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.

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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