Group Structure of Special Parabola and Its Application in Cryptography

Bin Li

doi:doi:10.11648/j.acm.20190806.11

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Group Structure of Special Parabola and Its Application in Cryptography

Bin Li

Published in Applied and Computational Mathematics (Volume 8, Issue 6)

Received: 5 November 2019 Accepted: 28 November 2019 Published: 9 December 2019

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Abstract

Public key cryptography is one of the most important research contents in modern cryptography. Curve-based public key cryptosystems have attracted widespread attention in recent years because they have more obvious advantages in speed and key length than general public key cryptosystems. People have done a lot of research on elliptic cryptosystem, among which the realization of elliptic cryptosystem is a key content. In this paper, the definition of special parabola in algebraic closed domain is proposed, the group structure of special parabola in finite field is studied, and several forms of public key cryptosystem based on this parabola are given. The results show that the parabola, together with the additive operations defined above, form an Abelian group. The radix of this parabola can be easily determined, so that the factors it contains can be large prime. The security of its public key cryptosystem is based on the difficulty of solving the discrete logarithm problem on this parabola. Moreover, these parabolic public key cryptosystems are easy to code and decode in plaintext, and easier to design and implement than elliptic curve public key cryptosystems.

Published in	Applied and Computational Mathematics (Volume 8, Issue 6)
DOI	10.11648/j.acm.20190806.11
Page(s)	88-94
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Special Parabola, Group Structure, Public Key Cryptosystem, Finite Field, Discrete Logarithm

References

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[4]	Y. C. Wang, Y. Ikematsu and D. N. Duong, “The secure parameters and efficient decryption algorithm for multivariate public key cryptosystem EFC” IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E102A (2019), 1028-1036.
[5]	V. S. Miller, “Use of elliptic curves in cryptography,” Advances in Cryptology-CRYPTO 1985, Berlin: Springer-Verlag, 417-426, 1986.
[6]	N. Koblitz, “Elliptic curve cryptosystems,” Mathematics of computiation, 48 (1987), 203-209.
[7]	S. Goldwasser and J. Kilian, “Almost all primes can be quickly certified,” Proceeding of the 18th STOC of the ACM. Berkeley: AFIPS press, 316-329, 1986.
[8]	H. W. Lenstra, “Factoring integers with elliptic curves,” Annals of Mathematatics, 126 (1987), 649-673.
[9]	H. C. Tilborg, “An introduction to cryptology,” Boston: Kluwer Academic Publishers, 1988.
[10]	G. Frey, M. Müller and H. Rück, “The tate pairing and the discrete logarithm applied to elliptic curve cryptosystems,” IEEE Transaction on Information Theory, 45 (1999), 1717-1719.
[11]	S. Erickson, M. J. Jacobson and N. Shang, “Explicit formulas for real hyperelliptic curves of genus 2 in affine representation,” WAIFI 2007, New York: Springer-Verlag, 202-218, 2007.
[12]	J. Miret, R. Moreno and J. Pujolas, “Halving for the 2-Sylow subgroup of genus 2 curves over binary fields,” Finite Fields Applicatae, 15 (2009), 569-579.
[13]	N. Smart, “A comparison of different finite fields for elliptic curve cryptosystems,” Computers and Mathematics with Applications, 42 (2001), 91-100.
[14]	J. Proos and C. Zalka, “Shor’s discrete logarithm quantum algorithm for elliptic curves,” Quantum Information and Computation, 3 (2003), 317-344.
[15]	N. Philip and D. Steven, “Point compression for Koblitz elliptic curves,” Advances in Mathematics of Communications, 5 (2011), 1-10.
[16]	H. B. Zhao, L. Y. Qian and L. F. Jin, “A new mceliece cryptosystem based on subfield subcode of elliptic curve code,” Computer Applications and Softwre, 36 (2019), 317-322.
[17]	C. Y. Wang, G. A. Xu and J. Sun, “An enhanced three-factor user authentication scheme using elliptic curve cryptosystem for wireless sensor networks,” Sensors, 17 (2017), 1018-1027.
[18]	Z. Y. Liu, T. C. Xia and J. B. Wang, “Fractional two-dimensional discrete chaotic map and its applications to the information security with elliptic-curve public key cryptography,” Journal of Vibration and Control, 24 (2018), 4797-4824.
[19]	L. D. Han, X. Tan and S. B. Wang, “An efficient and secure three-factor based authenticated key exchange scheme using elliptic curve cryptosystems,” Peer-to-peer Networking and Applications, 11 (2018), 63-73.
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APA Style

Bin Li. (2019). Group Structure of Special Parabola and Its Application in Cryptography. Applied and Computational Mathematics, 8(6), 88-94. https://doi.org/10.11648/j.acm.20190806.11

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ACS Style

Bin Li. Group Structure of Special Parabola and Its Application in Cryptography. Appl. Comput. Math. 2019, 8(6), 88-94. doi: 10.11648/j.acm.20190806.11

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AMA Style

Bin Li. Group Structure of Special Parabola and Its Application in Cryptography. Appl Comput Math. 2019;8(6):88-94. doi: 10.11648/j.acm.20190806.11

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@article{10.11648/j.acm.20190806.11,
  author = {Bin Li},
  title = {Group Structure of Special Parabola and Its Application in Cryptography},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {6},
  pages = {88-94},
  doi = {10.11648/j.acm.20190806.11},
  url = {https://doi.org/10.11648/j.acm.20190806.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190806.11},
  abstract = {Public key cryptography is one of the most important research contents in modern cryptography. Curve-based public key cryptosystems have attracted widespread attention in recent years because they have more obvious advantages in speed and key length than general public key cryptosystems. People have done a lot of research on elliptic cryptosystem, among which the realization of elliptic cryptosystem is a key content. In this paper, the definition of special parabola in algebraic closed domain is proposed, the group structure of special parabola in finite field is studied, and several forms of public key cryptosystem based on this parabola are given. The results show that the parabola, together with the additive operations defined above, form an Abelian group. The radix of this parabola can be easily determined, so that the factors it contains can be large prime. The security of its public key cryptosystem is based on the difficulty of solving the discrete logarithm problem on this parabola. Moreover, these parabolic public key cryptosystems are easy to code and decode in plaintext, and easier to design and implement than elliptic curve public key cryptosystems.},
 year = {2019}
}

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TY  - JOUR
T1  - Group Structure of Special Parabola and Its Application in Cryptography
AU  - Bin Li
Y1  - 2019/12/09
PY  - 2019
N1  - https://doi.org/10.11648/j.acm.20190806.11
DO  - 10.11648/j.acm.20190806.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 88
EP  - 94
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20190806.11
AB  - Public key cryptography is one of the most important research contents in modern cryptography. Curve-based public key cryptosystems have attracted widespread attention in recent years because they have more obvious advantages in speed and key length than general public key cryptosystems. People have done a lot of research on elliptic cryptosystem, among which the realization of elliptic cryptosystem is a key content. In this paper, the definition of special parabola in algebraic closed domain is proposed, the group structure of special parabola in finite field is studied, and several forms of public key cryptosystem based on this parabola are given. The results show that the parabola, together with the additive operations defined above, form an Abelian group. The radix of this parabola can be easily determined, so that the factors it contains can be large prime. The security of its public key cryptosystem is based on the difficulty of solving the discrete logarithm problem on this parabola. Moreover, these parabolic public key cryptosystems are easy to code and decode in plaintext, and easier to design and implement than elliptic curve public key cryptosystems.
VL  - 8
IS  - 6
ER  -

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Author Information

Bin Li

School of Mathematics, Chengdu Normal University, Chengdu, China

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APA Style

Bin Li. (2019). Group Structure of Special Parabola and Its Application in Cryptography. Applied and Computational Mathematics, 8(6), 88-94. https://doi.org/10.11648/j.acm.20190806.11

Copy | Download

ACS Style

Bin Li. Group Structure of Special Parabola and Its Application in Cryptography. Appl. Comput. Math. 2019, 8(6), 88-94. doi: 10.11648/j.acm.20190806.11

Copy | Download

AMA Style

Bin Li. Group Structure of Special Parabola and Its Application in Cryptography. Appl Comput Math. 2019;8(6):88-94. doi: 10.11648/j.acm.20190806.11

Copy | Download

@article{10.11648/j.acm.20190806.11,
  author = {Bin Li},
  title = {Group Structure of Special Parabola and Its Application in Cryptography},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {6},
  pages = {88-94},
  doi = {10.11648/j.acm.20190806.11},
  url = {https://doi.org/10.11648/j.acm.20190806.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190806.11},
  abstract = {Public key cryptography is one of the most important research contents in modern cryptography. Curve-based public key cryptosystems have attracted widespread attention in recent years because they have more obvious advantages in speed and key length than general public key cryptosystems. People have done a lot of research on elliptic cryptosystem, among which the realization of elliptic cryptosystem is a key content. In this paper, the definition of special parabola in algebraic closed domain is proposed, the group structure of special parabola in finite field is studied, and several forms of public key cryptosystem based on this parabola are given. The results show that the parabola, together with the additive operations defined above, form an Abelian group. The radix of this parabola can be easily determined, so that the factors it contains can be large prime. The security of its public key cryptosystem is based on the difficulty of solving the discrete logarithm problem on this parabola. Moreover, these parabolic public key cryptosystems are easy to code and decode in plaintext, and easier to design and implement than elliptic curve public key cryptosystems.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Group Structure of Special Parabola and Its Application in Cryptography
AU  - Bin Li
Y1  - 2019/12/09
PY  - 2019
N1  - https://doi.org/10.11648/j.acm.20190806.11
DO  - 10.11648/j.acm.20190806.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 88
EP  - 94
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20190806.11
AB  - Public key cryptography is one of the most important research contents in modern cryptography. Curve-based public key cryptosystems have attracted widespread attention in recent years because they have more obvious advantages in speed and key length than general public key cryptosystems. People have done a lot of research on elliptic cryptosystem, among which the realization of elliptic cryptosystem is a key content. In this paper, the definition of special parabola in algebraic closed domain is proposed, the group structure of special parabola in finite field is studied, and several forms of public key cryptosystem based on this parabola are given. The results show that the parabola, together with the additive operations defined above, form an Abelian group. The radix of this parabola can be easily determined, so that the factors it contains can be large prime. The security of its public key cryptosystem is based on the difficulty of solving the discrete logarithm problem on this parabola. Moreover, these parabolic public key cryptosystems are easy to code and decode in plaintext, and easier to design and implement than elliptic curve public key cryptosystems.
VL  - 8
IS  - 6
ER  -

Copy | Download