scholarly journals Division by $2$ on odd-degree hyperelliptic curves and their Jacobians

2019 ◽  
Vol 83 (3) ◽  
pp. 501-520
Author(s):  
Yu. G. Zarhin
Keyword(s):  
Hyperelliptic Curves ◽  
Odd Degree

scholarly journals Rational points on hyperelliptic curves having a marked non-Weierstrass point

2017 ◽  
Vol 154 (1) ◽  
pp. 188-222
Author(s):  
Arul Shankar ◽  
Xiaoheng Wang
Keyword(s):  
Weierstrass Point ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Selmer Groups ◽  
Automorphic Representations ◽  
The Family ◽  
Odd Degree ◽  
Fundamental Research ◽  
Average Size ◽  
Fixed Genus

In this paper, we consider the family of hyperelliptic curves over$\mathbb{Q}$having a fixed genus$n$and a marked rational non-Weierstrass point. We show that when$n\geqslant 9$, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as$n$tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of$5/2$on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2)180(2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, inAutomorphic representations and$L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

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 Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Jennifer S. Balakrishnan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Hyperelliptic Curves ◽  
Open Book ◽  
Line Integral ◽  
Book Series ◽  
Numerical Examples ◽  
Lecture Notes ◽  
Integration Algorithms ◽  
Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

scholarly journals Integral points on hyperelliptic curves

2008 ◽  
Vol 2 (8) ◽  
pp. 859-885 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek ◽  
Michael Stoll ◽  
Szabolcs Tengely
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

N-Flips in Triangulations with Two Odd Degree Vertices

2020 ◽  
Vol 36 (3) ◽  
pp. 469-490
Author(s):  
Yoshihiro Asayama ◽  
Ryo Matsukawa ◽  
Naoki Matsumoto ◽  
Atsuhiro Nakamoto
Keyword(s):  
Odd Degree
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