EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, by Dr. Herong Yang
Reduced Point Additive Operation Improved
This section describes the improved version of the reduced point additive operation by applying the same modular arithmetic reduction on the parameter m as the reduced elliptic curve equation.
In previous tutorials, we learned how to apply the same modular arithmetic reduction as the reduced elliptic equation on coordinates (xR, yR) of the resulting point R of the point addition operation:
Reduced addition operation based rule of chord operation: For any two given points on the curve: P = (xP, yP) Q = (xQ, yQ) 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) If P != Q, m is determined by: yP - yQ m = --------- (10) xP - xQ If P = Q, m is determined by: 3(xP)2 + a m = --------- (6) 2(yP)
Because of the associativity of modular arithmetic, we can actually apply the same modular arithmetic reduction on the parameter m, to bring it to integer.
Improved reduced addition operation based rule of chord operation: For any two given points on the curve: P = (xP, yP) Q = (xQ, yQ) 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) If P != Q, m is determined by: yP - yQ m = --------- (mod p) (16) xP - xQ If P = Q, m is determined by: 3(xP)2 + a m = --------- (mod p) (17) 2(yP)
If you are not comfortable with modular arithmetic division, we can rewrite those equations as in modular arithmetic multiplication:
Improved reduced addition operation based rule of chord operation: For any two given points on the curve: P = (xP, yP) Q = (xQ, yQ) 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) If P != Q, m is determined by: m(xP - xQ) = yP - yQ (mod p) (18) If P = Q, m is determined by: 2m(yP) = 3(xP)2 + a (mod p) (19)
You can prove that equations (11), (12), (18) and (19) will provide the same (xR, yR) as questions (11), (12), (10) and (6). But equations (11), (12), (10) and (6) are better, because only 1 possible modular multiplicative inverse operation is needed on m. And equations (11), (12), (18) and (19) requires 2 possible modular multiplicative inverse operations on xR and yR.
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)