A Mathematical Perspective on Post-Quantum Cryptography

Abstract

In 2016, the National Institute of Standards and Technology (NIST) announced an open competition with the goal of finding and standardizing suitable algorithms for quantum-resistant cryptography. This study presents a detailed, mathematically oriented overview of the round-three finalists of NIST’s post-quantum cryptography standardization consisting of the lattice-based key encapsulation mechanisms (KEMs) CRYSTALS-Kyber, NTRU and SABER; the code-based KEM Classic McEliece; the lattice-based signature schemes CRYSTALS-Dilithium and FALCON; and the multivariate-based signature scheme Rainbow. The above-cited algorithm descriptions are precise technical specifications intended for cryptographic experts. Nevertheless, the documents are not well-suited for a general interested mathematical audience. Therefore, the main focus is put on the algorithms’ corresponding algebraic foundations, in particular LWE problems, NTRU lattices, linear codes and multivariate equation systems with the aim of fostering a broader understanding of the mathematical concepts behind post-quantum cryptography.

Keywords

• post-quantum cryptography
• lattices
• learning with errors
• linear codes
• multivariate cryptography
• Kyber