1,551 research outputs found
Computing Hilbert class polynomials with the Chinese Remainder Theorem
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 construct pairing-friendly elliptic curves
of prime order, using the CM method.Comment: 37 pages, corrected a typo that misstated the heuristic complexit
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . The first is a p-adic lifting algorithm for inert primes p
in the order of discriminant D < 0. The second is an improved Chinese remainder
algorithm which uses the class group action on CM-curves over finite fields.
Our run time analysis gives tighter bounds for the complexity of all known
algorithms for computing , and we show that all methods have comparable
run times
Modular polynomials via isogeny volcanoes
We present a new algorithm to compute the classical modular polynomial Phi_n
in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m.
Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p
for many primes p of a suitable form, and then applies the Chinese Remainder
Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an
expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m
using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to
compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also
consider several modular functions g for which Phi_n^g is smaller than Phi_n,
allowing us to handle n over 60000.Comment: corrected a typo in equation (14), 31 page
Parallelization of Modular Algorithms
In this paper we investigate the parallelization of two modular algorithms.
In fact, we consider the modular computation of Gr\"obner bases (resp. standard
bases) and the modular computation of the associated primes of a
zero-dimensional ideal and describe their parallel implementation in SINGULAR.
Our modular algorithms to solve problems over Q mainly consist of three parts,
solving the problem modulo p for several primes p, lifting the result to Q by
applying Chinese remainder resp. rational reconstruction, and a part of
verification. Arnold proved using the Hilbert function that the verification
part in the modular algorithm to compute Gr\"obner bases can be simplified for
homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be
adapted to the local case, i.e. for local orderings and not necessarily
homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In
this paper we prove the corresponding theorem for non-homogeneous ideals in
case of a global ordering.Comment: 16 page
On the evaluation of modular polynomials
We present two algorithms that, given a prime ell and an elliptic curve E/Fq,
directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the
j-invariants of the elliptic curves that are ell-isogenous to E. We do not
assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be
adapted to handle other types of modular polynomials, and we consider
applications to point counting and the computation of endomorphism rings. We
demonstrate the practical efficiency of the algorithms by setting a new
point-counting record, modulo a prime q with more than 5,000 decimal digits,
and by evaluating a modular polynomial of level ell = 100,019.Comment: 19 pages, corrected a typo in equation (8) and added equation (9
Constructing Class invariants
Shimura reciprocity law allows us to verify that a modular function is a
class invariant. Here we present a new method based on Shimura reciprocity that
allows us not only to verify but to find new class invariants from a modular
function of level .Comment: 12 page
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