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The Mathematical Foundation of Post-Quantum Cryptography.

Chuanming Zong1

  • 1Center for Applied Mathematics, Tianjin University, Tianjin 300072, China.

Research (Washington, D.C.)
|August 28, 2025
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Summary

Quantum computing threatens current encryption. This paper explores the mathematical foundations of post-quantum cryptography, linking lattice problems like SVP and CVP to ball packing and quadratic forms.

Area of Science:

  • Cryptography
  • Quantum Computing
  • Number Theory

Background:

  • Quantum algorithms developed by P. Shor in 1994 and the advent of quantum computers pose a significant threat to current secret communication methods like RSA and ElGamal.
  • The National Institute of Standards and Technology (NIST) is standardizing post-quantum cryptography (PQC) to address this crisis, with candidates based on lattice theory and hash functions.

Purpose of the Study:

  • To provide a review article on the mathematical foundations of post-quantum cryptography complexity theory.
  • To demonstrate the mathematical roots of PQC in fundamental problems such as ball packing, ball covering, and positive definite quadratic forms.

Main Methods:

  • Introduction to post-quantum cryptography (PQC).
  • Demonstration of the mathematical connections between lattice-based cryptosystems and computational problems.

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  • Explanation of the relationship between the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and positive definite quadratic forms.

    • Main Results:

    • NIST has announced initial PQC standards (FIPS 203, 204, 205) based on CRYSTALS-Kyber, CRYSTALS-Dilithium, and Sphincs+.
    • The security of lattice-based cryptosystems is fundamentally linked to the hardness of SVP and CVP.
    • SVP and CVP can be understood as ball packing and ball covering problems, respectively, and are equivalent to arithmetic problems involving positive definite quadratic forms.

    Conclusions:

    • Understanding the mathematical underpinnings of lattice problems is crucial for developing and analyzing post-quantum cryptographic systems.
    • This work bridges the gap between abstract mathematical concepts and their practical application in securing communications against quantum threats.