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Abstract:
We describe how a shortest vector of a 2-dimensional integral lattice
corresponds to a best approximation of a unique rational number
defined by the lattice. This rational number and its best
approximations can be computed with
the euclidean algorithm and its speedup by Schoenhage (1971)
from any basis of the lattice.
The described correspondence allows,
on the one hand, to reduce a basis of a
2-dimensional integral lattice with the euclidean algorithm, up to
a single normalization step.
On the other hand, one can use the classical result
of Schoenhage (1971) to obtain a shortest vector of a 2-dimensional
integral lattice with respect to the $\ell_\infty$-norm.
It follows that in two dimensions, a fast basis-reduction algorithm
can be solely based on Schönhage's algorithm
and the reduction algorithm of Gauss (1801).
