# Random Feedback Shift Registers, and the Limit Distribution for Largest Cycle Lengths

## TL;DR

This paper proves that the scaled cycle lengths of a random binary feedback shift register follow a Poisson-Dirichlet distribution, similar to random permutations, confirming a long-standing conjecture from 1959.

## Contribution

It establishes the limiting distribution of cycle lengths in random feedback shift registers, connecting them to the Poisson-Dirichlet distribution and confirming a conjecture from 1959.

## Key findings

- Cycle lengths follow Poisson-Dirichlet distribution
- Convergence in distribution for scaled cycle lengths
- Confirms conjecture by Golomb, Welch, and Goldstein

## Abstract

For a random binary noncoalescing feedback shift register of width $n$, with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$, converges in distribution to the same Poisson-Dirichlet limit as holds for random permutations in $\mathcal{S}_N$, with all $N!$ possible permutations equally likely. Such behavior was conjectured by Golomb, Welch, and Goldstein in 1959.