In this paper, we define the general form of the Moore-Penrose inverse for the matrix whose elements are Fibonacci numbers. We examine the states of the matrix  where F is a rectangular Fibonacci matrix based on the values of m and n. In the second part of this study, we introduce a novel coding theory using the Moore-Penrose inverse of the rectangular Fibonacci matrix and provide illustrative examples. The rectangular Fibonacci matrix plays a crucial role in the construction of the coding algorithm. This coding method is referred to as the “coding theory on rectangular Fibonacci matrix.”

Fibonacci matrix, the Moore-Penrose generalized inverse, pseudo-inverse, encryption, cryptology.

Received: August 12, 2023;  Accepted: October 19, 2023; Published: November 9, 2023

How to cite this article: Süleyman Aydınyüz and Mustafa Aşcı, The Moore-Penrose inverse of the rectangular Fibonacci matrix and applications to the cryptology, Advances and Applications in Discrete Mathematics 40(2) (2023), 195-211. http://dx.doi.org/10.17654/0974165823066

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References
[1] M. Asci and S. Aydinyuz, k-order Gaussian Fibonacci polynomials and applications to the coding/decoding theory, Journal of Discrete Mathematical Sciences and Cryptography 25(5) (2020), 1399-1416.
[2] M. Basu and B. Prasad, The generalized relations among the code elements for Fibonacci coding theory, Chaos, Solitons and Fractals 41(5) (2009), 2517-2525.
[3] M. Basu and M. Das, Tribonacci matrices and a new coding theory, Discrete Math. Algorithms Appl. 6(1) (2014), article ID: 1450008.
[4] M. Basu and M. Das, Coding theory on Fibonacci n-step numbers, Discrete Math. Algorithms Appl. 6(2) (2014), article ID: 145008.
[5] E. C. Boman, The Moore-Penrose pseudoinverse of an arbitrary, square k circulant matrix, Linear Multilinear Algebra 50(2) (2002), 175-179.
[6] S. L. Campbell and C. D. Meyer Jr., Generalized Inverses of Linear Transformations, London, Pitman Pub. Ltd., 1979.
[7] J. A. Fill and D. E. Fishkind, The Moore-Penrose generalized inverse for sums of matrices, SIAM J. Matrix Anal. Appl. 21(2) (2000), 629-635.
[8] H. W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart. 19(3) (1981), 250-257.
[9] D. H. Griffel, Linear Algebra and its Applications, Vols. 1 and 2, New York, Wiley, 1989.
[10] V. E. Hoggat, Fibonacci and Lucas Numbers, Houghton-Mifflin, Palo Alto, 1969.
[11] B. A. Israel, The Moore of the Moore-Penrose inverse, The Electronic Journal of Linear Algebra 9 (2002), 150-157.
[12] A. M. Kanan and A. Z. Zayd, Using the Moore-Penrose generalized inverse in cryptography, World Scientific News 148 (2020), 1-14.
[13] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, 2001.
[14] G. Y. Lee, J. S. Kim and S. G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart. 40(3) (2002), 203-211.
[15] P. W. Lewis, Matrix Theory, Singapore: World Scientific Press, 1991.
[16] P. Patrício, The Moore-Penrose inverse of a factorization, Linear Algebra Appl. 370 (2003), 227-235.
[17] K. Schmidt and G. Trenkler, The Moore-Penrose inverse of a semi-magic square is semi-magic, Int. J. Math. Educ. Sci. Technol. 32(4) (2001), 624-629.
[18] A. P. Stakhov, A generalization of the Fibonacci Q-matrix, Rep. Natl. Acad. Sci., Ukraine 9 (1999), 46-49.
[19] A. P. Stakhov, Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos, Solitons and Fractals 30(1) (2006), 56-66.
[20] S. Vajda, Fibonacci and Lucas Numbers and the Golden Section Theory and Applications, Ellis Harwood Limited, 1989.