Abstract
The factorization of polynomials is a fundamental computational problem in finite fields. Daqing [2] and von zur Gathen [3] have summarized prominent results for permutation polynomials in which interest was rekindled because of possible cryptographic applications. In this paper, we shall consider some factorizations in terms of linear recurrence relations. (For a detailed exposition of linear recurrence and relations and finite fields, the reader is referred to Selmer [5].) Here, we utilize Tang’s analog minimisation procedure for finding quadratic factors of a polynomial [7]. It is outlined as an application of generalized Fibonacci numbers {u n }. The sequence {U n } is defined by
with u 0 = 0,u 1 = 1, and {U n } and {V n } are sequences determined later. Clearly when U N = − V N = 1, the sequence of Fibonacci numbers is generated.
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References
Barakat, R. “The matrix operator e x and the Lucas polynomials”. Journal of Mathematics and Physics, Vol. 43, (1964): pp. 332–335.
Daqing, W. “On a conjecture of Carlitz, Part 3”. Journal of the Australian Mathematical Society (Series A), Vol. 43, (1987): pp. 375–384.
von zur Gathen, J. “Values of polynomials over finite fields, No. 1”. Bulletin of the Australian Mathematical Society, Vol. 43, (1991): pp. 141–146.
Horadam, A.F. “Basic properties of a certain generalized sequence of numbers”. The Fibonacci Quarterly, Vol. 3, (1965): pp. 161–176.
Selmer, E. Linear Recurrence Relations Over Finite Fields, University of Bergen, Norway, 1966.
Shannon, A.G., Loh, R.P., Melham, R.S. and Horadam, A.F. “A search for solutions of a functional equation”. Submitted.
Tang, I. “Finding quadratic factors by an analog minimisation procedure”. Simulation, Vol. 26, (1976): pp. 128–129.
Tang, I. “Simultaneous determination of quadratic factors by optimization methods”. Mathematics and Computers in Simulation, Vol. 19, (1977): pp. 57–59.
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© 1996 Kluwer Academic Publishers
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Shannon, A.G., Tang, I.C., Ollerton, R.L. (1996). A Use of Generalized Fibonacci Numbers in Finding Quadratic Factors. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0223-7_37
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DOI: https://doi.org/10.1007/978-94-009-0223-7_37
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6583-2
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