What question did this study set out to answer?

The aim is to explore the conditions under which SCR polynomials can construct permutation polynomials over finite fields.

February 26, 2026

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

Key Points

The aim is to explore the conditions under which SCR polynomials can construct permutation polynomials over finite fields.
Investigated degree 2 and degree 3 SCR polynomials for conditions avoiding roots in roots of unity.
Examined higher degree SCR polynomials reducible to degree 2 over both odd and even ordered fields.
Analyzed specific types of SCR polynomials involving parameters from finite fields.
Identified specific conditions under which certain SCR polynomials do not yield roots in the specified set.
Constructed new permutation polynomials from simpler forms based on identified SCR polynomial conditions.
Demonstrated the application of SCR polynomials of specific types with varying parameters.

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 Fₐ^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 ₐ+₁ (the set of (q+1) -th roots of unity), which helps in the determination of polynomials that permute Fₐ^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 ax^q+1+bx^q+bx+a^q taking both the cases under consideration viz. a Fₐ and aₐ^₂ₐ.

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

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