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Simultaneous Approximation for Lattice-Based Cryptography

Julia VanLandingham

TL;DR

These reductions show that instances of these problems in SA lattices are just as hard as general instances and thus are interesting new problems to consider for use in cryptography.

Abstract

We define two new problems called SIAP and CAP related to solving SIVP and CVP in a subset of lattices called Simultaneous Approximation (SA) lattices. We give dimension- and gap-preserving, deterministic polynomial-time and space reductions from SVP$_γ$, SIVP$_γ$, and CVP$_γ$ to their corresponding problems in SA lattices. These reductions show that instances of these problems in SA lattices are just as hard as general instances and thus are interesting new problems to consider for use in cryptography. We also show that the reductions are optimal in regards to integer inflation.

Simultaneous Approximation for Lattice-Based Cryptography

TL;DR

These reductions show that instances of these problems in SA lattices are just as hard as general instances and thus are interesting new problems to consider for use in cryptography.

Abstract

We define two new problems called SIAP and CAP related to solving SIVP and CVP in a subset of lattices called Simultaneous Approximation (SA) lattices. We give dimension- and gap-preserving, deterministic polynomial-time and space reductions from SVP, SIVP, and CVP to their corresponding problems in SA lattices. These reductions show that instances of these problems in SA lattices are just as hard as general instances and thus are interesting new problems to consider for use in cryptography. We also show that the reductions are optimal in regards to integer inflation.
Paper Structure (11 sections, 26 theorems, 34 equations, 4 figures, 4 algorithms)

This paper contains 11 sections, 26 theorems, 34 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Approximating with SA Lattices
  2. Intuition
  3. Algorithm
  4. Integer Inflation
  5. Gap-Preservation
  6. Time-Complexity
  7. Problem Reductions
  8. SVP to SAP
  9. SIVP to SIAP
  10. CVP to CAP
  11. Optimal Integer Inflation

Key Result

Theorem 1

Algorithms SVP_reduction, SIVP_reduction, and CVP_reduction are dimension- and gap-preserving, deterministic polynomial-time reductions from SVP$_\gamma$, SIVP$_\gamma$, and CVP$_\gamma$, to SAP$_\gamma$, SIAP$_\gamma$, and CAP$_\gamma$, respectively, with input length scaled by $\mathcal{O}(n^2\log

Figures (4)

  • Figure : Approximating a general lattice by an SA lattice
  • Figure : SVP$_\gamma$ to SAP$_\gamma$
  • Figure : SIVP$_\gamma$ to SIAP$_\gamma$
  • Figure : CVP$_\gamma$ to CAP$_\gamma$

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • ...and 36 more