## Abstract

Given a lattice*L* we are looking for a basis**B**=[**b**
_{1}, ...**b**
_{
n
}] of*L* with the property that both**B** and the associated basis**B**
^{*}=[**b**
^{*}_{1}
, ...,**b**
^{*}_{
n
}
] of the reciprocal lattice*L*
^{*} consist of short vectors. For any such basis**B** with reciprocal basis**B**
^{*} let\(S(B) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} (|b_i | \cdot |b_i^ * |)\). Håstad and Lagarias [7] show that each lattice*L* of full rank has a basis**B** with*S*(**B**)≤exp(*c*
_{1}·*n*
^{1/3}) for a constant*c*
_{1} independent of*n*. We improve this upper bound to*S*(**B**)≤exp(*c*
_{2}·(ln*n*)^{2}) with*c*
_{2} independent of*n*.

We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity\(\sum\limits_{i = 1}^n {|b|^2 } \cdot |b_i^ * |^2 \). In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.

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Seysen, M. Simultaneous reduction of a lattice basis and its reciprocal basis.
*Combinatorica* **13, **363–376 (1993). https://doi.org/10.1007/BF01202355

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### AMS subject classification code (1991)

- 11 H 55