54,363 research outputs found
Accelerating the CM method
Given a prime q and a negative discriminant D, the CM method constructs an
elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X)
modulo q. We consider an approach based on a decomposition of the ring class
field defined by H_D, which we adapt to a CRT setting. This yields two
algorithms, each of which obtains a root of H_D mod q without necessarily
computing any of its coefficients. Heuristically, our approach uses
asymptotically less time and space than the standard CM method for almost all
D. Under the GRH, and reasonable assumptions about the size of log q relative
to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D),
which may be as small as O(|D|^(1/4)log q). The practical efficiency of the
algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| >
10^15 and q ~ 2^33220. These examples are both an order of magnitude larger
than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation
and Mathematic
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
Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties
We obtain a convergent power series expansion for the first branch of the
dispersion relation for subwavelength plasmonic crystals consisting of
plasmonic rods with frequency-dependent dielectric permittivity embedded in a
host medium with unit permittivity. The expansion parameter is , where is the norm of a fixed wavevector, is the period of
the crystal and is the wavelength, and the plasma frequency scales
inversely to , making the dielectric permittivity in the rods large and
negative. The expressions for the series coefficients (a.k.a., dynamic
correctors) and the radius of convergence in are explicitly related to
the solutions of higher-order cell problems and the geometry of the rods.
Within the radius of convergence, we are able to compute the dispersion
relation and the fields and define dynamic effective properties in a
mathematically rigorous manner. Explicit error estimates show that a good
approximation to the true dispersion relation is obtained using only a few
terms of the expansion. The convergence proof requires the use of properties of
the Catalan numbers to show that the series coefficients are exponentially
bounded in the Sobolev norm
The Helioseismic and Magnetic Imager (HMI) Vector Magnetic Field Pipeline: Overview and Performance
The Helioseismic and Magnetic Imager (HMI) began near-continuous full-disk
solar measurements on 1 May 2010 from the Solar Dynamics Observatory (SDO). An
automated processing pipeline keeps pace with observations to produce
observable quantities, including the photospheric vector magnetic field, from
sequences of filtergrams. The primary 720s observables were released in mid
2010, including Stokes polarization parameters measured at six wavelengths as
well as intensity, Doppler velocity, and the line-of-sight magnetic field. More
advanced products, including the full vector magnetic field, are now available.
Automatically identified HMI Active Region Patches (HARPs) track the location
and shape of magnetic regions throughout their lifetime.
The vector field is computed using the Very Fast Inversion of the Stokes
Vector (VFISV) code optimized for the HMI pipeline; the remaining 180 degree
azimuth ambiguity is resolved with the Minimum Energy (ME0) code. The
Milne-Eddington inversion is performed on all full-disk HMI observations. The
disambiguation, until recently run only on HARP regions, is now implemented for
the full disk. Vector and scalar quantities in the patches are used to derive
active region indices potentially useful for forecasting; the data maps and
indices are collected in the SHARP data series, hmi.sharp_720s. Patches are
provided in both CCD and heliographic coordinates.
HMI provides continuous coverage of the vector field, but has modest spatial,
spectral, and temporal resolution. Coupled with limitations of the analysis and
interpretation techniques, effects of the orbital velocity, and instrument
performance, the resulting measurements have a certain dynamic range and
sensitivity and are subject to systematic errors and uncertainties that are
characterized in this report.Comment: 42 pages, 19 figures, accepted to Solar Physic
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