A Use of Generalized Fibonacci Numbers in Finding Quadratic Factors

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

$$ {u_{n\,}}\, = \,{U_{{N^u}n\, - \,1}}\, - \,V{\,_{{N^u}n\, - \,2\,}}n \geqslant \,2 $$

((1.1))

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.