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Designs, Codes and Cryptography

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16-10-2021

Existence of primitive normal pairs with one prescribed trace over finite fields

Authors: Hariom Sharma, R. K. Sharma

Published in: Designs, Codes and Cryptography | Issue 12/2021

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Abstract

Given \(m, n, q\in \mathbb {N}\) such that q is a prime power and \(m\ge 3\), \(a\in \mathbb {F}_q\), we establish a sufficient condition for the existence of a primitive pair \((\alpha , f(\alpha ))\) in \(\mathbb {F}_{q^m}\) such that \(\alpha \) is normal over \(\mathbb {F}_q\) and \(\text {Tr}_{\mathbb {F}_{q^m}/\mathbb {F}_q}(\alpha ^{-1})=a\), where \(f(x)\in \mathbb {F}_{q^m}(x)\) is a rational function of degree sum n. Further, when \(n=2\) and \(q=5^k\) for some \(k\in \mathbb {N}\), such a pair definitely exists for all (q, m) apart from at most 20 choices.

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Title: Existence of primitive normal pairs with one prescribed trace over finite fields
Authors: Hariom Sharma
R. K. Sharma
Publication date: 16-10-2021
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 12/2021
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-021-00956-7

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