EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, by Dr. Herong Yang
Python Program for Integer Elliptic Curves
This section provides simple Python program, IntegerEllipticCurve.py, that searches integer points on any given elliptic curve with integer coefficients.
If you are interested to find the element set of elliptic curve points in 2-dimensional integer space, here is my simple Python program, IntegerEllipticCurve.py, that searches integer points on any given elliptic curve with integer coefficients:
# IntegerEllipticCurve.py # Copyright (c) HerongYang.com. All Rights Reserved. # import sys if (len(sys.argv) < 3): print("Usage: python IntegerEllipticCurve.py a b") exit() aa = sys.argv[1] bb = sys.argv[2] a = int(aa) b = int(bb) r = 4*a**3 + 27*b**2 if (r == 0): print("Error: (a, b) = ("+aa+", "+bb+") and 4a^3 + 27b^2 == 0") exit() print("Elliptic equation: y^2 = x^3 + ax + b with (a, b) = (" +aa+", "+bb+")") # Find a x such that x^3 + ax + b < 0 x = 0 r = x**3 + a*x + b while (r >= 0): x = x - 1 r = x**3 + a*x + b print("Starting with: x = "+str(x)) # Find a x such that x^3 + ax + b >= 0 while ( r<0 ): x = x + 1 r = x**3 + a*x + b print("First candidate: x = "+str(x)) # Check every x < 100000 y = 0 while (x < 100000): r = x**3 + a*x + b l = y**2 while (l < r): y = y + 1 l = y**2 # print("(x, y) = ("+str(x)+", "+str(y)+")") # print("(r, l) = ("+str(r)+", "+str(l)+")") if (l == r): print("Point on curve: (x, y) = ("+str(x)+", "+str(y)+")") x = x + 1 print("Ended with (x, y, r, l) = " +str(x)+", "+str(y)+", "+str(r)+", "+str(l)+")") exit()
If you run it with (a,b) = (1,4), you will get:
herong> python IntegerEllipticCurve 1 4 Elliptic equation: y^2 = x^3 + ax + b with (a, b) = (1, 4) Starting with: x = -2 First candidate: x = -1 Point on curve: (x, y) = (0, 2) Point on curve: (x, y) = (0, -2) Point on curve: (x, y) = (4128, 265222) Point on curve: (x, y) = (4128, -265222) Ended with (x, y, r, l) = 100000, 31622303, 999970000400002, 999970047023809)
The program works and found 4 points in the range of (x,y) < (1000000, 31622303):
(x, y) = (0, 2) (x, y) = (0, -2) (x, y) = (4128, 265222) (x, y) = (4128, -265222)
Of course, you can increase the search range to a higher number. But the program may crash with integer overflow exceptions or run for a long long time.
Table of Contents
Geometric Introduction to Elliptic Curves
Algebraic Introduction to Elliptic Curves
Abelian Group and Elliptic Curves
Discrete Logarithm Problem (DLP)
Generators and Cyclic Subgroups
►Reduced Elliptic Curve Groups
Converting Elliptic Curve Groups
Elliptic Curves in Integer Space
►Python Program for Integer Elliptic Curves
Elliptic Curves Reduced by Modular Arithmetic
Python Program for Reduced Elliptic Curves
Point Pattern of Reduced Elliptic Curves
Integer Points of First Region as Element Set
Reduced Point Additive Operation
Modular Arithmetic Reduction on Rational Numbers
Reduced Point Additive Operation Improved
What Is Reduced Elliptic Curve Group
Reduced Elliptic Curve Group - E23(1,4)
Reduced Elliptic Curve Group - E97(-1,1)
Reduced Elliptic Curve Group - E127(-1,3)
Reduced Elliptic Curve Group - E1931(443,1045)
Finite Elliptic Curve Group, Eq(a,b), q = p^n
tinyec - Python Library for ECC
ECDH (Elliptic Curve Diffie-Hellman) Key Exchange
ECDSA (Elliptic Curve Digital Signature Algorithm)