Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

@article{Morain2007ImplementingTA,
  title={Implementing the asymptotically fast version of the elliptic curve primality proving algorithm},
  author={François Morain},
  journal={Math. Comput.},
  year={2007},
  volume={76},
  pages={493-505}
}
  • F. Morain
  • Published 4 February 2005
  • Computer Science, Mathematics
  • Math. Comput.
The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time O ((log N)^5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4). The aim of this article is to describe this version in more details, leading to actual implementations able to… 

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References

SHOWING 1-10 OF 60 REFERENCES
Proving the Primality of Very Large Numbers with fastECPP
TLDR
New ideas used when dealing with very large numbers are described, illustrated with the primality proofs of some numbers with more than 10,000 decimal digits.
ELLIPTIC CURVES AND PRIMALITY PROVING
TLDR
The Elliptic Curve Primality Proving algorithm - ECPP - is described, which can prove the primality of 100-digit numbers in less than five minutes on a SUN 3/60 workstation, and can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation.
Primality Proving Using Elliptic Curves: An Update
  • F. Morain
  • Mathematics, Computer Science
    ANTS
  • 1998
TLDR
An account of the recent theoretical and practical improvements of ECPP, as well as new benchmarks for integers of various sizes and a new primality record are given.
DISTRIBUTED PRIMALITY PROVING AND THE PRIMALITY OF
TLDR
The successful attempt at proving the primality of the 1065-digit (2 3539 +1)=3, the rst ordinary Titanic prime is described.
Primality testing using elliptic curves
TLDR
A primality proving algorithm—a probablistic primality test that produces short certificates of primality on prime inputs that is based on a new methodology for applying group theory to the problem of prime certification, and the application of this methodology using groups generated by elliptic curves over finite fields.
Distributed Primality Proving and the Primality of (23539+1)/3
  • F. Morain
  • Mathematics, Computer Science
    EUROCRYPT
  • 1990
TLDR
The successful attempt at proving the primality of the l065-digit (23539+1)/3, the first ordinary Titanic prime, is described.
Primality Proving via One Round in ECPP and One Iteration in AKS
  • Qi Cheng
  • Computer Science
    Journal of Cryptology
  • 2006
TLDR
This paper explores the possibility of designing a randomized primality-proving algorithm based on the AKS algorithm, and generalizes Berrizbeitia's algorithm to one which has higher density of primes whose primality can be proved in time complexity $\tilde{O}(\log^{4} n)$.
Fast Decomposition of Polynomials with Known Galois Group
TLDR
This work explains how fast polynomial arithmetic can be used to speed up the process of solving the equation f(X) = 0, and extends the algorithms to a more general case of extensions that are no longer Galois.
Solvability by radicals from an algorithmic point of view
TLDR
This paper reduces the problem to that of cyclic extensions of prime degree and work out the radicals, using the work of Girstmair, and applies the general framework to the construction of Hilbert Class fields of imaginary quadratic fields.
The complexity of class polynomial computation via floating point approximations
  • A. Enge
  • Computer Science, Mathematics
    Math. Comput.
  • 2009
TLDR
The complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots, is analysed, using a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean.
...
...