SciELO - Scientific Electronic Library Online

vol.39 issue1Weak convergence and weak compactness in the space of integrable functions with respect to a vector measureSome refinements to Hölder’s inequality and applications author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.39 no.1 Antofagasta Feb. 2020 


A cryptography method based on hyperbolicbalancing and Lucas-balancing functions

1 Sambalpur University, School of Mathematical Sciences, Sambalpur, OR, India. e-mail:


The goal is to study a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring balancing and Lucas-balancing numbers. These functions are indeed the extension of Binet formulas for both balancing and Lucas-balancing numbers in continuous domain. Some identities concerning hyperbolic balancing and Lucas-balancing functions are also established. Further, a new class of square matrices, a generalization of balancing QB-matrices for continuous domain, is considered. These matrices indeed enable us to develop a cryptography method for secrecy purpose.

Keywords: Balancing numbers; Lucas-balancing numbers; Hyperbolic balancing functions; Hyperbolic Lucas-balancing functions; Cryptography

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

[1] A. Behera and G. K. Panda, “On the square roots of triangular numbers”, The Fibonacci quarterly, vol. 37, no. 2, pp. 98-105, 1999. [On line]. Available: ]

[2] A. Bérczes, K. Liptai, I. Pink, “On generalized balancing numbers”, The Fibonacci quarterly , vol. 48, no. 2, pp. 121-128, May 2010. [On line]. Available: ]

[3] R. Keskin and O. Karaath, “Some new properties of balancing numbers and square triangular numbers”, Journal of integer sequences, vol. 15, no. 2, 2012. [On line]. Available: ]

[4] K. Liptai , “Fibonacci balancing numbers”, The Fibonacci quarterly , vol. 42, no. 4. pp. 330-340, Nov. 2004, [On line]. Available: ]

[5] K. Liptai , “Lucas balancing numbers”, Acta mathematica universitatis ostraviensis, vol. 14, no. 1, pp. 43-47, 2006. [On line]. Available: ]

[6] K. Liptai , F. Luca, A. Pintér, and L. Szalay, “Generalized balancing numbers”, Indagationes mathematicae, vol. 20, no. 1, pp. 87-100, Mar. 2009, doi: 10.1016/S0019-3577(09)80005-0. [ Links ]

[7] P. Olajos, “Properties of balancing, cobalancing and generalized balancing numbers”, Annales mathematicae et informaticae, vol. 37, pp. 125-138, 2010. [On line]. Available: ]

[8] G. K. Panda and P. K. Ray, “Some links of balancing and cobalancing numbers with Pell and associated Pell numbers”, Bulletin of the institute of mathematics, Academia sinica (New Series), vol. 6, no. 1, pp. 41-72, 2011. [On line]. Available: ]

[9] B. K. Patel, P.K. Ray, and M. Sahukar, “Positive integers solutions of certain Diophantine equations”, Proceeding: mathematical sciences, vol. 128, no. 1, Art. ID. 5, Feb. 2018, doi: 10.1007/s12044-018-0377-4. [ Links ]

[10] B. K. Patel , N. Irmak, and P. K. Ray, “Incomplete balancing and Lucas-balancing numbers”, Mathematical reports, vol. 20, no. 1, pp. 59- 72, 2018. [On line]. Available: ]

[11] P. K. Ray, “Balancing polynomials and their derivatives”, Ukrainian mathematical journal, vol. 69, no. 4, pp. 646-663, Sep. 2017, doi: 10.1007/s11253-017-1386-7. [ Links ]

[12] P. K. Ray, “On the properties of k-balancing and k-Lucas balancing numbers”, Acta et commentationes universitatis tartuensis de mathematica, vol. 21, no. 2, pp. 259-274, Dec. 2017, doi: 10.12697/ACUTM.2017.21.18. [ Links ]

[13] B. K. Patel and P. K. Ray, “The period, rank and order of the sequence of balancing numbers modulo m”, Mathematical reports , vol. 18, no. 3, pp. 395-401, 2016. [On line]. Available: ]

[14] P. K. Ray and B. K. Patel, “Uniform distribution of balancing numbers modulo m”, Uniform distribution theory, vol. 11, no. 1, pp. 15-21, 2016, doi: 10.1515/udt-2016-0002. [ Links ]

[15] P. K. Ray, “Certain diophantine equations involving balancing and Lucas-balancing numbers”, Acta et commentationes universitatis tartuensis de mathematica , vol. 20, no. 2, pp. 165-173, Dec. 2016, doi: 10.12697/ACUTM.2016.20.14. [ Links ]

[16] A. P. Stakhov and B. Rozin, “On a new class of hyperbolic function”, Chaos, solitons & fractals, vol. 23, no. 2, pp. 379-389, Jan. 2005, doi: 10.1016/j.chaos.2004.04.022. [ Links ]

[17] A. P. Stakhov, “Gazale formulas, a new class of hyperbolic Fibonacci and Lucas functions and the improved method of the ‘golden’ cryptography”, Akademiâ trinitarizma, no. 77-6567, Art, ID. 14098, Dec. 2006, [On line]. Available: ]

[18] A. P. Stakhov, “The ‘golden’ matrices and a new kind of cryptography”, Chaos, solutions & fractals, vol. 32, no. 3 pp. 1138-1146, May 2007, doi: 10.1016/j.chaos.2006.03.069. [ Links ]

[19] D. R. Stinson, Cryptography theory and practice, 3rd ed. Boca Raton, FL: Chapman & Hall/ CRC, 2006. [ Links ]

Received: December 2018; Accepted: December 2019

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License