Hi Guys , last article we spoke about addition , one of the most important “ ” operations on eliptic curve arithmetic . There’s been great feedback highlihghting that the rule might not be . invented “if a line crosses two points , it will cross a third point” absolute That’s is true , there’s a few exceptions and today I will try to explain one , what would happen if point and share the same position (x and y wise): P Q , well this is called “Point Doubling” also referred as adding a point to itself. So how do you add two points (P and Q ) that share the same location? This is nothing more than another invented operation and as we saw in EC addition it is represented by another mathematical identity , as well as a different “ ” equation. Slope Point Doubling Slope: To calculate the slope when we apply the following equation: P == Q s = (3 * Px ** 2 +a ) / (2 * Py) Slope in point doubling That will give you the value of the slope , remember that the slope plus some other domain parameters are curcial to then caculate addition or in this case point doubling. Point Doubling: Point doubling or “adding a point to itself” follows a very simple equation , and of course it requires the slope and the domain parameters that we’ve seen in previous articles. But basically it goes like this: Point Doubling in action RX = S ** 2 - 2 * PXRY = -1 * (PY + S * ( RX - PX)) And again the same catch , R becomes the negative reflection of R , so it looks something like: Reflection of R Conclusion: So point doubing is the second invented arithmetic operation we’ve talked about when it comes to Elliptic Curves , and some of the same rules apply than with addition: by knowing X you know Y R is the reflection of -R over the X axis Some gisted code so you can have a play: PS: Thanks for all the comments and questions
