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 Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

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.

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 Tight Closure and Plus Closure for Cones Over Elliptic Curves

2005 ◽  
Vol 177 ◽  
pp. 31-45 ◽  
Author(s):  
Holger Brenner
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Positive Characteristic ◽  
Primary Ideal ◽  
Coordinate Ring ◽  
Tight Closure

We characterize the tight closure of a homogeneous primary ideal in a normal homogeneous coordinate ring over an elliptic curve by a numerical condition and we show that it is in positive characteristic the same as the plus closure.

scholarly journals On Small Characteristic Algebraic Tori in Pairing-Based Cryptography

2006 ◽  
Vol 9 ◽  
pp. 64-85 ◽  
Author(s):  
R. Granger ◽  
D. Page ◽  
M. Stam
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Algebraic Tori ◽  
Simple Observation ◽  
Supersingular Elliptic Curves ◽  
Algebraic Torus ◽  
Pairing Based Cryptography

The value ot the late pairing on an elliptic curve over a finite field may be viewed as an element of an algebraic torus. Using this simple observation, we transfer techniques recently developed for torus-based cryptography to pairing-based cryptography, resulting in more efficient computations, and lower bandwidth requirements. To illustrate the efficacy of this approach, we apply the method to pairings on supersingular elliptic curves in characteristic three.

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.

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