Composition of random functions and word reconstruction

TL;DR

This paper investigates whether a single random function, formed by composing two random functions based on an unknown word, can reveal the word's length, structure, and distinguish it from others as the domain size grows.

Contribution

It introduces a method to recover word length and exponent with high probability and establishes conditions under which different words produce distinguishable functions.

Findings

Word length and exponent can be recovered with high probability.

Functions from different words are separated in total variation distance under certain conditions.

Explicit expression for a word-dependent constant c(w) is provided, linked to word structure.

Abstract

Given two functions $\mathbf{a}\!:\! [n] \rightarrow [n]$ and $\mathbf{b}\!:\! [n] \rightarrow [n]$ chosen uniformly at random, any word $w=w_1w_2\dots w_k\in \{a,b\}^k$ induces a random function $\mathbf{w}\!:\! [n] \rightarrow [n]$ by composition, i.e. $\mathbf{w}=\phi_{w_k}\circ \dots \circ \phi_{w_1}$ with $\phi_a=\mathbf{a}$ and $\phi_b=\mathbf{b}$. We study the following question: assuming $w$ is fixed but unknown, and $n$ goes to infinity, does one sample of $\mathbf{w}$ carry enough information to (partially) recover the word $w$ with good enough probability? We show that the length of $w$, and its exponent (largest $d$ such that $w={u}^d$ for some word ${u}$) can be recovered with high probability. We also prove that the random functions stemming from two different words are separated in total variation distance, provided that certain ``auto-correlation'' word-depending constant $c(w)$ is different for each of them. We give an explicit expression for $c(w)$ and conjecture that non-isomorphic words have different constants. We prove that this is the case assuming a major conjecture in transcendental number theory, Schanuel's conjecture.