@MISC{EC_PF_3.MIZ,
  AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
  TITLE = {Operations of Points on Elliptic Curve in Affine Coordinates},
  SECTION1 = {Set of points on elliptic curve in affine coordinates},
  SECTION2 = {Commutative property of operations of points on elliptic curve}
  DAY = {29},
  MONTH = {August},
  YEAR = {2019},
  ADDRESS1 = {Tokyo University of Technology, Tokyo, JAPAN},
  ADDRESS2 = {Shinshu University, Nagano, JAPAN},
  ADDRESS3 = {Shinshu University, Nagano, JAPAN},
  SUMMARY = {In this article, we formalize a binary operation of points on
  an elliptic curve over $\bf{GF}(p)$ in affine coordinates.
  We show that the operation is unital, complementable and commutative.
  Elliptic curve cryptography \cite{BSS99}, whose security is based on
  a difficulty of discrete logarithm problem of elliptic curves,
  is important for information security.},
  NOTE = {This work was supported by JSPS KAKENHI JP15K00183 and JP17K00182.}}

@BOOK{BSS99,
      AUTHOR = {I. Blake, G. Seroussi and N. Smart},
      TITLE = {Elliptic Curves in Cryptography},
      PUBLISHER = {Cambridge University Press},
      SERIES = {London Mathematical Society Lecture Note Series},
      NUMBER = {265},
      YEAR = {1999}}

