A 3-Dimensional Lattice Reduction Algorithm

Abstract

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 b ₁ , b₂, b₃ ε ℤⁿ, n ≥ 3 and |b₁|, |b₂|, |b₃ | ≤ M, our algorithm takes O(log² M) 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(log³ M) binary operations without using fast integer arithmetic.