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arXiv · 1703.02388

Maximal entries of elements in certain matrix monoids

Sandie Han·Ariane M. Masuda·Satyanand Singh·Johann Thiel
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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 $R_v$.

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Sandie Han, Ariane M. Masuda, Satyanand Singh, Johann Thiel. 2017-03-07. Maximal entries of elements in certain matrix monoids. https://arxiv.org/abs/1703.02388

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