Operations of Points on Elliptic Curve in Projective Coordinates

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.Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanMizushima Daichi - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.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.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 Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.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.Michał 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

Operations of Points on Elliptic Curve in Affine Coordinates

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, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, 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, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, 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 Korniłowicz. Recursive definitions. Part II. Formalized Mathematics, 12(2):167–172, 2004.27331532

Group law computations on Jacobians of hyperelliptic curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form

The geometry of efficient arithmetic on elliptic curves

The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripling, by considering families of elliptic curves admiting a $2$-torsion or $3$-torsion point