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\begin{document}

\title{Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation\footnote{Supported
by FWF (Austrian Science Fund) project Y757 
and by project MTM2017-88804-P (Spanish Ministry of Science and Innovation).}}
%\title{Modulo arithmetic-based algorithms for lattice basis reduction and for computing the Hermite normal form}
\author{Ralph Bottesch \and Jose Divas\'on \and Ren\'e Thiemann}
\maketitle

\begin{abstract}
We verify two algorithms for which modular arithmetic plays an essential role: Storjohann's variant of the LLL lattice basis reduction algorithm and Kopparty's algorithm for computing the Hermite normal form of a matrix. To do this, we also formalize some facts about the modulo operation with symmetric range. Our implementations are based on the original papers, but are otherwise efficient. For basis reduction we formalize two versions: one that includes all of the optimizations/heuristics from Storjohann's paper, and one excluding a heuristic that we observed to often decrease efficiency. We also provide a fast, self-contained certifier for basis reduction, based on the efficient Hermite normal form algorithm.
\end{abstract}

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