# Iteration of Quadratic Polynomials Over Finite Fields

## TL;DR

This paper investigates the iteration behavior of quadratic polynomials over finite fields, demonstrating that such sequences typically recur after a logarithmic number of steps relative to the field size, with specific results for certain polynomials.

## Contribution

It provides bounds on the recurrence time of quadratic polynomial iterates over finite fields and discusses limitations of the Birthday Paradox model for cubic polynomials.

## Key findings

- Quadratic polynomial iterates recur after O(q/log log q) steps
- For X^2+1, recurrence occurs for any starting value
- The Birthday Paradox model is unsuitable for X^3+c when q ≡ 2 mod 3

## Abstract

For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX^2+c$, starting at $0$, always recurs after $O(q/\log\log q)$ steps. For $X^2+1$ the same is true for any starting value. We suggest that the traditional "Birthday Paradox" model is inappropriate for iterates of $X^3+c$, when $q$ is 2 mod 3.