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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Acknowledgements
The authors thank the referee and the editor for their valuable comments and suggestions, which improved the presentation of the paper. Prof. R. K. Sharma is the ConsenSys Blockchain Chair Professor at IIT Delhi. He is grateful to ConsenSys AG for that privilege.
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Communicated by O. Ahmadi.
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Sharma, H., Sharma, R.K. Existence of primitive normal pairs with one prescribed trace over finite fields. Des. Codes Cryptogr. 89, 2841–2855 (2021). https://doi.org/10.1007/s10623-021-00956-7
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DOI: https://doi.org/10.1007/s10623-021-00956-7