Results 71 to 80 of about 216 (184)

Ate Pairing on Hyperelliptic Curves [PDF]

, 2007
In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller's algorithm can be up to gtimes shorter than for the Tate pairing, with gthe genus of the curve, and the pairing is ...
Robert Granger, Florian Hess, Roger Oyono, Nicolas Thériault, Frederik Vercauteren   +4 more
openaire   +2 more sources

Distortion maps for supersingular genus two curves

Journal of Mathematical Cryptology, 2009
Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated, since the full torsion subgroup has rank 2g.
Galbraith Steven D., Pujolàs Jordi, Ritzenthaler Christophe, Smith Benjamin   +3 more
doaj   +1 more source

The geometry and arithmetic of bielliptic Picard curves

Journal of the London Mathematical Society, Volume 112, Issue 5, November 2025.
Abstract We study the geometry and arithmetic of the curves C:y3=x4+ax2+b$C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces P$P$. We prove a Torelli‐type theorem in this context and give a geometric proof of the fact that P$P$ has quaternionic multiplication by the quaternion order of discriminant 6.
Jef Laga, Ari Shnidman
wiley   +1 more source

Quantum algorithm for solving binary hyperelliptic curve discrete logarithm problem

Cybersecurity
It is well-established that Shor’s algorithm can solve the discrete logarithm problem (DLP) in polynomial time. The hyperelliptic curve DLP (HCDLP) of genus 2 has found widespread industrial applications and remains an active research domain.
Yan Huang, Du Zeng, Chao Chen, Zijian Zhou, Fangguo Zhang   +4 more
doaj   +1 more source

On the finiteness of maps into simple abelian varieties satisfying certain tangency conditions

Bulletin of the London Mathematical Society, Volume 57, Issue 9, Page 2723-2730, September 2025.
Abstract We show that given a simple abelian variety A$A$ and a normal variety V$V$ defined over a finitely generated field K$K$ of characteristic zero, the set of non‐constant morphisms V→A$V \rightarrow A$ satisfying certain tangency conditions imposed by a Campana orbifold divisor Δ$\Delta$ on A$A$ is finite.
Finn Bartsch
wiley   +1 more source

Cyclic cubic points on higher genus curves

Journal of the London Mathematical Society, Volume 112, Issue 3, September 2025.
Abstract The distribution of degree d$d$ points on curves is well understood, especially for low degrees. We refine this study to include information on the Galois group in the simplest interesting case: d=3$d = 3$. For curves of genus at least 5, we show cubic points with Galois group C3$C_3$ arise from well‐structured morphisms, along with providing ...
James Rawson
wiley   +1 more source

Parity of ranks of Jacobians of curves

Proceedings of the London Mathematical Society, Volume 131, Issue 3, September 2025.
Abstract We investigate Selmer groups of Jacobians of curves that admit an action of a non‐trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks. Under the Shafarevich–Tate conjecture, we give an expression for the parity of the Mordell–Weil rank of an arbitrary Jacobian in terms of purely local invariants ...
Vladimir Dokchitser, Holly Green, Alexandros Konstantinou, Adam Morgan   +3 more
wiley   +1 more source

Hyperelliptic parametrizations of Q -curves

, 2020
For a square-free integer N, we present a procedure to compute Q-curves parametrized by rational points of the modular curve X∗0(N) when this is ...
Bars Cortina, Francesc, González Rovira, Josep, Xarles Ribas, Xavier   +2 more
openaire   +5 more sources

On quadratic residue codes and hyperelliptic curves

Discrete Mathematics & Theoretical Computer Science, 2008
For an odd prime p and each non-empty subset S⊂GF(p), consider the hyperelliptic curve X S defined by y 2 =f S (x), where f S (x) = ∏ a∈S (x-a).
David Joyner
doaj  

Finite‐Dimensional Reductions and Finite‐Gap‐Type Solutions of Multicomponent Integrable PDEs

Studies in Applied Mathematics, Volume 155, Issue 2, August 2025.
ABSTRACT The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well‐known equations such as the Korteweg–de Vries, coupled KdV, Harry Dym, coupled Harry Dym, Camassa–Holm, multicomponent Camassa–Holm, Dullin–Gottwald–Holm, and Kaup ...
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev   +2 more
wiley   +1 more source