Commutativity of Elliptic Curve Operations

EC Cryptography Tutorials - Herong's Tutorial Examples

∟Geometric Introduction to Elliptic Curves

∟Commutativity of Elliptic Curve Operations

This section describes the associativity of the addition operation on an elliptic curve. P + (Q + S) = (P + Q) + S is true.

With the "infinity point" identified as the "identity element" of our elliptic curve addition and subtraction operations, we can look at the "commutativity" property of those operations. In other words, is the following statement true or not: true or not:

P + Q = Q + P

It turns out that the above statement is absolutely true.

If we look at the first step in our addition operation definition, "Draw a straight line passing P and Q". It doesn't matter if you draw a straight line "from P to Q", or "from Q to P", you will end up with the same straight line, because there is only one straight line passing through P and Q.

If P and Q are the same point, the first step of our addition operation definition says "Draw a straight line passing P and tangent to the elliptic curve". You will end up with the same straight line too, no matter how you draw it.