The review on elliptic curves as cryptographic pairing groups

Document Type : Original Article

Author

Department of Mathematics, Islamic Azad university, Shahr-e-Qods Branch, Tehran, Iran.

Abstract

Elliptic curve is a set of two variable points on polynomials of degree 3 over a field acted
by an addition operation that forms a group structure. The motivation of this study is that the
mathematics behind that elliptic curve to the applicability within a cryptosystem. Nowadays, pair-
ings bilinear maps on elliptic curve are popular to construct cryptographic protocol pairings help to
transform a discrete logarithm problem on an elliptic curve to the discrete logarithm problem in nite
elds. The purpose of this paper is to introduce elliptic curve, bilinear pairings on elliptic curves as
based on pairing cryptography. Also this investigation serves as a basis in guiding anyone interested
to understand one of the applications of group theory in cryptosystem.

Keywords


[1] G. Adj, O. Ahmadi, A. Menezes, On isogeny graphs of supersingular elliptic curves over  nite  elds,
Finite Fields and Their Applications, 55 2019, 268-283.
[2] S. Akleylek, B.B. Kirlar, O. Sever and Z. Yuce, Pairing-based cryptography: A Survey, 3rd information
security and cryptology conference, 2008.
[3] R. Balasubramanian, N. Koblitz, The improbability that an elliptic curve has subexponential discrete
log problem under the Menezes-Dkamoto-Vanstone algorithm, Journal of cryptology, 11(2) 1998, 141-
145.
[4] P. Barreto, B. Lynn, M. Scott, Efficient implemention of pairing-based cryptosystem, Journal of
Cryptology, 17(4) 2004, 321-334.
[5] P.S.L.M. Barreto, M. Naehrig, Pairing-friendly elliptic curves of prime order, International Workshop
on Selected Areas in Cryptography. Springer, Berlin, Heidelberg, 2005.
[6] B. Den Boer, Diffie-Hellman is as strong as discrete log for certain primes, Lecture Notes in Computer
Science, 403 1996, 530{539.
[7] L. Chen, Z. Cheng, N. P. Smart, Identity-based key agreement protocols from pairings, International
Journal of Information Security, 6(4) 2007, 213-241.
[8] C. Cocks, R.G.E. Pinch, Identity-based cryptosystems based on the Weil pairing, unpublished
manuscript, 2001.
[9] W. Diffie, M. Hellman, New directions in cryptography, IEEE Transactions on Information Theory,
22(6) 1976.
[10] C. Costello , Pairing for beginners, A Note, 2013.
[11] P. Duan, S. Cui, C. Chan, Finding More Non-Supersingular Elliptic Curves for Pairing-Based Cryp-
tosystems, Technology, 2(2) 2005, 157-163.
[12] A. Enge, J. Milan, Implementing cryptographic pairings at standard security levels, International
Conference on Security, Privacy, and Applied Cryptography Engineering. Springer, Cham, 2014.
[13] D. Freeman, M. Scott, E. Teske, A taxonomy of pairing-friendly elliptic curves, Journal of cryptology,
23(2) 2010, 224-280.
[14] D. Freeman, Constructing pairing-friendly elliptic curves with embedding degree 10, International
Algorithmic Number Theory Symposium, Springer, Berlin, Heidelberg, 2006.
[15] G. Frey, H. Ruck, A remark concerning m-advisibility and the discrete logarithm in the divisor class
group of curves, Mathematics of computation, 62(206) 1994, 865-874.
[16] S.D. Galbraith, K. G. Paterson, P.N. Smart, Pairings for cryptographers, Discrete Applied Mathe-
matics, 156(16) 2008, 3113-3121.
[17] S.D. Galbraith, F. Vercauteren, Computational problems in supersingular elliptic curve, Quantum
Information Processing, 17(10) 2018, 1-22.
[18] S.D. Galbraith, K. Paterson, editors, Pairing Based Cryptography-Pairing 2008, Second International
Conference, Egham, UK, September 1-3, 2008, Proceedings. Vol. 5209. Springer, 2008.
[19] F. Hess, Efficient Identity Based Signature Schemes Based on Pairings, Lecture Notes in Computer
Science, 2595 2003, 310-324.
[20] F. Hess, N.P. Smart, F. Vercauteren, The eta pairing revisited, IEEE Transactions on Information
Theory, 52(10) 2006, 4595-4602.
[21] A. Joux, A one round protocol for tripartite Diffie-Hellman, Journal of cryptology, 17(4) 2004, 263-276.
[22] A. Menezes, T. Okamoto, S. Vanstone, Reducing elliptic curve logarithms to logarithms in a  nite
 eld, IEEE Transactions on Information Theory, 39(5) 1993, 1639-1646.
[23] V. Miller, The Weil pairing, and its efficient calculation, Journal of cryptology, 17(4) 2004, 235-261.
[24] A. Miyaji, M. Nakabayashi, S. Takano, New explicit conditions of elliptic curves traces for FR-
reduction, IEICE transactions on fundamentals of electronics, communications and computer sciences,
84(5) 2001, 1234-1243.
[25] J. Pollard, Monte Carlo methods for index computation mod p, Mathematics of computation,
32(143)1978, 918-924.
[26] H. Shacham, New Paradigms in Signature Schemes, PhD thesis, Stanford, 2006.
[27] H. Silverman Joseph, The arithmetic of elliptic curves, Graduate texts in Mathematics, Springer
Verlag , 2008.
[28] O. Uzunkol, M.S. Kiraz, Still wrong use of pairing in cryptography, Applied Mathematics and Com-
putation, 333(C) 2018, 467-479.
[29] F. Vercauteren, Optimal pairings, IEEE Transactions on Information Theory, 56(1) 2009, 455-461.