EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, by Dr. Herong Yang
Reduced Elliptic Curve Group - E127(-1,3)
This section provides an example of a reduced Elliptic Curve group E127(-1,3). An example of addition operation is also provided.
Let's take a look at another reduced elliptic curve group, E127(-1,3), as discussed in "Elliptic Curve Cryptography: finite fields and discrete logarithms" by Andrea Corbellini at andrea.corbellini.name/2015/05/23/elliptic-curve-cryptography-finite-fields-and-discrete-logarithms/.
Here is the reduced elliptic curve group using modular arithmetic of prime number 127, E127(-1,3):
y2 = x3 - x + 3 (mod 127)
The above diagram provides all points in this group. It also illustrates an example of the reduced addition operation:
Given two points on the curve: P = (16,20) Q = (41,120) Draw a straight line passing through A and B, And wrap the line around when it reaches the boundary of the region. It will reach another point -R on the curve. Take the symmetric point R of -R: R = A + B R = (86,81)
Let's verify R = P+Q using algebraic equations given by the reduced elliptic curve group definition:
For any two given points on the curve: P = (xP, yP) = (16,20) Q = (xQ, yQ) = (41,120) R = P + Q is a third point on the curve: R = (xR, yR) Where: xR = m2 - xP - xQ (mod p) (11) yR = m(xP - xR) - yP (mod p) (12) m(xP - xQ) = yP - yQ (mod p) (18) Calculation: m*(16-41) = 20-120 (mod 127) m*(-25) = -100 (mod 127) 102*m = 27 (mod 127) m = 27 * 1/102 (mod 127) m = 27*66 (mod 127) m = 1782 (mod 127) m = 4 xR = 4*4 - 16 - 41 = -41 (mod 127) xR = 86 yR = 4*(16 - 113) - 20 = -300 (mod 127) yR = 81 C = (86,81)
The result from algebraic equations matches the geometrical result!
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)