Abstract
In this paper, we consider the finite groups
We obtain the generalized order-k Pell sequences and study their periods. We prove the period of the order-k Pell sequence divided the period of the generalized order-k Pell sequence in the Heisenberg group. Then, the generalized order-k Pell sequence in Heisenberg group are used to define new cyclic groups. As an application, these groups are used in encryption algorithms.
Similar content being viewed by others
References
Deveci, Ö.: The \(k\)-nacci sequences and the generalized order-\(k\) Pell sequences in the semi-direct product of finite cyclic groups. Chiang Mai J. Sci. 40(1), 89–98 (2013)
Deveci, Ö., Aküzüm, Y.: The cyclic groups and the semigroups via MacWilliams and Chebyshev matrices. J. Math. Res. 6(2), 55–58 (2014)
Deveci, Ö., Karaduman, E.: The cyclic groups via the Pascal matrices and generalized Pascal matrices. Linear Algebra Appl. 473(10), 2538–2545 (2012)
Deveci, Ö., Karaduman, E.: The Pell sequences in finite groups. Util. Math. 96, 263–276 (2015)
Deveci, Ö., Karaduman, E.: On the Padovan p-numbers. Hacettepe J. Math. Stat. 46(4), 579–592 (2017)
Deveci, Ö., Shannon, A.G.: The quaternion-Pell sequence. Commun. Algebra 46(12), 5403–5409 (2018)
Hashemi, M., Mehraban, E.: On the generalized order 2-Pell sequence of some classes of groups. Commun. Algebra 46(9), 4104–4119 (2018)
Hashemi, M., Mehraban, E.: The generalized order \(k\)-Pell sequences in some special groups of nilpotency class 2. Commun. Algebra 50(4), 1768–1784 (2021)
Hashemi, M., Mehraban, E.: Fibonacci length and the generalized order \(k\)-Pell sequences of the 2-generator \(p\)-groups of nilpotency class 2. J. Algebra Appl. 22(3), 2350061 (2023)
Horadam, A.F.: Pell identities. Fibonacci Quart. 9(3), 245–252 (1971)
Kiliç, E.: The generalized order-\(k\) Fibonacci–Pell sequence by matrix methods. J. Comput. Appl. Math. 209, 133–145 (2007)
Kiliç, E.: The generalized Pell \((p, i)-\)numbers and their Binnet formulas, combinatorial representations, sums. Chaos Solitions Fract. 40(4), 2047–2063 (2009)
Kiliç, E., Taşci, D.: The generalized Binet formula, representation and sum of the generalizied order \(k\)-Pell numbers. Taiwan J. Math. 10(6), 1661–1670 (2006)
Osipov, D.V.: The discrete Heisenberg group and its automorphism group. Math. Notes 98(1–2), 185–188 (2015)
Shannon, A.G., Erdağ, Ö., Deveci, Ö.: On the connections between Pell numbers and Fibonacci \(p\)-numbers. Notes Number Theory Discrete Math. 27(1), 148–160 (2021)
Szynal-Liana, A., Wloch, I.: The Pell quaternions and the Pell octonions. Adv. Appl. Clifford Algebras 26, 435–440 (2016)
Yılmaz, N., Cetinalp, E.K., Deveci, Ö., Özta, E.: The quaternion-type cyclic-Pell sequences in finite groups. Bull. Int. Math. Virtual Inst. 13(1), 169–178 (2023)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mehraban, E., Gulliver, T.A. & Hincal, E. New cyclic groups based on the generalized order-k Pell sequences in the Heisenberg group and their application in cryptography. AAECC (2024). https://doi.org/10.1007/s00200-024-00649-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00200-024-00649-3