12,554 research outputs found
Constructing elliptic curves of prime order
We present a very efficient algorithm to construct an elliptic curve E and a
finite field F such that the order of the point group E(F) is a given prime
number N. Heuristically, this algorithm only takes polynomial time Otilde((\log
N)^3), and it is so fast that it may profitably be used to tackle the related
problem of finding elliptic curves with point groups of prime order of
prescribed size. We also discuss the impact of the use of high level modular
functions to reduce the run time by large constant factors and show that recent
gonality bounds for modular curves imply limits on the time reduction that can
be obtained.Comment: 13 page
Computing the cardinality of CM elliptic curves using torsion points
Let E be an elliptic curve having complex multiplication by a given quadratic
order of an imaginary quadratic field K. The field of definition of E is the
ring class field Omega of the order. If the prime p splits completely in Omega,
then we can reduce E modulo one the factors of p and get a curve Ep defined
over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a
fast way to find this sign. For this, we propose to use the action of the
Frobenius on torsion points of small order built with class invariants a la
Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for
computing the cardinality of a given elliptic curve modulo p. We apply our
results to the Elliptic Curve Primality Proving algorithm (ECPP).Comment: Revised and shortened version, including more material using
discriminants of curves and division polynomial
Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
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 handle numbers with several thousands of decimal
digits
A simple approach towards the sign problem using path optimisation
We suggest an approach for simulating theories with a sign problem that
relies on optimisation of complex integration contours that are not restricted
to lie along Lefschetz thimbles. To that end we consider the toy model of a
one-dimensional Bose gas with chemical potential. We identify the main
contribution to the sign problem in this case as coming from a nearest
neighbour interaction and approximately cancel it by an explicit deformation of
the integration contour. We extend the obtained expressions to more general
ones, depending on a small set of parameters. We find the optimal values of
these parameters on a small lattice and study their range of validity. We also
identify precursors for the onset of the sign problem. A fast method of
evaluating the Jacobian related to the contour deformation is proposed and its
numerical stability is examined. For a particular choice of lattice parameters,
we find that our approach increases the lattice size at which the sign problem
becomes serious from to . The efficient
evaluation of the Jacobian ( for a sweep) results in running times that
are of the order of a few minutes on a standard laptop.Comment: V1: 25 pages, 8 figures; V2: 28 pages, 8 figures, the methods used
for finding the contour parameters are clarified, further discussion added,
typos corrected, refs adde
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