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An Algebraic Proof of the Associative Law of Elliptic Curves
International College of Arts and Sciences, Yokohama City University, Yokohama, Japan
Takado, Yamagata, Japan
- 1 International College of Arts and Sciences, Yokohama City University, Yokohama, Japan
- 2 Takado, Yamagata, Japan
Advances in Pure Mathematics·Volume 07 (2017)·Pages 649–659·Published 11 December 2017·DOI10.4236/apm.2017.712040
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Abstract
In this paper we revisit the addition of elliptic curves and give an algebraic proof to the associative law by use of MATHEMATICA. The existing proofs of the associative law are rather complicated and hard to understand for beginners. An ‘‘elementary” proof to it based on algebra has not been given as far as we know. Undergraduates or non-experts can master the addition of elliptic curves through this paper. After mastering it they should challenge the elliptic curve cryptography.
KeywordsElliptic CurveAdditionAssociative LawMATHEMATICAElliptic Curve Cryptography
- Diffie, W. and Hellman, M. (1976) New Directions in Cryptography. IEEE Transactions on Information Theory, 22, 644-654. https://doi.org/10.1109/TIT.1976.1055638
- Rivest, R.L., Shamir, A. and Adleman, L. (1978) A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM, 21, 120-126. https://doi.org/10.1145/359340.359342
- ELGamal, T. (1985) A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. IEEE Transactions on Information Theory, 31, 469-472. https://doi.org/10.1109/TIT.1985.1057074
- Nakanishi, T. (2017) Mechanisms of Modern Cryptography. Kyoritsu Smart Selection 12, Kyoritsu Shuppan.
- Silverman, J.H. (2006) A Friendly Introduction to NUMBER THEORY. 3rd Edition, Pearson Education, London.
- Silverman, J.H. and Tate, J. (1992) Rational Points on Elliptic Curves. Springer-Verlag, Berlin. https://doi.org/10.1007/978-1-4757-4252-7
- Koblitz, N. (1987) Elliptic Curve Cryptosystems. Mathematics of Computation, 48, 203-209. https://doi.org/10.1090/S0025-5718-1987-0866109-5
- Fujii, K. (2014-2016) Public-Key Cryptography and Its Decoding by Quantum Computation (in Japanese). Lecture Note at Yokohama City University, Yokohama, 39.
- Shor, P.W. (1999) Polynomial—Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Review, 41, 303-332. https://doi.org/10.1137/S0036144598347011