arXiv subjects

Phong Q. Nguyen

Publications and source records attributed to Phong Q. Nguyen.

4 recordsLinked to original sources

arXiv2026

Adelic reduction of module lattices

We give a strict generalization of the LLL algorithm over number fields, based on the reduction theory of $\GL(n)$ over the adele ring of a number field. Our algorithm is free of heuristics, with rigorous bounds on output quality and complexity. As a consequence, we obtain a hierarchy of reductions from module-(H)SVP to ideal-HSVP, an example of which has runtime and approximation factors subexponential in the field degree. More importantly, we uncover a close connection between structured lattice reduction and a Diophantine approximation over number fields.

math.NT↗

arXiv2025

On counterexamples to the Mertens conjecture

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp(1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp(5.15 \times 10^{23})$ by Kotnik and van de Lune [KvdL04].

math.NT↗

arXiv2019

Slide Reduction, Revisited---Filling the Gaps in SVP Approximation

We show how to generalize Gama and Nguyen's slide reduction algorithm [STOC '08] for solving the approximate Shortest Vector Problem over lattices (SVP). As a result, we show the fastest provably correct algorithm for $δ$-approximate SVP for all approximation factors $n^{1/2+\varepsilon} \leq δ\leq n^{O(1)}$. This is the range of approximation factors most relevant for cryptography.

cs.DS↗

arXiv2015

Counting Co-Cyclic Lattices

There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mathbb{Z}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq \mathbb{Z}^n$ such that $\mathbb{Z}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $\left(ζ(6) \prod_{k=4}^n ζ(k)\right)^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.

math.NT↗