Citation: Richter, M.; Bertram, M.;
Seidensticker, J.; Tschache, A. A
Mathematical Perspective on
Post-Quantum Cryptography.
Mathematics 2022, 10, 2579. https://
doi.org/10.3390/math10152579
Academic Editor: Angel
Martín-del-Rey
Received: 27 June 2022
Accepted: 21 July 2022
Published: 25 July 2022
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mathematics
Article
A Mathematical Perspective on Post-Quantum Cryptography
Maximilian Richter
1,
* , Magdalena Bertram
1
, Jasper Seidensticker
1
and Alexander Tschache
2
1
Secure Systems Engineering, Fraunhofer AISEC, 14199 Berlin, Germany;
magdalena.bertram@aisec.fraunhofer.de (M.B.); jasper.seidensticker@aisec.fraunhofer.de (J.S.)
2
Volkswagen AG, 38440 Wolfsburg, Germany; alexander.tschache@volkswagen.de
* Correspondence: maximilian.richter@aisec.fraunhofer.de
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; Saber; Dilithium; NTRU; Falcon; Classic McEliece; Rainbow; NIST
MSC: 11T71
1. Introduction
In recent years, significant progress in researching and building quantum computers
has been made. The existence of such computers threatens the security of many modern
cryptographic systems. This affects, in particular, asymmetric cryptography, i.e., KEMs
and digital signatures. By leveraging Shor’s quantum algorithm to find the period of a
function in a large group, a quantum computer can solve a distinct set of mathematical
problems. In particular, this includes integer factorization and the discrete logarithm, which
are the basis for a wide range of cryptographic schemes. Therefore, a fully fledged quantum
computer would be able to efficiently break the security of many modern cryptosystems.
To defend against this threat, the need for novel mathematical problems which are resistant
to Shor’s algorithm arises. Such problems are thereby promising candidates to withstand
the superior computing possibilities of quantum computers.
In 2016, NIST announced an open competition with the goal of finding and standard-
izing suitable algorithms for quantum-resistant cryptography. The standardization effort
by NIST is aimed at KEMs and digital signatures [
1
]. This process is currently in its third
round of candidate selection (April 2022).
At this point, the submitted algorithms are complex technical specifications without a
presentation of the underlying mathematical fundamentals and therefore do not allow an
easy access to these novel post-quantum algorithm approaches. As some of these algorithms
will probably become widely used in industrial areas very soon, it is vital to foster a broad
understanding of these mathematical concepts. In this document, we therefore address
the described lack of educational presentation. As we do not intend to give a detailed
Mathematics 2022, 10, 2579. https://doi.org/10.3390/math10152579 https://www.mdpi.com/journal/mathematics
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