Short Bases of Lattices over Number Fields

  • Claus Fieker
  • Damien Stehlé
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


Lattices over number fields arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of ℤ-lattices, the choice of a nice, “short” (pseudo)-basis is important in many applications. In this article, we provide the first algorithm that computes such a “short” (pseudo)-basis. We utilize the LLL algorithm for ℤ-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to find a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a field invariant. As it runs in polynomial time, this provides an effective variant of Minkowski’s second theorem for lattices over number fields.


Polynomial Time Prime Ideal Integral Basis Dedekind Domain Fractional Ideal 
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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Claus Fieker
    • 1
  • Damien Stehlé
    • 1
    • 2
  1. 1.Magma Computer Algebra Group, School of Mathematics and StatisticsUniversity of SydneyAustralia
  2. 2.CNRS and Macquarie University 

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