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Construction of self-reciprocal normal polynomials over finite fields of even characteristic

In this paper, a computationally simple and explicit construction of some sequences of normal polynomials and self-reciprocal normal polynomials over finite fields of even characteristic are presented. | Turk J Math (2015) 39 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1407-32 Research Article Construction of self-reciprocal normal polynomials over finite fields of even characteristic 1 Mahmood ALIZADEH1,∗, Saeid MEHRABI2 Department of Mathematics, College of Science, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran 2 Department of Mathematics, Farhangian University, Tehran, Iran Received: 11.07.2014 • Accepted: 14.01.2015 • Published Online: 23.02.2015 • Printed: 20.03.2015 Abstract: In this paper, a computationally simple and explicit construction of some sequences of normal polynomials and self-reciprocal normal polynomials over finite fields of even characteristic are presented. Key words: Finite fields, normal polynomial, self-reciprocal 1. Introduction Let Fq , be the Galois field of order q = ps , where p is a prime and s is a natural number, and F∗q be its multiplicative group. Let P (x) be a monic irreducible polynomial of degree n over Fq and β be a root of P (x). The field Fq (β) = Fqn is an n -dimensional extension of Fq and can be considered as a vector space of dimension n over Fq . The Galois group of Fqn over Fq is cyclic and is generated by the Frobenius mapping σ(α) = αq , α ∈ Fqn . A normal basis of Fqn over Fq is a basis of the form N = {α, αq , ., αq basis that consists of the algebraic conjugates of a fixed element α ∈ F∗qn . n−1 } , i.e. a Recall that an element α ∈ Fqn is said to generate a normal basis over Fq if its conjugates form a basis of Fqn as a vector space over Fq . For our convenience we call a generator of a normal basis a normal element. A monic irreducible polynomial F (x) ∈ Fq [x] is called normal polynomial or N -polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over Fq . The elements in a normal basis are exactly the roots of some N polynomial. Hence, an N -polynomial is just another way of describing