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(2)
Elliptic Curve
Finite Field
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(1)
Endomorphism Rings of Elliptic Curves over Finite Fields
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Computing the endomorphism ring of an ordinary elliptic curve over a finite field
Computing the endomorphism ring of an ordinary elliptic curve over a finite field,10.1016/j.jnt.2009.11.003,Journal of Number Theory,Gaetan Bisson,And
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Computing the endomorphism ring of an ordinary elliptic curve over a finite field
(
Citations: 6
)
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Gaetan Bisson
,
Andrew V. Sutherland
We present two algorithms to compute the endomorphism ring of an ordinary
elliptic curve
E defined over a
finite field
Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of logq, while our bound for the second algorithm depends primarily on logDE, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.
Journal:
Journal of Number Theory  J NUMBER THEOR
, vol. 131, no. 5, pp. 815831, 2011
DOI:
10.1016/j.jnt.2009.11.003
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Citation Context
(3)
...Recently, more applications of isogeny volcanoes were found: the computation of Hilbert class polynomials [1,23], of modular polynomials [4] and of endomorphism rings of elliptic curves [
2
]...
Sorina Ionica
,
et al.
Pairing the Volcano
...We note that with the CM method, the correctness of the final result can be efficiently and unconditionallyconfirmed [
5
], so we are generally happy to work with a heuristic bound...
Andreas Enge
,
et al.
Class Invariants by the CRT Method
...To verify the endomorphism ring in Step 3 we may use the algorithm in [
9
], or Kohel’s algorithm [40]...
Andrew V. Sutherland
.
Computing Hilbert class polynomials with the Chinese Remainder Theorem
References
(41)
On the coe cients of the transformation polynomials for the elliptic modular function
(
Citations: 13
)
Paula Cohen
Published in 1984.
Computing modular polynomials in quasilinear time
(
Citations: 13
)
Andreas Enge
Journal:
Computing Research Repository  CORR
, vol. abs/0704.3, 2007
Isogeny Volcanoes and the SEA Algorithm
(
Citations: 18
)
Mireille Fouquet
,
François Morain
Conference:
Algorithmic Number Theory Symposium  ANTS
, pp. 276291, 2002
Smooth numbers: computational number theory and beyond
(
Citations: 29
)
ANDREW GRANVILLE
Endomorphism Rings of Elliptic Curves over Finite Fields
(
Citations: 52
)
D. Kohel
Published in 1996.
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Citations
(6)
A lowmemory algorithm for finding short product representations in finite groups
Gaetan Bisson
,
Andrew V. Sutherland
Journal:
Designs, Codes and Cryptography  DCC
, vol. abs/1101.0, pp. 113, 2011
Modular polynomials via isogeny volcanoes
(
Citations: 5
)
Reinier Broker
,
Kristin Lauter
,
Andrew V. Sutherland
Published in 2010.
Pairing the Volcano
(
Citations: 3
)
Sorina Ionica
,
Antoine Joux
Conference:
Algorithmic Number Theory Symposium  ANTS
, pp. 201218, 2010
Class Invariants by the CRT Method
(
Citations: 1
)
Andreas Enge
,
Andrew V. Sutherland
Conference:
Algorithmic Number Theory Symposium  ANTS
, pp. 142156, 2010
Constructing elliptic curve isogenies in quantum subexponential time
(
Citations: 1
)
Andrew M. Childs
,
David Jao
,
Vladimir Soukharev
Journal:
Computing Research Repository  CORR
, vol. abs/1012.4, 2010