New Korkin-Zolotarev inequalities

Korkin and Zolotarev showed that if \sum_i A_i\Big(x_i-\sum_{j>i} \alpha_{ij}x_j\Big)^2 is the Lagrange expansion of a Korkin–Zolotarev (KZ-) reduced positive definite quadratic form, then $A_{i+1}\geq \frac{3}{4} A_i$ and $A_{i+2}\geq \frac{2}{3}A_i$. They showed that the implied bound $A_{5}\geq \frac{4}{9}A_1$ is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange expansions of KZ-reduced quadratic forms using a semidefinite relaxation combined with a branch and bound process. We use a rounding technique to derive exact results from the numerical data. Applying these methods, we prove several new linear inequalities on the $A_i$ of any KZ-reduced form, one of them being $A_{i+4}\geq (\frac{15}{32}-2 \cdot 10^{-5})A_i$. We also give a form with $A_{5}= \frac{15}{32}A_1$. These new inequalities are then used to study the cone of outer coefficients of KZ-reduced forms, to find bounds on Hermite's constant, and to give better estimates on the quality of $k$-block KZ-reduced lattice bases

Some Inequalities Related to the Seysen Measure of a Lattice

Given a lattice $L$, a basis $B$ of $L$ together with its dual $B^*$, the orthogonality measure $S(B)=\sum_i ||b_i||^2 ||b_i^*||^2$ of $B$ was introduced by M. Seysen in 1993. This measure is at the heart of the Seysen lattice reduction algorithm and is linked with different geometrical properties of the basis. In this paper, we explicit different expressions for this measure as well as new inequalities.Comment: Typos correcte

New Korkin-Zoloratev inequalities : implementation and numerical data

This technical report discusses the mathematical details and the implementation of the methods discussed in the accompanying paper [PZ06]. In particular a method to find a finite list of inequalities that certify Korkin–Zolotarev reducedness of a quadratic form is presented. Moreover a semidefinite programming relaxation of the space of KZ-reduced quadratic forms is described in detail, together with a branching strategy to optimize over this space. Finally the implementation of these methods is discussed, together with some hints on how to compile and use the programs. The two digital appendices, which can be obtained from the SPOR reports website†, contain an implementation of the methods discussed and numerical data that prove the theorems in [PZ06]

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

New practical algorithms for the approximate shortest lattice vector

We present a practical algorithm that given an LLL-reduced lattice basis of dimension n, runs in time O(n3(k=6)k=4+n4) and approximates the length of the shortest, non-zero lattice vector to within a factor (k=6)n=(2k). This result is based on reasonable heuristics. Compared to previous practical algorithms the new method reduces the proven approximation factor achievable in a given time to less than its fourthth root. We also present a sieve algorithm inspired by Ajtai, Kumar, Sivakumar [AKS01]