The Mathematical Foundation of Post-Quantum Cryptography

TL;DR

This paper explores the mathematical foundations of post-quantum cryptography, focusing on lattice-based schemes, their relation to sphere packing and covering problems, and quadratic forms, highlighting the underlying complexity assumptions.

Contribution

It provides a concise overview of the mathematical structures underpinning lattice-based post-quantum cryptography and their connections to classical geometric and algebraic problems.

Findings

Lattice cryptography security relies on SVP and CVP hardness.

SVP and CVP are equivalent to sphere packing and covering problems.

Connections between lattice problems and quadratic forms are established.

Abstract

On July 5, 2022, the National Institute of Standards and Technology announced four possible post-quantum cryptography standards, three of them are based on lattice theory and the other one is based on Hash function. It is well-known that the security of the lattice cryptography relies on the hardness of the shortest vector problem (SVP) and the closest vector problem (CVP). In fact, the SVP is a sphere packing problem and the CVP is a sphere covering problem. Furthermore, both SVP and CVP are equivalent to arithmetic problems of positive definite quadratic forms. This paper will briefly introduce the post-quantum cryptography and show its connections with sphere packing, sphere covering, and positive definite quadratic forms.