Redei permutations with cycles of length $1$ and $p$

Abstract: Let $\mathbb F_q$ be the finite field of odd characteristic with $q$ elements and $\mathbb P^1(\mathbb F_q):=\mathbb F_q\cup \{\infty\}$. Consider the binomial expansion $\displaystyle (x+\sqrt y)^n = N(x,y)+D(x,y)\sqrt{y}.$ For $n\in\mathbb N$ and $a \in \mathbb F_q$, the Rédei function $R_{n,a}\colon \mathbb P^1(\mathbb F_q) \to \mathbb P^1(\mathbb F_q)$ is defined by $$ R_{n,a}(x)= \begin{cases} \dfrac{N(x,a)}{D(x,a)} & \text{ if } D(x,a)\neq 0, x\neq\infty\\ \infty & \text{ if } D(x,a)=0, x\neq\infty, \text{ or if } x=\infty. \end{cases} $$ Rédei functions have been used in several applications such as cryptography and coding theory as well as in the generation of pseudorandom numbers and Pell equations. In this talk we will present results on R\'edei permutations that decompose in cycles of length $1$ and $p$, where $p$ is prime. In particular, we will describe all Rédei functions that are involutions.

Joint work with Juliane Capaverde and Virgínia Rodrigues.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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