Cryptography using Fibonacci–Mersenne and Fibonacci-balancing p-sequences with a self-invertible matrix and the Affine–Hill cipher

Elahe Mehraban, T. Aaron Gulliver, Ömür Deveci and Evren Hincal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 792–818
DOI: 10.7546/nntdm.2025.31.4.792-818
Full paper (PDF, 313 Kb)

Details

Authors and affiliations

Elahe Mehraban
¹ Mathematics Research Center, Near East University TRNC
Mersin 10, 99138 Nicosia, Turkey
² Department of Mathematics, Near East University TRNC
Mersin 10, 99138 Nicosia, Turkey

T. Aaron Gulliver
³ Department of Electrical and Computer Engineering, University of Victoria
Victoria, BC V8W 2Y2, Canada

Ömür Deveci
⁴ Department of Mathematics, Faculty of Science and Letters, Kafkas University
36100, Turkey

Evren Hincal
¹ Mathematics Research Center, Near East University TRNC
Mersin 10, 99138 Nicosia, Turkey
² Department of Mathematics, Near East University TRNC
Mersin 10, 99138 Nicosia, Turkey
⁵ Research Center of Applied Mathematics, Khazar University
Baku, Azerbaijan

Abstract

In this paper, we define two new sequences using the Fibonacci -numbers, the generalized Mersenne numbers, and -balancing numbers. These sequences are obtained from the corresponding characteristic polynomials. The determinants and combinatorial and exponential representations of these new sequences are given. As an application, we provide two algorithms using these new sequences with self-invertible matrices and the Affine–Hill cipher.

Keywords

• Mersenne numbers
• -balancing numbers
• Fibonacci -numbers
• Cryptography
• Self-invertible matrix
• Affine–Hill cipher

2020 Mathematics Subject Classification

• 11K31
• 11C20
• 68P25
• 68R01
• 68P30
• 15A15

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Manuscript history

• Received: 18 July 2025
• Revised: 3 November 2025
• Accepted: 5 November 2025
• Online First: 8 November 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).