A New Criterion on k -Normal Elements over Finite Fields
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
A New Criterion on k-Normal Elements over Finite Fields∗ Aixian ZHANG1
Keqin FENG2
Abstract The notion of normal elements for finite fields extension was generalized as k-normal elements by Huczynska et al. (2013). Several methods to construct k-normal elements were presented by Alizadah et al. (2016) and Huczynska et al. (2013), and the criteria on k-normal elements were given by Alizadah et al. (2016) and Antonio et al. (2018). In the paper by Huczynska, S., Mullen, G., Panario, D. and Thomson, D. (2013), the number of k-normal elements for a fixed finite field extension was calculated and estimated. In this paper the authors present a new criterion on k-normal elements by using idempotents and show some examples. Such criterion was given for usual normal elements before by Zhang et al. (2015). Keywords Normal basis, Finite field, Idempotent, Linearized polynomial, Gauss period 2000 MR Subject Classification 11T71, 13M06, 97H40
1 Introduction Let q = pm , where p is a prime number, m ≥ 1, Fq a finite field with q elements, F∗q = Fq \{0}. For n ≥ 1 and Q = q n , α ∈ F∗Q is called a normal element for extension FQ /Fq if 2
n−1
N = {α, αq , αq , · · · , αq } is a basis of FQ over Fq ( N is called a normal basis for FQ /Fq ). For a normal element α of FQ /Fq , the minimal polynomial fα (x) ∈ Fq [x] of α is called a normal polynomial for FQ /Fq , which is a monic irreducible polynomial in Fq [x] with degree n. Normal bases have many applications including coding theory, cryptography and communication theory due to the efficiency of exponentiation (see [5–6]). It is proved that α ∈ F∗Q is a normal element for FQ /Fq if and only if gcd(gα (x), xn − 1) = 1,
gα (x) =
n−1 X
i
αq xn−i−1
(1.1)
i=0
(see [5, Theorem 2.39]). The following definition given by Huczynska et al. [3] is a generalization of normal elements. Definition 1.1 (see [3]) Let q = pm , Q = q n and 0 ≤ k ≤ n − 1. An element α ∈ F∗Q is called a k-normal element for FQ /Fq if the degree of gcd(gα (x), xn − 1) is k. Manuscript received September 25, 2018. of Mathematical Sciences, Xi’an University of Technology, Xi’an 710054, China. E-mail: [email protected] 2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. E-mail: [email protected] ∗ This work was supported by the National Natural Science Foundation of China (No. 11571107) and the Natural Science Basic Research Plan of Shaanxi Province of China (No. 2019JQ-333). 1 Department
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A. X. Zhang and K. Q. Feng
With this terminology, a normal element is just 0-normal. As shown in the normal element case (see [6]), the k-normal elements can be used to reduce the multiplication process in finite fields. And another motivation for studying k-normal elements is due to the observation that they implicitly arise during the process of constructing quasi-normal bases of finite fields (see [7]). The number of k-normal elements for extension FQ /Fq was
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