The trace of primitive and 2-primitive elements in finite fields,\n revisited

2021/08/18 by Stephen D. Cohen, Cohen, Stephen D., Giorgos Kapetanakis +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Primary 11T30 #Secondary 11T06 #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2108.08066

openalex publication_date 2021/08/18 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

By definition primitive and 2-primitive elements of a finite field\nextension mathbbF_qⁿ have order qⁿ-1 and (qⁿ-1)/2, respectively.\nWe have already shown that, with minor reservations, there exists a primitive\nelement and a 2-primitive element \ξ \∈ mathbbF_qⁿ with prescribed\ntrace in the ground field mathbbF_q. Here we amend our previous proofs of\nthese results, firstly, by a reduction of these problems to extensions of prime\ndegree n and, secondly, by deriving an exact expression for the number of\nsquares in mathbbF_qⁿ whose trace has prescribed value in\n mathbbF_q. The latter corrects an error in the proof in the case of\n2-primitive elements. We also streamline the necessary computations.\n

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