On the Probability of Generating a Lattice

TL;DR

This paper analyzes the probability that randomly chosen vectors from a lattice generate the entire lattice, providing rigorous bounds and conjectures that could improve quantum algorithms for number theory problems.

Contribution

It offers the first rigorous proof that 2n+1 vectors suffice for lattice generation and conjectures that only n+1 vectors may be enough, potentially enhancing quantum algorithm success rates.

Findings

2n+1 vectors suffice with high probability

Conjecture: n+1 vectors may be enough for lattice generation

Simulation results support the conjecture

Abstract

We study the problem of determining the probability that m vectors selected uniformly at random from the intersection of the full-rank lattice L in R^n and the window [0,B)^n generate $\Lambda$ when B is chosen to be appropriately large. This problem plays an important role in the analysis of the success probability of quantum algorithms for solving the Discrete Logarithm Problem in infrastructures obtained from number fields and also for computing fundamental units of number fields. We provide the first complete and rigorous proof that 2n+1 vectors suffice to generate L with constant probability (provided that B is chosen to be sufficiently large in terms of n and the covering radius of L and the last n+1 vectors are sampled from a slightly larger window). Based on extensive computer simulations, we conjecture that only n+1 vectors sampled from one window suffice to generate L with constant success probability. If this conjecture is true, then a significantly better success probability of the above quantum algorithms can be guaranteed.