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Module lattices and their shortest vectors

Nihar Gargava, Vlad Serban, Maryna Viazovska, Ilaria Viglino

TL;DR

This work investigates shortest-vector phenomena for module lattices over number fields, focusing on cyclotomic fields, by extending Rogers-style second-moment analysis to the algebraic setting. The authors develop height-based controls and unit-count bounds to bound the second moment $\\mathbb{E}[\\rho_V(\\Lambda)^2]$ of the number of nonzero lattice vectors in a ball of volume $V$, obtaining the leading term $V^2+\\omega_K V$ with exponentially small errors. They specialize to cyclotomic fields to derive explicit bounds and, in favorable regimes, Poisson-type convergence for the vector counts with mean $V/\\omega_K$, enabling probabilistic estimates for the shortest vector through the volume-minimum $\\mathcal{V}_1(\\Lambda)$. The results yield practically sharp probabilistic guarantees for SVP across various random module lattice models, including discrete-lattice constructions from codes, and provide a concrete workflow for fixed fields (e.g., a 256-dimensional case) to assess cryptographic-security-relevant parameters.

Abstract

We study the shortest vector lengths in module lattices over arbitrary number fields, with an emphasis on cyclotomic fields. In particular, we sharpen the techniques of arXiv:2308.15275v2 to establish improved results for the variance of the number of lattice vectors of bounded Euclidean norm in a random module lattice. We then derive tight probabilistic bounds for the shortest vector lengths for several notions of random module lattice.

Module lattices and their shortest vectors

TL;DR

This work investigates shortest-vector phenomena for module lattices over number fields, focusing on cyclotomic fields, by extending Rogers-style second-moment analysis to the algebraic setting. The authors develop height-based controls and unit-count bounds to bound the second moment of the number of nonzero lattice vectors in a ball of volume , obtaining the leading term with exponentially small errors. They specialize to cyclotomic fields to derive explicit bounds and, in favorable regimes, Poisson-type convergence for the vector counts with mean , enabling probabilistic estimates for the shortest vector through the volume-minimum . The results yield practically sharp probabilistic guarantees for SVP across various random module lattice models, including discrete-lattice constructions from codes, and provide a concrete workflow for fixed fields (e.g., a 256-dimensional case) to assess cryptographic-security-relevant parameters.

Abstract

We study the shortest vector lengths in module lattices over arbitrary number fields, with an emphasis on cyclotomic fields. In particular, we sharpen the techniques of arXiv:2308.15275v2 to establish improved results for the variance of the number of lattice vectors of bounded Euclidean norm in a random module lattice. We then derive tight probabilistic bounds for the shortest vector lengths for several notions of random module lattice.
Paper Structure (13 sections, 18 theorems, 86 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Random module lattices
  3. Embeddings and unit structure
  4. The space of module lattices and components
  5. Lattices from codes and reduction to principal component
  6. Second moment results
  7. Additional symmetry and moments
  8. Integral formula and heights
  9. Unit counts and unit contribution
  10. Summing contributions over ideals
  11. Second moment for cyclotomic fields
  12. Shortest vector bounds
  13. Remarks on fixed number fields

Key Result

Theorem 1

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

Figures (1)

  • Figure 1: Log second moment errors for cyclotomic fields $\mathbb{Q}(\zeta_m)$ for $m=8,10,12,13,15,16$ and varying ranks $15\leq t\leq 32$.

Theorems & Definitions (37)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Definition 4
  • Remark 5
  • Remark 6
  • Theorem 7: R1956
  • Remark 8
  • Lemma 9
  • Lemma 10
  • ...and 27 more