Abelian Group on Elliptic Curve
This section demonstrates that an Abelian Group can be defined with all points on an elliptic curve with the "rule of chord" operation.
The addition operation we have defined earlier on an elliptic curve
actually helps to create an Abelian Group on the curve:
- The set of elements is the set of all points on the curve and the infinity point.
- The binary operation is the addition operation we have defined earlier.
- The identity element is the infinity point.
You can verify that all 5 Abelian Group conditions are satisfied.
P + Q = R The "closure" condition
P + Q = Q + P The "commutativity" condition
(P + Q) + S = P + (Q + S) The "associativity" condition
P + 0 = P The "identity" condition
P + (-P) = 0 The "symmetry" condition
Table of Contents
About This Book
Geometric Introduction to Elliptic Curves
Algebraic Introduction to Elliptic Curves
►Abelian Group and Elliptic Curves
What Is Abelian Group
Niels Henrik Abel and Abelian Group
Multiplicative Notation of Abelian Group
Additive Notation of Abelian Group
Modular Addition of 10 - Abelian Group
Modular Multiplication of 10 - Not Abelian Group
Modular Multiplication of 11 - Abelian Group
►Abelian Group on Elliptic Curve
Discrete Logarithm Problem (DLP)
Generators and Cyclic Subgroups
Reduced Elliptic Curve Groups
Elliptic Curve Subgroups
tinyec - Python Library for ECC
EC (Elliptic Curve) Key Pair
ECDH (Elliptic Curve Diffie-Hellman) Key Exchange
ECDSA (Elliptic Curve Digital Signature Algorithm)
ECES (Elliptic Curve Encryption Scheme)
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