New sequences from the generalized Pell $ p- $numbers and mersenne numbers and their application in cryptography

Elahe Mehraban; T. Aaron Gulliver; Salah Mahmoud Boulaaras; Kamyar Hosseini; Evren Hincal; Elahe Mehraban; T. Aaron Gulliver; Salah Mahmoud Boulaaras; Kamyar Hosseini; Evren Hincal

doi:10.3934/math.2024660

AIMS Mathematics

2024, Volume 9, Issue 5: 13537-13552. doi: 10.3934/math.2024660

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New sequences from the generalized Pell $ p- $numbers and mersenne numbers and their application in cryptography

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
3.
Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, 99320 Kyrenia, Turkey
4.
Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W 2Y2, Canada
5.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
6.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

Received: 20 January 2024 Revised: 29 March 2024 Accepted: 03 April 2024 Published: 12 April 2024
MSC : 11K31, 11C20, 68P25, 68R01, 68P30, 15A15

This paper presents the generalized Pell $ p- $numbers and provides some related results. A new sequence is defined using the characteristic polynomial of the Pell $ p- $numbers and generalized Mersenne numbers. Two algorithms for Diffie-Hellman key exchange are given as an application of these sequences. They are illustrated via numerical examples and shown to be secure against attacks. Thus, these new sequences are practical for encryption and constructing private keys.
- Mersenne numbers,
- Pell $ p- $numbers,
- Diffie-Hellman key exchange,
- public key,
- private key,
- sequence analysis
Citation: Elahe Mehraban, T. Aaron Gulliver, Salah Mahmoud Boulaaras, Kamyar Hosseini, Evren Hincal. New sequences from the generalized Pell $ p- $numbers and mersenne numbers and their application in cryptography[J]. AIMS Mathematics, 2024, 9(5): 13537-13552. doi: 10.3934/math.2024660

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Abstract

This paper presents the generalized Pell $ p- $numbers and provides some related results. A new sequence is defined using the characteristic polynomial of the Pell $ p- $numbers and generalized Mersenne numbers. Two algorithms for Diffie-Hellman key exchange are given as an application of these sequences. They are illustrated via numerical examples and shown to be secure against attacks. Thus, these new sequences are practical for encryption and constructing private keys.

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