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Effective module lattices and their shortest vectors

Nihar Gargava, Vlad Serban, Maryna Viazovska, Ilaria Viglino

TL;DR

The work extends Rogers’ mean-value framework to the moduli space of rank-$t$ module lattices over number fields and to discrete constructions obtained from lifts of algebraic codes. By developing an effective Rogers integral formula for lifts and controlling rank- and height-based errors, it transfers Haar-random lattice moment bounds to broad families of algebraic lattices, yielding sharp probabilistic bounds for the shortest vector and related statistics. The authors obtain explicit cyclotomic-field results with uniform constants, and show that the same moment formulas hold for lifts of codes, bridging continuous and discrete lattice models with potential cryptographic and coding-theoretic applications. Overall, the paper strengthens the connection between Haar averages and structured module lattices, enabling precise SVP-type analyses in algebraic settings.

Abstract

We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.

Effective module lattices and their shortest vectors

TL;DR

The work extends Rogers’ mean-value framework to the moduli space of rank- module lattices over number fields and to discrete constructions obtained from lifts of algebraic codes. By developing an effective Rogers integral formula for lifts and controlling rank- and height-based errors, it transfers Haar-random lattice moment bounds to broad families of algebraic lattices, yielding sharp probabilistic bounds for the shortest vector and related statistics. The authors obtain explicit cyclotomic-field results with uniform constants, and show that the same moment formulas hold for lifts of codes, bridging continuous and discrete lattice models with potential cryptographic and coding-theoretic applications. Overall, the paper strengthens the connection between Haar averages and structured module lattices, enabling precise SVP-type analyses in algebraic settings.

Abstract

We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.
Paper Structure (12 sections, 20 theorems, 99 equations)

This paper contains 12 sections, 20 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Moduli space of module lattices
  3. Steinitz classes
  4. Lifts of codes and Haar measure
  5. Class of test functions
  6. Rogers integral formula using lifts of codes
  7. Matrices of fixed rank
  8. How many matrices are we summing up?
  9. Artificially injecting lower rank matrices
  10. Error estimates for the difference between the integrals and Riemann sums
  11. Lower rank matrices
  12. Bounds for the shortest vector

Key Result

Theorem 1

LN20 Let $\Lambda\subseteq \mathbb{R}^{d}$ be a Haar-random lattice chosen from the space $\mathop{\mathrm{SL}}\nolimits_{d}(\mathbb{R})/\mathop{\mathrm{SL}}\nolimits_{d}(\mathbb{Z})$. Then, as $d \rightarrow \infty$, with probability $1- o(1)$ we have where $\gamma(d)$ is radius of a ball of unit volume in $d$ dimensions.

Theorems & Definitions (48)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Remark 8
  • Proposition 9
  • Remark 10
  • ...and 38 more