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Some Properties of Higher Order Mersenne and Gaussian Higher Order Mersenne Polynomials

  • 2026
  • OriginalPaper
  • Chapter
Published in:
Authors
Rabiranjan Mohanta
Kamalesh Acharya
Published in
Security and Privacy
Publisher
Springer Nature Switzerland

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Abstract

This chapter delves into the fascinating world of higher order Mersenne and Gaussian higher order Mersenne polynomials, exploring their properties and applications. The text begins by defining these polynomials recursively and establishing their relation to classical Mersenne and Gaussian Mersenne polynomials. It then presents Binet's formulas for these polynomials, followed by a series of identities such as Halton's, Honsberger's, Catalan's, Vajda's, Cassini's, Gelin-Cesaro's, and d'Ocagne identity. The chapter also discusses sum formulas and generating functions, providing a comprehensive analysis of these mathematical constructs. One of the most intriguing aspects of this chapter is its application of these polynomials in cryptography. The authors define Hadamard-type Fibonacci-Higher order Mersenne p-sequences and use them to generate keys for the Affine-Hill cipher. The chapter concludes with a security analysis of the proposed scheme, demonstrating the strength of the key space and the intractability of the problem for an adversary. This chapter is a must-read for anyone interested in the intersection of algebra, number theory, and cryptography.
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previous chapter Dynamic and Adaptive Broadcast Encryption for VANETs Supporting Inclusive-Exclusive Properties
next chapter Privacy-Preserving Auditable Authentication Scheme for Vehicular Ad-hoc Networks
Title
Some Properties of Higher Order Mersenne and Gaussian Higher Order Mersenne Polynomials
Authors
Rabiranjan Mohanta
Kamalesh Acharya
Copyright Year
2026
Book
Security and Privacy

Print ISBN: 978-3-032-12833-1

Electronic ISBN: 978-3-032-12834-8

Copyright Year: 2026

DOI
https://doi.org/10.1007/978-3-032-12834-8_5
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