Operations of Points on Elliptic Curve in Affine Coordinates

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

doi:10.2478/forma-2019-0026

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Operations of Points on Elliptic Curve in Affine Coordinates

Formalized Mathematics

Volume 27 (2019): Issue 3 (October 2019)

By: Yuichi Futa, Hiroyuki Okazaki and Yasunari Shidama

Open Access

|Feb 2020

Abstract

In this article, we formalize in Mizar [1], [2] a binary operation of points on an elliptic curve over GF(p) in affine coordinates. We show that the operation is unital, complementable and commutative. Elliptic curve cryptography [3], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.

References

[1] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.10.1007/978-3-319-20615-8_17
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[2] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.10.1007/s10817-017-9440-6604425130069070
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[4] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Set of points on elliptic curve in projective coordinates. Formalized Mathematics, 19(3):131–138, 2011. doi:10.2478/v10037-011-0021-6.10.2478/v10037-011-0021-6
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[5] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Operations of points on elliptic curve in projective coordinates. Formalized Mathematics, 20(1):87–95, 2012. doi:10.2478/v10037-012-0012-2.10.2478/v10037-012-0012-2
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Articles in this issue

DOI: https://doi.org/10.2478/forma-2019-0026 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 315 - 320

Accepted on: Aug 29, 2019

Published on: Feb 17, 2020

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

elliptic curve,

commutative operation

Related subjects:

Computer sciences,

Computer sciences, other,

Mathematics,

General mathematics

© 2020 Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 License.

Volume 27 (2019): Issue 3 (October 2019)