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Variations of the Primitive Normal Basis Theorem

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Abstract

The celebrated Primitive Normal Basis Theorem states that for any \(n\ge 2\) and any finite field \({\mathbb {F}}_q\), there exists an element \(\alpha \in {\mathbb {F}}_{q^n}\) that is simultaneously primitive and normal over \({\mathbb {F}}_q\). In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014).

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Acknowledgements

We are grateful to the anonymous referees for their suggestions and comments. Giorgos Kapetanakis was supported by TÜBİTAK Project Number 114F432 and Lucas Reis was supported by CAPES-PDSE (process - 88881.134747/2016-01).

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Correspondence to Giorgos Kapetanakis.

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Communicated by D. Jungnickel.

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Appendix A: Pseudocodes for search for 2-primitive elements with special properties

Appendix A: Pseudocodes for search for 2-primitive elements with special properties

Following the approach of [11], Algorithm 1 presents a search routine for 2-primitive, k-normal elements of \({\mathbb {F}}_{q^n}\) over \({\mathbb {F}}_q\). This search is based on the original characterization of k-normal elements from [7].

Theorem A.1

Let \(\alpha \in {\mathbb {F}}_{q^n}\) and let \(g_\alpha (x) = \sum _{i=0}^{n-1} \alpha ^{q^i}x^{n-1-i} \in {\mathbb {F}}_{q^n}[x]\). Then \(\gcd (x^n-1, g_\alpha (x))\) has degree k if and only if \(\alpha \) is a k-normal element of \({\mathbb {F}}_{q^n}\) over \({\mathbb {F}}_q\).

Algorithm 1 proceeds as follows. Let \({\mathbb {F}}_{q^n}\cong {\mathbb {F}}_q[x]/(f)\) with f a primitive polynomial, and let g be a root of f. Hence, g is a generator of \({\mathbb {F}}_{q^n}^*\) and \(g^i\) is 2-primitive if and only if \(\gcd (i, q^n-1) = 2\). For each 2-primitive element, check its k-normality using Theorem A.1. If \(k=1\), the resulting element is 2-primitive, 1-normal and is returned. The algorithm returns “Fail” if no 2-primitive 1-normal is found after \(q^n-2\) iterations; that is, if all of \({\mathbb {F}}_{q^n}^*\) is traversed.

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Kapetanakis, G., Reis, L. Variations of the Primitive Normal Basis Theorem. Des. Codes Cryptogr. 87, 1459–1480 (2019). https://doi.org/10.1007/s10623-018-0543-9

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