Theoretical and Natural Science

- The Open Access Proceedings Series for Conferences


Theoretical and Natural Science

Vol. 12, 17 November 2023


Open Access | Article

Security of elliptic curve cryptosystems over Z_n

Ruoxi Hu 1 , Weihong Wu * 2
1 The Vanguard school
2 University of California

* Author to whom correspondence should be addressed.

Advances in Humanities Research, Vol. 12, 38-45
Published 17 November 2023. © 2023 The Author(s). Published by EWA Publishing
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Citation Ruoxi Hu, Weihong Wu. Security of elliptic curve cryptosystems over Z_n. TNS (2023) Vol. 12: 38-45. DOI: 10.54254/2753-8818/12/20230429.

Abstract

Elliptic curves over Galois fields are widely used in modern cryptography. Cryptosystems based on elliptic curves are commonly deemed more secure than RSA for a given key size. However, with the rapid progress of quantum computing, the security of this traditional systems faces unprecedented challenge. To address this concern, this paper explores the resilience of a generalization of traditional elliptic curve cryptography. That is, we explore elliptic curves over non-prime rings (Zn), instead of fields. Elliptic curves over Zn for a composite integer n has been considered by researchers on information security. However, it is unclear how they stand against the unparalleled power of quantum computers. This article investigates quantum attacks on cryptosystems based on this new paradigm. The conclusion sheds light on the pressing and important task of searching for post-quantum cryptographic systems. In particular, the effectiveness of Shor’s algorithm (or its variation) on such systems is analyzed.

Keywords

elliptic curve, cryptosystems, security

References

1. Sala, Massimiliano, and Daniele Taufer. ”The group structure of elliptic curves over Z/NZ.” arXiv:2010.15543 (2020).

2. Pradella, S. ”Introduction to Elliptic Curve Cryptography.” (2000).

3. Koyama, Kenji, et al. ”New public-key schemes based on elliptic curves over the ring Z n.” Annual International Cryptology Conference. Springer, Berlin, Heidelberg, 1991.

4. Silverman, J. H., and Tate, J. T. (1992). Rational points on elliptic curves (Vol. 9). New York: Springer-Verlag.

5. Blake, I., Seroussi, G., Seroussi, G., & Smart, N. (1999). Elliptic curves in cryptography (Vol. 265). Cambridge university press.

6. Nielsen, Michael A., and Isaac Chuang. ”Quantum computation and quantum information.” (2002): 558-559.

7. Preskill, John. ”Lecture notes for Physics 219: Quantum computation.” Caltech Lecture Notes (1999).

Data Availability

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

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Volume Title
Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation
ISBN (Print)
978-1-83558-135-3
ISBN (Online)
978-1-83558-136-0
Published Date
17 November 2023
Series
Theoretical and Natural Science
ISSN (Print)
2753-8818
ISSN (Online)
2753-8826
DOI
10.54254/2753-8818/12/20230429
Copyright
© 2023 The Author(s)
Open Access
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Copyright © 2023 EWA Publishing. Unless Otherwise Stated