Computing Hilbert class polynomials with the Chinese remainder theorem

@article{Sutherland2009ComputingHC,
  title={Computing Hilbert class polynomials with the Chinese remainder theorem},
  author={Andrew V. Sutherland},
  journal={Math. Comput.},
  year={2009},
  volume={80},
  pages={501-538}
}
We present a space-efficient algorithm to compute the Hilbert class polynomial H_D(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 10^13 and h(D) up to 10^6. We apply these results to… 

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References

SHOWING 1-10 OF 83 REFERENCES

A $p$-adic algorithm to compute the Hilbert class polynomial

Classically, the Hilbert class polynomial PΔ ∈ Z[X] of an imaginary quadratic discriminant A is computed using complex analytic techniques. In 2002, Couveignes and Henocq suggested a p-adic algorithm

A p-adic algorithm to compute the Hilbert class polynomial

TLDR
A detailed description of the p-adic algorithm to compute the Hilbert class polynomial P∆ of an imaginary quadratic discriminant ∆ is given, and a careful study of the complexity shows that, if the Generalized Riemann Hypothesis holds true, the expected runtime is O(|∆|(log | ∆|)8+ε) instead of O( |∆ |1+ε).

Computing Hilbert Class Polynomials

TLDR
A p-adic lifting algorithm forinert primes p in the order of discriminant D < 0.1 and an improved Chinese remainder algorithm which uses the class group action onCM-curves over finite fields are presented.

Numerical Results on Class Groups of Imaginary Quadratic Fields

TLDR
Using techniques described in [3], the class number and class group structure of all imaginary quadratic fields with discriminant Δ for 0 < |Δ| < 1011 is computed.

Factoring integers with elliptic curves

TLDR
This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2.

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.

CONSTRUCTING ELLIPTIC CURVES WITH A KNOWN NUMBER OF POINTS OVER A PRIME FIELD

TLDR
A modified version of the Chinese remainder theorem (CRT) is presented to compute HD(X) modulo n directly from the knowledge of HD( X)modulo enough small primes, suggesting that asymptotically the algorithm is an improvement over previously known methods.

Computing modular polynomials in quasi-linear time

  • A. Enge
  • Computer Science
    Math. Comput.
  • 2009
TLDR
It is shown that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially linear in the size of the computed polynomials.

A rigorous subexponential algorithm for computation of class groups

Let C(-d) denote the Gauss Class Group of quadratic forms of a negative discriminant -d (or equivalently, the class group of the imaginary quadratic field Q(A/=') ). We give a rigorous proof that

Explicit bounds for primality testing and related problems

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits
...