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Constructing supersingular elliptic curves with a given endomorphism ring

Part of: Arithmetic problems. Diophantine geometry Geometry of numbers Algebraic number theory: global fields Arithmetic algebraic geometry Computational number theory Forms and linear algebraic groups

Published online by Cambridge University Press:  01 August 2014

Ilya Chevyrev  and
Steven D. Galbraith
Ilya Chevyrev
Affiliation:
Mathematical Institute, University of Oxford, United Kingdom email [email protected]
Steven D. Galbraith
Affiliation:
Mathematics Department, University of Auckland, New Zealand email [email protected]

Abstract

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Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ 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.

MSC classification

Secondary: 11G20: Curves over finite and local fields 11Y16: Algorithms; complexity 11R52: Quaternion and other division algebras 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11H55: Quadratic forms (reduction theory, extreme forms, etc.) 14G15: Finite ground fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 71 - 91
Copyright
© The Author(s) 2014 

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