Construction of permutation polynomials over finite fields with the help of SCR polynomials

Abstract

In this paper, we undertake a deeper study of self-conjugate reciprocal (SCR) polynomials, which ultimately contributes to the construction of new classes of permutation polynomials of simpler forms over $\mathbb{F}_{q^{2}}$. The primary focus is on identifying the conditions under which certain degree 2 and degree 3 SCR polynomials have no roots in $\mu_{q+1}$, the set of $(q+1)^{\rm th}$ roots of unity-which plays a key role in determining polynomials that permute $\mathbb{F}_{q^{2}}$. Along the way, we also examine certain higher-degree SCR polynomials that can be reduced to degree 2 SCR polynomials over both odd and even characteristic fields. Furthermore, we investigate SCR polynomials of the form $ax^{q+1} + bx^{q} + bx + a^{q}$, considering both cases where $a \in \mathbb{F}_{q}$ and where $a \in \mathbb{F}_{q^{2}} \setminus \mathbb{F}_{q}$.

Keywords

• permutation polynomial
• trinomial
• quadrinomial
• self conjugate reciprocal (SCR) polynomial

Thanks

The authors would like to express their sincere gratitude to Michael E. Zieve for his careful remarks and comments that helped in improving the quality of the paper.

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