File:Double Point on Edwards Curve.svg
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Original file (SVG file, nominally 540 × 540 pixels, file size: 26 KB)
Summary
| Description |
English: The plot presents the geometry meaning of point doubling on the Edwards curves .
Here you can see doubling a point on the curve with . The point has x-coordinate -0.6. Unlike the traditional elliptic curves where points and lay on the tangent line to the curve at the point , in the case of the Edwards curves the points and lay on a conic that touches the curve at the point . The graph was created using the following script: import matplotlib.pyplot as plt
import numpy as np
from collections import namedtuple
# Utility type
Point = namedtuple('Point', ['x', 'y'])
d = -30
def edwards_y(x):
return np.sqrt((x*x - 1)/(d*x*x - 1))
# Draw Edwards curve
x = np.linspace(-1,1,200)
ypos = edwards_y(x)
yneg = -ypos
plt.figure(figsize=[6, 6])
plt.plot(x,ypos, 'b')
plt.plot(x,yneg, 'b')
# Draw neutral point
plt.scatter(0,1)
plt.annotate("O", (0.01, 1.01))
# Draw order 2 point
plt.scatter(0,-1)
plt.annotate("O'", (0.01, -1.05))
# Draw the point P
P=Point(-0.6, edwards_y(-0.6))
plt.scatter(*P)
plt.annotate("P", (P.x-0.05, P.y+0.05))
# Compute and draw 2P
def edwards_sum(x1,y1,x2,y2):
return ( (x1*y2+x2*y1)/(1+d*x1*x2*y1*y2) , (y1*y2 - x1*x2)/(1-d*x1*x2*y1*y2) )
P2 = Point(*edwards_sum(*P, *P))
plt.scatter(*P2)
plt.annotate("2P", (P2.x-0.05, P2.y+0.05))
P2_ = Point(-P2.x, P2.y)
plt.scatter(*P2_)
plt.annotate("-2P", (P2_.x+0.01, P2_.y+0.05))
# Draw the line that connects 2P and -2P
plt.axhline(P2.y, linestyle='--', color="grey")
# Draw the conic that P1, P2 and -(P1+P2) belong to
def conic_coefs(x,y):
"Computes coeffitiens of the quadratic form Axy + Bx + Cx + D"
return (d*x*x*y - 1,
y - x*x,
x*(1-y),
x*(1-y)
)
def conic_y(x, A,B,C,D):
return -(B*x + D)/(A*x + C)
A,B,C,D = conic_coefs(*P)
# Left and right branches of the hyperbole
xleft = np.linspace(-1,-0.3,50)
xright = np.linspace(-0.01, 1, 50)
yleft = conic_y(xleft, A,B,C,D)
yright = conic_y(xright, A,B,C,D)
plt.plot(xleft, yleft,"--", color="green")
plt.plot(xright, yright,"--", color="green")
# Draw axis lines
plt.axhline(0, color='black')
plt.axvline(0, color='black')
# Set same scale on x and y
plt.gca().set_aspect('equal', adjustable='box')
plt.savefig("Double_Point_on_Edwards_Curve.svg")
Русский: График иллюстрирует геометрический смысл удвоения точек на кривых Эрдвадса .
На графике изображено удвоение точки на кривой с параметром . Точка с x-координатой -0.6. В отличие от традиционных эллиптических кривых, где точки и лежат на касательной к эллиптической кривой в точке , на кривых Эдвардса точки и лежат на гиперболе , которая касается график кривой в точке . |
| Date | |
| Source | Own work |
| Author | Pakuula |
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 05:29, 20 December 2020 | | 540 × 540 (26 KB) | commons>Pakuula | Uploaded own work with UploadWizard |
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