A course in computational algebraic number theory
A course in computational algebraic number theory
An affine point of view on minima finding in integer lattices of lower dimensions
EUROCAL '87 Proceedings of the European Conference on Computer Algebra
Fast Reduction of Ternary Quadratic Forms
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Sparse Non-negative Stencils for Anisotropic Diffusion
Journal of Mathematical Imaging and Vision
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The aim of this paper is a reduction algorithm for a basis b1, b2, b3 of a 3-dimensional lattice in Rn for fixed n ≥ 3. We give a definition of the reduced basis which is equivalent to that of the Minkowski reduced basis of a 3-dimensional lattice. We prove that for b1, b2, b3 ∈ Zn, n = 3 and |b1|, |b2|, |b3| ≤ M, our algorithm takes O(log2M) binary operations, without using fast integer arithmetic, to reduce this basis and so to find the shortest vector in the lattice. The definition and the algorithm can be extended to any dimension. Elementary steps of our algorithm are rather different from those of the LLL-algorithm, which works in O(log3M) binary operations without using fast integer arithmetic.