On the KZ Reduction

Abstract

The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around 7 8 times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is polynomially smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even polynomially and exponentially smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr-Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang et al.Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang et al., especially when the basis matrix is ill conditioned.

Improved Upper Bounds on the Hermite and KZ Constants

“…The proof for (7) first gives an upper bound on Γ 2 + n 2 and then shows the right-hand side of (7) is larger than this upper bound, while the proof for (5) here shows φ(t) is a monotonically increasing function by using an upper bound on the digamma function (see (6)).…”

“…Remark 2. The improved linear upper bound (4) on γ n can be used to improve the lower bound on the decoding radius of the LLL-aided SIC decoder that was given in [7], which is an improvement of the one given in [9,Lemma 1]. Since the derivation for the new lower bound on the decoding radius is straightforward by following the proof of [9, Lemma 1] and using (4), we do not provide details.…”

“…In this section, we develop an upper bound on the KZ constant that is sharper than that given by [7,Theorem 2].…”

“…It also has applications in bounding the KZ constant from above [5]. Furthermore, it can be used to derive lower bounds on the decoding radius of the LLL-aided SIC decoders [7], [9], and upper bounds on the orthogonality defect of KZ reduced matrices [6], [7], [12]. Although the Hermite constant is important, its exact values are known for dimension 1 ≤ n ≤ 8 and n = 24 only.…”

On Low-Complexity Lattice Reduction Algorithms for Large-Scale MIMO Detection: The Blessing of Sequential Reduction

“…Regarding KZ, refs. [15]- [17] give some practical implementations and improve the performance bounds. As for blockwise KZ, its faster implementations and the expected basis properties are given in [18].…”