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Security of elliptic curve cryptosystems over Z_n
- 1 The Vanguard school
- 2 University of California
* Author to whom correspondence should be addressed.
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
[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).
Cite this article
Hu,R.;Wu,W. (2023).Security of elliptic curve cryptosystems over Z_n.Theoretical and Natural Science,12,38-45.
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Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation
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