12,929 research outputs found

    Set of Points on Elliptic Curve in Projective Coordinates

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    In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Józef BiaƂas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.CzesƂaw ByliƄski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesƂaw ByliƄski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesƂaw ByliƄski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaƂ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.G. Seroussi I. Blake and N. Smart. Elliptic Curves in Cryptography. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.Eugeniusz Kusak, Wojciech LeoƄczuk, and MichaƂ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.RafaƂ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Konrad Raczkowski and PaweƂ Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.MichaƂ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990

    Operations of Points on Elliptic Curve in Affine Coordinates

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    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.This work was supported by JSPS KAKENHI Grant Numbers JP15K00183 and JP17K00182.Yuichi Futa - Tokyo University of Technology, Tokyo, JapanHiroyuki Okazaki - Shinshu University, Nagano, JapanYasunari Shidama - Shinshu University, Nagano, JapanGrzegorz Bancerek, CzesƂaw Bylinski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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.Grzegorz Bancerek, CzesƂaw Bylinski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. 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.I. Blake, G. Seroussi, and N. Smart. Elliptic Curves in Cryptography. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.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.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.Artur KorniƂowicz. Recursive definitions. Part II. Formalized Mathematics, 12(2):167–172, 2004.27331532

    Isoptic curves of conic sections in constant curvature geometries

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    In this paper we consider the isoptic curves on the 2-dimensional geometries of constant curvature \bE^2,~\bH^2,~\cE^2. The topic is widely investigated in the Euclidean plane \bE^2 see for example \cite{CMM91} and \cite{Wi} and the references given there, but in the hyperbolic and elliptic plane there are few results in this topic (see \cite{CsSz1} and \cite{CsSz2}). In this paper we give a review on the preliminary results of the isoptics of Euclidean and hyperbolic curves and develop a procedure to study the isoptic curves in the hyperbolic and elliptic plane geometries and apply it for some geometric objects e.g. proper conic sections. We use for the computations the classical models which are based on the projectiv interpretation of the hyperbolic and elliptic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer
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