# On Inverses of Permutation Polynomials of Small Degree over Finite Fields

## TL;DR

This paper provides explicit formulas for the inverses of permutation polynomials of degree up to 6 over finite fields and degree 7 over fields of characteristic 2, enhancing understanding of their structure and applications.

## Contribution

It systematically derives the inverses of all permutation polynomials of degree ≤6 over any finite field and degree 7 over fields of characteristic 2, including a new explicit inverse for a class of fifth degree PPs.

## Key findings

- Explicit inverses for all degree ≤6 PPs over finite fields.
- Explicit inverse for degree 7 PPs over fields of characteristic 2.
- Main result includes a new inverse formula for a class of fifth degree PPs.

## Abstract

Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree $\leq 6$ over finite fields $\mathbb{F}_{q}$ for all $q$ and the inverses of all PPs of degree $7$ over $\mathbb{F}_{2^n}$. The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas' theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields.