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On the Maximum Distance Sublattice Problem and Closest Vector Problem

Rajendra Kumar, Shashank K Mehta, Mahesh Sreekumar Rajasree

TL;DR

The paper defines the Maximum Distance Sublattice Problem (MDSP) and reveals its relationship to the Closest Vector Problem (CVP), including a reduction to CVP on the dual lattice. It further develops a novel dual-free reduction from MDSP to CVP that preserves both rank and dimension, providing a polynomial-time, Karp reduction in either direction. The main technical contribution is a constructive, geometry-based reduction between MDSP and CVP that avoids dual lattice concepts while maintaining equivalence of optimal solutions. These results deepen our understanding of the MDSP–CVP connection and may influence algorithmic approaches and cryptanalytic considerations for lattice problems.

Abstract

In this paper, we introduce the Maximum Distance Sublattice Problem (MDSP). We observed that the problem of solving an instance of the Closest Vector Problem (CVP) in a lattice $\mathcal{L}$ is the same as solving an instance of MDSP in the dual lattice of $\mathcal{L}$. We give an alternate reduction between the CVP and MDSP. This alternate reduction does not use the concept of dual lattice.

On the Maximum Distance Sublattice Problem and Closest Vector Problem

TL;DR

The paper defines the Maximum Distance Sublattice Problem (MDSP) and reveals its relationship to the Closest Vector Problem (CVP), including a reduction to CVP on the dual lattice. It further develops a novel dual-free reduction from MDSP to CVP that preserves both rank and dimension, providing a polynomial-time, Karp reduction in either direction. The main technical contribution is a constructive, geometry-based reduction between MDSP and CVP that avoids dual lattice concepts while maintaining equivalence of optimal solutions. These results deepen our understanding of the MDSP–CVP connection and may influence algorithmic approaches and cryptanalytic considerations for lattice problems.

Abstract

In this paper, we introduce the Maximum Distance Sublattice Problem (MDSP). We observed that the problem of solving an instance of the Closest Vector Problem (CVP) in a lattice is the same as solving an instance of MDSP in the dual lattice of . We give an alternate reduction between the CVP and MDSP. This alternate reduction does not use the concept of dual lattice.

Paper Structure

This paper contains 5 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Our Contributions:
  3. Organisation:
  4. Preliminaries
  5. New Reduction between $\mathsf{MDSP}$ and $\mathsf{CVP}$

Key Result

Theorem 1

There exists a polynomial time rank-preserving dimension-preserving many-one (Karp) reduction between $\mathsf{MDSP}$ and $\mathsf{CVP}$.

Theorems & Definitions (20)

  • Theorem 1
  • Definition 1: Lattice
  • Definition 2: Unimodular Matrix
  • Theorem 2
  • Definition 3: Dual Lattice
  • Claim 1
  • Definition 4: Shortest Vector Problem ($\mathsf{SVP}$)
  • Definition 5: Closest Vector Problem ($\mathsf{CVP}$)
  • Definition 6
  • Definition 7: Maximum Distance Sublattice Problem ($\mathsf{MDSP}$)
  • ...and 10 more