Simultaneous reduction of a lattice basis and its reciprocal basis

Abstract

Given a latticeL we are looking for a basisB=[b ₁, ...b _n ] ofL with the property that bothB and the associated basisB ^*=[b ^*₁ , ...,b ^* _n ] of the reciprocal latticeL ^* consist of short vectors. For any such basisB with reciprocal basisB ^* let$S(B) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} (|b_i | \cdot |b_i^ * |)$. Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)≤exp(c ₁·n ^1/3) for a constantc ₁ independent ofn. We improve this upper bound toS(B)≤exp(c ₂·(lnn)²) withc ₂ independent ofn.

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.