## Algorithms to construct Minkowski reduced and Hermite reduced lattice bases.(English)Zbl 0601.68034

In 1983, Kannan presented an algorithm to construct a Hermite reduced basis of a lattice $$L=\sum^{n}_{i=1}b_ i {\mathbb{Z}}$$ in $${\mathbb{R}}^ d$$ $$(b_ i$$ linear independent vectors of $${\mathbb{R}}^ d)$$. In this paper, which is an outgrow of the author’s Ph. D. thesis under the guidance of C. P. Schnorr, an idea of C. P. Schnorr is used to reduce the complexity from $$(4n)^{1.5n+O(1)}$$ to $$n^{0.5n+O(n)}$$. Using the fact that a basis $$(b_ 1,...,b_{i-1})$$ is Hermite reduced if $$(b_ 1,...,b_ n)$$ is, the author also improves Kannan’s recursion algorithm to solve the closest lattice point problem.
Reviewer: A.Bachem

### MSC:

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

### Keywords:

basis reduction; closest lattice point
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### References:

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