Faster MapToPoint on Supersingular Elliptic Curves in Characteristic 3

2011 ◽  
Vol E94-A (1) ◽  
pp. 150-155 ◽  
Author(s):  
Yuto KAWAHARA ◽  
Tetsutaro KOBAYASHI ◽  
Gen TAKAHASHI ◽  
Tsuyoshi TAKAGI
Keyword(s):  
Elliptic Curves ◽  
Supersingular Elliptic Curves ◽  
Characteristic 3

scholarly journals Reduction of Elliptic Curves in Equal Characteristic 3 (and 2)

2005 ◽  
Vol 48 (3) ◽  
pp. 428-444
Author(s):  
Roland Miyamoto ◽  
Jaap Top
Keyword(s):  
Elliptic Curves ◽  
Fibre Type ◽  
Valued Fields ◽  
Characteristic 2 ◽  
Characteristic 3

AbstractWe determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

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 Fast Architectures for the \eta_T Pairing over Small-Characteristic Supersingular Elliptic Curves

2011 ◽  
Vol 60 (2) ◽  
pp. 266-281 ◽  
Author(s):  
Jean-Luc Beuchat ◽  
Jeremie Detrey ◽  
Nicolas Estibals ◽  
Eiji Okamoto ◽  
Francisco Rodriguez Henriquez
Keyword(s):  
Elliptic Curves ◽  
Supersingular Elliptic Curves
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