Construction of Permutation Polynomials over Finite Fields with the help of SCR polynomials

Abstract

In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over ${\mathbb{F}}_{q^2}$. The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in ${\mu }_{q+1}$ (the set of $(q+1)-\text{th}$ roots of unity), which helps in the determination of polynomials that permute ${\mathbb{F}}_{q^2}$. In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type $a{x}^{q+1}+b{x}^q+bx+a^q$ taking both the cases under consideration viz. $a\in {\mathbb{F}}_q$ and $a\in {\mathbb{F}}_{q^2}\setminus {\mathbb{F}}_q$ both.

13 pages