Simultaneous Approximation for Lattice-Based Cryptography

TL;DR

This paper introduces new lattice problems related to simultaneous approximation, demonstrating their computational hardness and potential cryptographic applications through polynomial-time reductions from well-known lattice problems.

Contribution

It defines the SIAP and CAP problems, establishes their equivalence in difficulty to classical lattice problems via reductions, and shows these are promising for cryptography.

Findings

SIAP and CAP are as hard as SIVP and CVP in general lattices.

Reductions are dimension- and gap-preserving, deterministic, polynomial-time.

Reductions are optimal concerning integer inflation.

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$_\gamma$, SIVP$_\gamma$, and CVP$_\gamma$ 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.