Another view of the Gaussian algorithm

We introduce here a rewrite system in the group of unimodular matrices, \emph{i.e.}, matrices with integer entries and with determinant equal to $\pm 1$. We use this rewrite system to precisely characterize the mechanism of the Gaussian algorithm, that finds shortest vectors in a two–dimensional lattice given by any basis. Putting together the algorithmic of lattice reduction and the rewrite system theory, we propose a new worst–case analysis of the Gaussian algorithm. There is already an optimal worst–case bound for some variant of the Gaussian algorithm due to Vall'ee \cite {ValGaussRevisit}. She used essentially geometric considerations. Our analysis generalizes her result to the case of the usual Gaussian algorithm. An interesting point in our work is its possible (but not easy) generalization to the same problem in higher dimensions, in order to exhibit a tight upper-bound for the number of iterations of LLL–like reduction algorithms in the worst case. Moreover, our method seems to work for analyzing other families of algorithms. As an illustration, the analysis of sorting algorithms are briefly developed in the last section of the paper.

💡 Research Summary

The paper presents a novel algebraic framework for analysing the classic Gaussian algorithm, which computes a shortest vector in a two‑dimensional lattice given an arbitrary basis. The authors begin by defining the group of unimodular 2 × 2 integer matrices, GL₂(ℤ), i.e., matrices with determinant ±1. Within this group they introduce a rewrite system consisting of two elementary operations: (1) swapping rows (the transpose operation) and (2) adding an integer multiple of one row to another (row i ← row i ± k·row j, k ∈ ℤ). These operations correspond exactly to the elementary basis transformations performed by the Gaussian algorithm (often called “Bézout moves”).

By representing a lattice basis B =