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A hierarchy of polynomial time lattice basis reduction algorithms. (English) Zbl 0642.10030
We present a hierarchy of polynomial time lattice basis reduction algorithms that stretch from Lenstra, Lenstra, Lovász reduction to Korkine-Zolotareff reduction. Let $$\lambda$$ (L) be the length of a shortest nonzero element of a lattice L. We present an algorithm which for $$k\in {\mathbb{N}}$$ finds a nonzero lattice vector b so that $$| b|^ 2\leq (6k^ 2)^{n/k}\lambda (L)^ 2.$$ This algorithm uses $$O(n^ 2(\sqrt{k^{k+o(k)}}+n^ 2)\log B$$ arithmetic operations on O(n log B)-bit integers. This holds provided that the given basis vectors $$b_ 1,...,b_ n\in {\mathbb{Z}}^ n$$ are integral and have the length bound B. This algorithm successively applies Korkine-Zolotareff reduction to blocks of length k of the lattice basis. We also improve Kannan’s algorithm for Korkine-Zolotareff reduction.

##### MSC:
 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 68Q25 Analysis of algorithms and problem complexity 11H06 Lattices and convex bodies (number-theoretic aspects)
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