An efficient algorithm for integer lattice reduction

TL;DR

The paper tackles lattice reduction by introducing a simple, fast iterative scheme that monotonically reduces the Euclidean norms of a lattice basis via nearest-integer projections and Gram-matrix updates. While it does not guarantee a global optimum, it can significantly polish the output of other reduction methods such as LLL and does so with per-iteration costs of $\mathcal{O}(m n + n^2)$ and overall runtimes near $\mathcal{O}(n^3)$, often outperforming LLL in speed. The main contributions include a concrete, provably monotone reduction procedure, a detailed cost analysis, and extensive experiments showing effective post-processing improvements and practical applicability to cryptographic-style bases. The approach offers a practical, efficient tool for rapidly refining lattice bases when used in conjunction with stronger reduction algorithms.

Abstract

A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the problem of finding a set of vectors in a given lattice such that the collection of all integer linear combinations of this subset is still the entire original lattice and so that the Euclidean norms of the subset are reduced. The present paper proposes simple, efficient iterations for lattice reduction which are guaranteed to reduce the Euclidean norms of the basis vectors (the vectors in the subset) monotonically during every iteration. Each iteration selects the basis vector for which projecting off (with integer coefficients) the components of the other basis vectors along the selected vector minimizes the Euclidean norms of the reduced basis vectors. Each iteration projects off the components along the selected basis vector and efficiently updates all information required for the next iteration to select its best basis vector and perform the associated projections.

Figures (20)

• Figure 1: $\delta = 1-10^{-15}$, $p = 2$; the upper plots are for $q = 2^{13} - 1$, the lower plots are for $q = 2^{31} - 1$
• Figure 2: $\delta = 1-10^{-15}$, $p = 2$, $q = 2^{13} - 1$
• Figure 3: $\delta = 1-10^{-15}$, $p = 2$, $q = 2^{13} - 1$
• Figure 4: $\delta = 1-10^{-15}$, $p = 2$, $q = 2^{31} - 1$
• Figure 5: $\delta = 1-10^{-15}$, $p = 2$, $q = 2^{31} - 1$

Theorems & Definitions (9)

• Remark 1
• Remark 2
• Remark 3
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof