A 3-Dimensional Lattice Reduction Algorithm

  • Igor Semaev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2146)

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 b1, b2, b3 ε 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.

Keywords

3-dimensional lattice lattice reduction problem shortest vector in a lattice Gaussian algorithm LLL-algorithm 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Igor Semaev
    • 1
  1. 1.Laboratory of Mathematical Problems of CryptologyMoscow State UniversityMoscowRussia

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