathematical Sciences and Cryptography, 18(1-2)(2015), 129–138.
38. Yosh, H., The key exchange cryptosystem used with higher order Diophantine equations, International Journal of Network Security & Its Applications, 3(2)(2011), 43–50.