Loops of isogeny graphs of supersingular elliptic curves at j = 0

2019 ◽  
Vol 58 ◽  
pp. 174-176 ◽  
Author(s):  
Yi Ouyang ◽  
Zheng Xu
Keyword(s):  
Elliptic Curves ◽  
Supersingular Elliptic Curves ◽  
Isogeny Graphs

scholarly journals Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs

2020 ◽  
Vol 4 (1) ◽  
pp. 215-232
Author(s):  
Kirsten Eisenträger ◽  
Sean Hallgren ◽  
Chris Leonardi ◽  
Travis Morrison ◽  
Jennifer Park
Keyword(s):  
Elliptic Curves ◽  
Path Finding ◽  
Endomorphism Rings ◽  
Supersingular Elliptic Curves ◽  
Isogeny Graphs

scholarly journals Identifying supersingular elliptic curves

2012 ◽  
Vol 15 ◽  
pp. 317-325 ◽  
Author(s):  
Andrew V. Sutherland
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Positive Characteristic ◽  
Simple Algorithm ◽  
New Approach ◽  
Complexity Bounds ◽  
Supersingular Elliptic Curves ◽  
Structural Differences ◽  
Isogeny Graphs ◽  
Field Of Positive Characteristic

AbstractGiven an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in 𝔽p2, determines the supersingularity of E in O(n3log 2n) time and O(n) space, where n=O(log p) . Both these complexity bounds are significant improvements over existing methods, as we demonstrate with some practical computations.

scholarly journals Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves

2021 ◽  
Vol 15 (1) ◽  
pp. 454-464
Author(s):  
Guanju Xiao ◽  
Lixia Luo ◽  
Yingpu Deng
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Endomorphism Rings ◽  
Principal Ideals ◽  
Supersingular Elliptic Curves ◽  
Isogeny Graphs ◽  
Quadratic Order

Abstract Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽 p 2 , if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.

scholarly journals Orienting supersingular isogeny graphs

2020 ◽  
Vol 14 (1) ◽  
pp. 414-437
Author(s):  
Leonardo Colò ◽  
David Kohel
Keyword(s):  
Elliptic Curves ◽  
Diffie Hellman ◽  
Supersingular Elliptic Curves ◽  
Isogeny Graphs

AbstractWe introduce a category of 𝓞-oriented supersingular elliptic curves and derive properties of the associated oriented and nonoriented ℓ-isogeny supersingular isogeny graphs. As an application we introduce an oriented supersingular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.

scholarly journals Constructing supersingular elliptic curves with a given endomorphism ring

2014 ◽  
Vol 17 (A) ◽  
pp. 71-91 ◽  
Author(s):  
Ilya Chevyrev ◽  
Steven D. Galbraith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Quaternion Algebra ◽  
Maximal Order ◽  
Running Time ◽  
Successive Minima ◽  
Supersingular Elliptic Curves ◽  
Class Polynomials ◽  
Theoretical Results

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

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