File:Sum of two points on an Edwards curve.svg
![File:Sum of two points on an Edwards curve.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Sum_of_two_points_on_an_Edwards_curve.svg/540px-Sum_of_two_points_on_an_Edwards_curve.svg.png)
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Summary
DescriptionSum of two points on an Edwards curve.svg |
English: The plot presents the geometry meaning of point addition on the Edwards curves
Here you can see the sum of two points on the curve with Unlike the traditional elliptic curves where points The graph was created using the following script: import matplotlib.pyplot as plt
import numpy as np
import math
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 points P1 and P2
P1=Point(-0.6, edwards_y(-0.6))
P2=Point(0.1, edwards_y(0.1))
plt.scatter(*P1)
plt.annotate("P1", (P1.x-0.05, P1.y+0.05))
plt.scatter(*P2)
plt.annotate("P2", P2)
# Compute and draw P1 + P2
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) )
P3 = Point(*edwards_sum(*P1, *P2))
plt.scatter(*P3)
plt.annotate("P3", (P3.x-0.05, P3.y+0.05))
P3_ = Point(-P3.x, P3.y)
plt.scatter(*P3_)
plt.annotate("-(P1+P2)", (P3_.x+0.01, P3_.y+0.05))
# Draw the line that connects P3 and -P3
plt.axhline(P3.y, linestyle='--', color="grey")
# Draw the conic that P1, P2 and -(P1+P2) belong to
def conic_coefs(x1,y1,x2,y2):
"Computes coeffitiens of the quadratic form Axy + Bx + Cx + D"
return (x1-x2 + (x1*y2-x2*y1),
(x2*y2-x1*y1)+y1*y2*(x2-x1),
x1*x2*(y1-y2),
x1*x2*(y1-y2)
)
def conic_y(x, A,B,C,D):
return -(B*x + D)/(A*x + C)
A,B,C,D = conic_coefs(*P1,*P2)
# Left and right branches of the hyperbole
xleft = np.linspace(-1,0.003,50)
xright = np.linspace(P2[0] - 0.02, 1.1, 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("Add_points_Edwards.svg")
Русский: График иллюстрирует геометрический смысл сложения точек на кривых Эрдвадса На графике изображено сложение двух точек на кривой с параметром |
Date | |
Source | Own work |
Author | Pakuula |
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20 December 2020
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current | 08:24, 20 December 2020 | ![]() | 540 × 540 (26 KB) | wikimediacommons>Pakuula | Uploaded own work with UploadWizard |
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Width | 432pt |
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Height | 432pt |