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Deterministic Hardness of Approximation For SVP in all Finite $\ell_p$ Norms

Isaac M Hair, Amit Sahai

Abstract

We show that, assuming NP $\not\subseteq$ $\cap_{δ> 0}$DTIME$\left(\exp{n^δ}\right)$, the shortest vector problem for lattices of rank $n$ in any finite $\ell_p$ norm is hard to approximate within a factor of $2^{(\log n)^{1 - o(1)}}$, via a deterministic reduction. Previously, for the Euclidean case $p=2$, even hardness of the exact shortest vector problem was not known under a deterministic reduction.

Deterministic Hardness of Approximation For SVP in all Finite $\ell_p$ Norms

Abstract

We show that, assuming NP DTIME, the shortest vector problem for lattices of rank in any finite norm is hard to approximate within a factor of , via a deterministic reduction. Previously, for the Euclidean case , even hardness of the exact shortest vector problem was not known under a deterministic reduction.

Paper Structure

This paper contains 52 sections, 30 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Deterministic Hardness for $\ell_2$?
  3. A Barrier: Locally Dense Lattices.
  4. Another Barrier: Anti-Concentration Gadgets.
  5. Our Results.
  6. Technical Overview
  7. A Different Starting Problem
  8. A New Tensor Product
  9. Why are VF Tensor Products Useful?
  10. Shortcomings of Direct Tensoring.
  11. Constructing the VF Tensor Product.
  12. The GapSVP$_2$ Reduction
  13. Setting Up the Main Reduction
  14. Formal Statement of the Main Theorem.
  15. An Intermediate Lattice Problem.
  16. ...and 37 more sections

Key Result

Theorem 1.1

For all finite $p \geq 1$, $\gamma$-GapSVP$_p$ on lattices of rank $n$ is hard $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: (Left) An example 0/1 matrix $\mathbf{P}$ with $N = 3$ columns and $M = 3$ rows. (Right) The $2$-fold VF tensor product of $\mathbf{P}$, denoted as $\mathbf{T} = [\mathbf{A} \| \mathbf{Q}]$. Each (empty) white square is a zero entry, and each red bar is a row taken from a reduced Vandermonde matrix.
  • Figure 2: An example sequence of matrices $\mathbf{P}, \mathbf{Q}$, and $\mathbf{R}$. Each (empty) white square is a zero entry, and each red bar is a row taken from a reduced Vandermonde matrix.
  • Figure 3: An example matrix $\mathbf{C}$. As in Figure \ref{['fig:PQR']}, each (empty) white square is a zero entry, and each red bar is a row taken from a reduced Vandermonde matrix.

Theorems & Definitions (81)

  • Theorem 1.1: Due to khot2005hardnesshaviv2007tensor
  • Conjecture 1: Conjecture 7.1 from hecht2025deterministic
  • Theorem 1.2: Informal
  • Theorem 2.1: Informal
  • Remark 2.2
  • Definition 2.3: Reduced Vandermonde Matrix
  • Theorem 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm:main']} using Theorem \ref{['thm:constantgap']}.
  • Claim 3.3
  • ...and 71 more