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New cyclic groups based on the generalized order-k Pell sequences in the Heisenberg group and their application in cryptography

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Abstract

In this paper, we consider the finite groups

$$\begin{aligned} H_{(t,l,m)}=\langle a,b,c | a^t=b^l=c^m=1, [a,b]=c, [a,c]=[b,c]=1\rangle . \end{aligned}$$

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.

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Authors and Affiliations

  1. Mathematics Research Center, Near East University TRNC, Mersin 10, 99138, Nicosia, Turkey

    Elahe Mehraban & Evren Hincal

  2. Department of Mathematics, Near East University TRNC, Mersin 10, 99138, Nicosia, Turkey

    Elahe Mehraban & Evren Hincal

  3. Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, 99320, Kyrenia, Turkey

    Elahe Mehraban & Evren Hincal

  4. Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W 2Y2, Canada

    T. Aaron Gulliver

Authors
  1. Elahe Mehraban
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  2. T. Aaron Gulliver
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Correspondence to Elahe Mehraban.

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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

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  • DOI: https://doi.org/10.1007/s00200-024-00649-3

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