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Gaussian Integers

Yuichi Futa;  Hiroyuki Okazaki;  Daichi Mizushima;  Yasunari Shidama — 2013

Formalized Mathematics

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Operations of Points on Elliptic Curve in Projective Coordinates

Yuichi Futa;  Hiroyuki Okazaki;  Daichi Mizushima;  Yasunari Shidama — 2012

Formalized Mathematics

In this article, we formalize operations of points on an elliptic curve over GF(p). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo and addellProjCo are unary and binary operations of a point over the elliptic curve.

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