Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases

Guillaume Hanrot 1 Damien Stehlé 2
1 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
2 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least reduced. We prove that for such bases, Kannan's algorithm solving the shortest lattice vector problem requires~$d^{\frac{d}{2\e}(1+o(1))}$ bit operations in dimension~$d$. This matches the best complexity upper bound known for this algorithm. These bases also provide lower bounds on Schnorr's constants~$\alpha_d$ and~$\beta_d$ that are essentially equal to the best upper bounds. Finally, we also show the existence of particularly bad bases for Schnorr's hierarchy of reductions.
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Submitted on : Thursday, January 24, 2008 - 4:12:59 PM
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  • HAL Id : inria-00211875, version 2
  • ARXIV : 0801.3331

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Guillaume Hanrot, Damien Stehlé. Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases. [Research Report] RR-6422, INRIA. 2008, pp.25. ⟨inria-00211875v2⟩

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