TORSION POINTS ON CURVES

2007 ◽  
pp. 85-92
Author(s):  
Andrew Granville ◽  
Zeév Rudnick
Keyword(s):  
Torsion Points

scholarly journals Optimized CSIDH Implementation Using a 2-Torsion Point

Cryptography ◽  
2020 ◽  
Vol 4 (3) ◽  
pp. 20 ◽  
Author(s):  
Donghoe Heo ◽  
Suhri Kim ◽  
Kisoon Yoon ◽  
Young-Ho Park ◽  
Seokhie Hong
Keyword(s):  
Elliptic Curve ◽  
Group Action ◽  
Large Degree ◽  
Transitive Group ◽  
Optimization Method ◽  
Torsion Point ◽  
Torsion Points ◽  
Diffie Hellman ◽  
Odd Degree ◽  
Montgomery Curve

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

scholarly journals Mazur’s rational torsion result for pointless genus one curves: examples

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Arjan Dwarshuis ◽  
Majken Roelfszema ◽  
Jaap Top
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Rational Point ◽  
Torsion Points ◽  
Arbitrary Genus ◽  
Mazur’S Theorem ◽  
Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

scholarly journals G-linear sets and torsion points in definably compact groups

2009 ◽  
Vol 48 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Margarita Otero ◽  
Ya’acov Peterzil
Keyword(s):  
Compact Groups ◽  
Torsion Points ◽  
Linear Sets

scholarly journals Genus-2 Jacobians with torsion points of large order

2014 ◽  
Vol 47 (1) ◽  
pp. 127-135 ◽  
Author(s):  
Everett W. Howe
Keyword(s):  
Torsion Points ◽  
Large Order ◽  
Genus 2

scholarly journals Torsion points of elliptic curves with good reduction

2008 ◽  
Vol 31 (3) ◽  
pp. 385-403
Author(s):  
Masaya Yasuda
Keyword(s):  
Elliptic Curves ◽  
Good Reduction ◽  
Torsion Points

scholarly journals The number of maximal torsion cosets in subvarieties of tori

2019 ◽  
Vol 2019 (755) ◽  
pp. 103-126
Author(s):  
César Martínez
Keyword(s):  
Newton Polytope ◽  
Torsion Point ◽  
Torsion Points ◽  
Sharp Bounds ◽  
Maximal Torsion ◽  
Algebraic Torus

AbstractWe present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus {\mathbb{G}_{\mathrm{m}}^{n}}. Our first main result gives a bound in terms of the degree of the defining polynomials. We also give a bound for the number of isolated torsion point, that is maximal torsion cosets of dimension 0, in terms of the volume of the Newton polytope of the defining polynomials. This result proves the conjectures of Ruppert and of Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.

Sign in / Sign up

Export Citation Format

Share Document