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arXiv · 0801.3331

Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases

Guillaume Hanrot·Damien Stehlé
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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 $α_d$ and $β_d$ 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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Guillaume Hanrot, Damien Stehlé. 2008-01-24. Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases. https://arxiv.org/abs/0801.3331

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