Primitive Normal Values of Rational Functions over Finite Fields

2021/12/14 by Avnish K. Sharma, Mamta Rani, Sharma, Avnish K. +3
Computer Science · Social Sciences · #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2112.07410

openalex publication_date 2021/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider rational functions f with some minor restrictions over the finite field \mathbbF_qⁿ, where q=p^k for some prime p and positive integer k. We establish a sufficient condition for the existence of a pair (α,f(α)) of primitive normal elements in \mathbbF_qⁿ over \mathbbF_q. Moreover, for q=2^k and rational functions f with quadratic numerators and denominators, we explicitly find that there are at most 55 finite fields \mathbbF_qⁿ in which such a pair (α,f(α)) of primitive normal elements may not exist.

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