Torsion points, Pell’s equation, and integration in elementary terms

David Masser; Umberto Zannier

doi:10.4310/acta.2020.v225.n2.a2

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

Journal of the European Mathematical Society ◽

10.4171/jems/560 ◽

2015 ◽

Vol 17 (9) ◽

pp. 2379-2416 ◽

Cited By ~ 10

Author(s):

David Masser ◽

Umberto Zannier

Keyword(s):

Polynomial Rings ◽

Abelian Surfaces ◽

Torsion Points ◽

Pell’S Equation

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The Solvability of Polynomial Pell’s Equation

Journal of Institute of Science and Technology ◽

10.3126/jist.v25i2.33749 ◽

2020 ◽

Vol 25 (2) ◽

pp. 125-132

Author(s):

Bal Bahadur Tamang ◽

Ajay Singh

Keyword(s):

Trivial Solution ◽

Continued Fraction ◽

Laurent Series ◽

Quadratic Polynomial ◽

Continued Fraction Expansion ◽

Pell’S Equation ◽

Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

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Optimized CSIDH Implementation Using a 2-Torsion Point

Cryptography ◽

10.3390/cryptography4030020 ◽

2020 ◽

Vol 4 (3) ◽

pp. 20 ◽

Cited By ~ 1

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.

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

Research in Number Theory ◽

10.1007/s40993-020-00231-z ◽

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.

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