Schoof’s point counting algorithm

2010 ◽  
pp. 11-63
Author(s):  
Nicolas Thériault
Keyword(s):  
Point Counting ◽  
Counting Algorithm

scholarly journals On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

2015 ◽  
Vol 18 (1) ◽  
pp. 308-322 ◽  
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Log P ◽  
Point Counting ◽  
Generalized Riemann Hypothesis ◽  
Expected Time ◽  
Almost All ◽  
Counting Algorithm

For an elliptic curve$E/\mathbb{Q}$without complex multiplication we study the distribution of Atkin and Elkies primes$\ell$, on average, over all good reductions of$E$modulo primes$p$. We show that, under the generalized Riemann hypothesis, for almost all primes$p$there are enough small Elkies primes$\ell$to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in$(\log p)^{4+o(1)}$expected time.

scholarly journals Computing cardinalities of -curve reductions over finite fields

2016 ◽  
Vol 19 (A) ◽  
pp. 115-129
Author(s):  
François Morain ◽  
Charlotte Scribot ◽  
Benjamin Smith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Lower Degree ◽  
Point Counting ◽  
Low Degree ◽  
Asymptotic Complexity ◽  
Specialized Point ◽  
Counting Algorithm

We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$-curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

A New System For Optomanual Semiautomatic Quantitative Image Analysis

1977 ◽  
Vol 35 ◽  
pp. 210-211
Author(s):  
H.P. Rohr
Keyword(s):  
Image Analysis ◽  
Volume Density ◽  
List Type ◽  
Distribution Analysis ◽  
Type Number ◽  
Point Counting ◽  
Human Eye ◽  
Quantitative Image ◽  
Point Density ◽  
Electronic Counting

Today, in image analysis the broadest possible rationalization and economization have become desirable. Basically, there are two approaches for image analysis: The image analysis through the so-called scanning methods which are usually performed without the human eye and the systems of optical semiautomatic analysis completely relying on the human eye.The new MOP AM 01 opto-manual system (fig.) represents one of the very promising approaches in this field. The instrument consists of an electronic counting and storing unit, which incorporates a microprocessor and a keyboard for choice of measuring parameters, well designed for easy use.Using the MOP AM 01 there are three possibilities of image analysis:the manual point counting,the opto-manual point counting andthe measurement of absolute areas and/or length (size distribution analysis included).To determine a point density for the calculation of the corresponding volume density the intercepts lying within the structure are scanned with the light pen.

A Comparison Between Sterological Point Counting and Digitizing Tablets

1985 ◽  
Vol 43 ◽  
pp. 666-667
Author(s):  
John M. Basgen ◽  
Eileen N. Ellis ◽  
S. Michael Mauer ◽  
Michael W. Steffes
Keyword(s):  
Electron Microscopy ◽  
Kidney Tissue ◽  
Volume Density ◽  
Diabetic Patients ◽  
Normal Kidney ◽  
Point Counting ◽  
Normal Kidney Tissue ◽  
Biopsy Specimens ◽  
Mitochondrial Volume ◽  
Kidney Lesions

To determine the efficiency of methods of quantitation of the volume density of components within kidney biopsies, techniques involving a semi-automatic digitizing tablet and stereological point counting were compared.Volume density (Vv) is a parameter reflecting the volume of a component to the volume that contains the component, e.g., the fraction of cell volume that is made up of mitochondrial volume. The units of Vv are μm3 /μm3.Kidney biopsies from 15 patients were used. Five were donor biopsies performed at the time of kidney transplantation (patients 1-5, TABLE 1) and were considered normal kidney tissue. The remaining biopsies were obtained from diabetic patients with a spectrum of diabetic kidney lesions. The biopsy specimens were fixed and embedded according to routine electron microscogy protocols. Three glomeruli from each patient were selected randomly for electron microscopy. An average of 12 unbiased and systematic micrographs were obtained from each glomerulus and printed at a final magnification of x18,000.

3D box-counting algorithm for calculating fractal dimension of cities

2010 ◽  
Vol 30 (8) ◽  
pp. 2070-2072
Author(s):  
Le-shan ZHANG ◽  
Ge CHEN ◽  
Yong HAN ◽  
Tao ZHANG
Keyword(s):  
Fractal Dimension ◽  
Box Counting ◽  
Counting Algorithm
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