Abelian Group on Elliptic Curve

EC Cryptography Tutorials - Herong's Tutorial Examples

∟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. For example:

P + Q = R                   The "closure" condition
P + Q = Q + P               The "commutativity" condition
(P + Q) + S = P + (Q + S)   The "associativity" condition
P + ∞ = P                   The "identity" condition
P + (-P) = ∞                The "symmetry" condition