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Faster Lattice Basis Computation via a Natural Generalization of the Euclidean Algorithm

Kim-Manuel Klein, Janina Reuter

TL;DR

This paper shows that the classical Euclidean algorithm can be adapted in a very natural way to compute a basis of a lattice that is generated by vectors A1, … , An ∈ ℤd with n> rank(A1, … ,An).

Abstract

The Euclidean algorithm is the oldest algorithms known to mankind. Given two integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd) of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it computes a basis of the sum of two one-dimensional lattices $a_1 \mathbb{Z}$ and $a_2 \mathbb{Z}$ as $\gcd(a_1,a_2) \mathbb{Z} = a_1 \mathbb{Z} + a_2 \mathbb{Z}$. In this paper, we show that the classical Euclidean algorithm can be adapted in a very natural way to compute a basis of a general lattice $L(A_1, \ldots , A_n)$ given vectors $A_1, \ldots , A_n \in \mathbb{Z}^d$ with $n> \mathrm{rank}(a_1, \ldots ,a_d)$. Similar to the Euclidean algorithm, our algorithm is very easy to describe and implement and can be written within 12 lines of pseudocode. As our main result, we obtain an algorithm to compute a lattice basis for given vectors $A_1, \ldots , A_n \in \mathbb{Z}^d$ in time (counting bit operations) $LS + \tilde{O}((n-d)d^2 \cdot \log(||A||)$, where $LS$ is the time required to obtain the exact fractional solution of a certain system of linear equalities. The analysis of the running time of our algorithms relies on fundamental statements on the fractionality of solutions of linear systems of equations. So far, the fastest algorithm for lattice basis computation was due to Storjohann and Labhan [SL96] having a running time of $\tilde{O}(nd^ω\log ||A||)$. For current upper bounds of $LS$, our algorithm has a running time improvement of a factor of at least $d^{0.12}$ over [SL96]. Our algorithm is therefore the first general algorithmic improvement to this classical problem in nearly 30 years. At last, we present a postprocessing procedure which yields an improved size bound of $\sqrt{d} ||A||$ for vectors of the resulting basis matrix.

Faster Lattice Basis Computation via a Natural Generalization of the Euclidean Algorithm

TL;DR

This paper shows that the classical Euclidean algorithm can be adapted in a very natural way to compute a basis of a lattice that is generated by vectors A1, … , An ∈ ℤd with n> rank(A1, … ,An).

Abstract

The Euclidean algorithm is the oldest algorithms known to mankind. Given two integral numbers and , it computes the greatest common divisor (gcd) of and in a very elegant way. From a lattice perspective, it computes a basis of the sum of two one-dimensional lattices and as . In this paper, we show that the classical Euclidean algorithm can be adapted in a very natural way to compute a basis of a general lattice given vectors with . Similar to the Euclidean algorithm, our algorithm is very easy to describe and implement and can be written within 12 lines of pseudocode. As our main result, we obtain an algorithm to compute a lattice basis for given vectors in time (counting bit operations) , where is the time required to obtain the exact fractional solution of a certain system of linear equalities. The analysis of the running time of our algorithms relies on fundamental statements on the fractionality of solutions of linear systems of equations. So far, the fastest algorithm for lattice basis computation was due to Storjohann and Labhan [SL96] having a running time of . For current upper bounds of , our algorithm has a running time improvement of a factor of at least over [SL96]. Our algorithm is therefore the first general algorithmic improvement to this classical problem in nearly 30 years. At last, we present a postprocessing procedure which yields an improved size bound of for vectors of the resulting basis matrix.
Paper Structure (13 sections, 14 theorems, 49 equations, 5 figures, 1 table, 4 algorithms)

This paper contains 13 sections, 14 theorems, 49 equations, 5 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Lattice Basis Computation
  3. Our Contribution
  4. A Multi-Dimensional Generalization of the Euclidean Algorithm
  5. The Algorithm
  6. Formal Description of the Algorithm
  7. Speeding Up the Generalized Euclidean Algorithm
  8. The Fractionality of the Parallelepipped
  9. The Bit Complexity of the fast Generalized Euclidean Algorithm
  10. Matrix Multiplication
  11. Linear System Solving
  12. Lower Rank Lattices
  13. Reducing the Size of the Solution Basis

Key Result

Lemma 1

Let $A\in\mathbb{Z}^{\Tilde{d}\times \Tilde{n}}$ have full column rank. There exists an algorithm that finds indices $i_1, \ldots, i_{\Tilde{d}}$ such that $A_{i_1}, \ldots, A_{i_{\Tilde{d}}}$ are linearly independent using $\mathop{\mathrm{\tilde{O}}}\nolimits(\Tilde{n}\Tilde{d}^{\omega-1}\cdot \lo

Figures (5)

  • Figure 1: The parallelepiped of $B=(B_1B_2)$.
  • Figure 2: The modulo operation with respect to a lattice and the exchange operation depending on $\nint{x_1}$.
  • Figure 3: An application of the algorithm.
  • Figure 4: Application of Algorithm \ref{['alg:fasteuclidean']}
  • Figure 5: Translates of the Subspaces

Theorems & Definitions (23)

  • Lemma 1: DBLP:conf/issac/LiS22
  • Theorem 1
  • proof
  • Theorem 3
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • ...and 13 more