Maximal entries of elements in certain matrix monoids

TL;DR

This paper extends previous results on the maximal entries in matrices generated by specific monoids, providing bounds that ensure collision resistance in related hashing functions, and generalizes known sequences like Fibonacci and Lucas.

Contribution

It generalizes the analysis of maximal entries in matrix monoids for all $u,v \, \geq 1$, extending prior specific cases and connecting to Fibonacci and Lucas sequences.

Findings

Derived bounds for maximal matrix entries for all $u,v \geq 1$

Extended the connection to Fibonacci and Lucas sequences

Provided collision resistance guarantees for related hash functions

Abstract

Let $L_u=\begin{bmatrix}1 & 0\\u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v\\0 & 1\end{bmatrix}$ be matrices in $SL_2(\mathbb Z)$ with $u, v\geq 1$. Since the monoid generated by $L_u$ and $R_v$ is free, we can associate a depth to each element based on its product representation. In the cases where $u=v=2$ and $u=v=3$, Bromberg, Shpilrain, and Vdovina determined the depth $n$ matrices containing the maximal entry for each $n\geq 1$. By using ideas from our previous work on $(u,v)$-Calkin-Wilf trees, we extend their results for any $u, v\geq 1$ and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on $L_u$ and $R_v$.