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Mathematics of Computation

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Constructing elliptic curves over finite fields with prescribed torsion

Author: Andrew V. Sutherland
Journal: Math. Comp. 81 (2012), 1131-1147
MSC (2010): Primary 11G05, 11G07; Secondary 11-04, 14H10
Published electronically: August 4, 2011
MathSciNet review: 2869053
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Abstract: We present a method for constructing optimized equations for the modular curve $ X_1(N)$ using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field $ \mathbb{F}_q$ to efficiently generate elliptic curves with nontrivial $ N$-torsion by searching for affine points on $ X_1(N)(\mathbb{F}_q)$, and we give a fast method for generating curves with (or without) a point of order $ 4N$ using $ X_1(2N)$.

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Andrew V. Sutherland
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): September 21, 2010
Received by editor(s) in revised form: February 20, 2011
Published electronically: August 4, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.