Post-Quantum Cryptography: Mathematical Foundations and Future Challenges
Research Article
Open Access
CC BY

Post-Quantum Cryptography: Mathematical Foundations and Future Challenges

Yiqing Jiang 1*
1 Wake Forest University, 1834 Wake Forest Rd, Winston-Salem, NC, 27109
*Corresponding author: yiqingjiang21@gmail.com
Published on 24 July 2025
Volume Cover
TNS Vol.125
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-80590-233-1
ISBN (Online): 978-1-80590-234-8
Download Cover

Abstract

Modern public-key cryptography relies on the hardness of mathematical problems such as integer factorization and discrete logarithms. However, the development of quantum computing poses an imminent threat to these assumptions. Shor’s algorithm, in particular, can factor large semiprimes exponentially faster than classical algorithms, compromising systems like RSA, DSA, and ECC. This paper explores the mathematical foundations of pre-quantum cryptography, discusses the limitations of classical security models when confronted with quantum capabilities, and then pays attention to post-quantum cryptography (PQC), a field dedicated to developing cryptographic schemes resilient against both classical and quantum attacks. Among the proposed families, this paper focuses specifically on hash function–based cryptography for its simplicity and minimal reliance on algebraic structure. This study focuses in particular on SPHINCS+, a stateless hash-based digital signature scheme currently under consideration by NIST. Through detailed mathematical explanation and a visual example, we analyze its construction using Winternitz One-Time Signatures and Merkle trees. The results highlight SPHINCS+ as a robust candidate for post-quantum security due to its reliance on well-understood hash primitives and its resistance to known quantum algorithms such as Grover’s. Finally, this paper discusses ongoing challenges such as performance trade-offs, standardization, and real-world deployment. This research underscores the urgency of adopting quantum-resistant cryptographic systems before large-scale quantum computers become a reality.

Keywords:

Post-Quantum Cryptography, Shor’s Algorithm, Lattice-Based Cryptography, Hash-Based Signatures, Cryptographic Security

View PDF
Jiang,Y. (2025). Post-Quantum Cryptography: Mathematical Foundations and Future Challenges. Theoretical and Natural Science,125,65-72.

References

[1]. Evgeny Milanov. (2009). The RSA Algorithm. RSA Laboratories, 1.11.

[2]. M. Guru Vimal Kumar and U. S. Ragupathy. (2016). “A Survey on Current Key Issues and Status in Cryptography.” In: IEEE WiSPNET Conference. doi: 10.1109/WiSPNET.2016.7566435.

[3]. Jon R. Lindsay. (2020). Surviving the Quantum Cryptocalypse. Strategic Studies Quarterly, 14(2): 49–73. https: //www.jstor.org/stable/26915277

[4]. P. W. Shor. (1994). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. doi: 10.1109/SFCS.1994.365700

[5]. Ruben Niederhagen and Michael Waidner. (2017). “Practical Post-Quantum Cryptography.” Fraunhofer SIT.

[6]. Ritik Bavdekar et al. (2022). “Post-Quantum Cryptography: Techniques, Challenges, Standardization, and Directions for Future Research.” arXiv preprint arXiv: 2202.02826.

[7]. Matthew Edward Briggs. (1998). “An Introduction to the General Number Field Sieve.” PhD Thesis, Virginia Tech.

[8]. Eric W. Weisstein. (2002). Number Field Sieve. MathWorld—A Wolfram Web Resource. https: //mathworld.wolfram.com/NumberFieldSieve.html

[9]. Matthew Hayward. (2008). “Quantum Computing and Shor’s Algorithm.” Technical Report No. 1, Macquarie University Mathematics Department, Sydney.

[10]. David Beckman, Amalavoyal N. Chari, Srikrishna Devabhaktuni, and John Preskill. (1996). “Efficient Networks for Quantum Factoring.” Physical Review A, 54(2), 1034–1063.

[11]. Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman. (2008). An Introduction to Mathematical Cryptography. Springer, Vol. 1.

[12]. Daniel J. Bernstein and Tanja Lange. (2017). “Post-Quantum Cryptography.” Nature, 549(7671), 188–194.

References

[1]. Evgeny Milanov. (2009). The RSA Algorithm. RSA Laboratories, 1.11.

[2]. M. Guru Vimal Kumar and U. S. Ragupathy. (2016). “A Survey on Current Key Issues and Status in Cryptography.” In: IEEE WiSPNET Conference. doi: 10.1109/WiSPNET.2016.7566435.

[3]. Jon R. Lindsay. (2020). Surviving the Quantum Cryptocalypse. Strategic Studies Quarterly, 14(2): 49–73. https: //www.jstor.org/stable/26915277

[4]. P. W. Shor. (1994). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. doi: 10.1109/SFCS.1994.365700

[5]. Ruben Niederhagen and Michael Waidner. (2017). “Practical Post-Quantum Cryptography.” Fraunhofer SIT.

[6]. Ritik Bavdekar et al. (2022). “Post-Quantum Cryptography: Techniques, Challenges, Standardization, and Directions for Future Research.” arXiv preprint arXiv: 2202.02826.

[7]. Matthew Edward Briggs. (1998). “An Introduction to the General Number Field Sieve.” PhD Thesis, Virginia Tech.

[8]. Eric W. Weisstein. (2002). Number Field Sieve. MathWorld—A Wolfram Web Resource. https: //mathworld.wolfram.com/NumberFieldSieve.html

[9]. Matthew Hayward. (2008). “Quantum Computing and Shor’s Algorithm.” Technical Report No. 1, Macquarie University Mathematics Department, Sydney.

[10]. David Beckman, Amalavoyal N. Chari, Srikrishna Devabhaktuni, and John Preskill. (1996). “Efficient Networks for Quantum Factoring.” Physical Review A, 54(2), 1034–1063.

[11]. Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman. (2008). An Introduction to Mathematical Cryptography. Springer, Vol. 1.

[12]. Daniel J. Bernstein and Tanja Lange. (2017). “Post-Quantum Cryptography.” Nature, 549(7671), 188–194.

Cite this article

Jiang,Y. (2025). Post-Quantum Cryptography: Mathematical Foundations and Future Challenges. Theoretical and Natural Science,125,65-72.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of CONF-APMM 2025 Symposium: Multi-Qubit Quantum Communication for Image Transmission over Error Prone Channels

ISBN: 978-1-80590-233-1(Print) / 978-1-80590-234-8(Online)
Editor: Anil Fernando
Conference website: https://2025.confapmm.org/
Conference date: 29 August 2025
Series: Theoretical and Natural Science
Volume number: Vol.125
ISSN: 2753-8818(Print) / 2753-8826(Online)

© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Authors who publish this series agree to the following terms:

1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.

2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.

3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open access policy for details).

For authors
For editors
For reviewers

Cookies are used by this site.
Copyright © 2024 EWA Publishing, its licensors, and contributors. All rights reserved, including text and data mining, AI training, and similar technologies.