# Elliptic Curve Crypto , Addition

Hello! , this will be the sequel of:

Elliptic Curve Crypto , The Basics

Today let’s try to tackle **operations** as in addition and multiplication , keep in mind that these are “invented” operations , the fact that it is called addition it’s a mere coincidence , They should’ve been called something different but such is life.

I will split this into a few concepts and then lately write about some of the exceptions.

#### The Slope

The Slope is calculated between two **points** , and it’s got it’s own formulas for it , depending on the location of **P** and **Q** (two points on the curve).

So let’s see these formulas:

Slope is also called **λ** (lambda) , but i think slope makes more sense to me .

The Slope is calculated differently if **P** = **Q**, let’s look at that example:

Alright so that’s pretty simple , **S** or **λ ,** are crucial to then do point doubling and point addition.

#### Point Addition:

So this is one of the most basic elliptic curve operations , and remember that when we say addition **we’re not talking** about addition as **1+1** , it’s a different kind of operation that follows and algorithm as well.

So let’s look at the formula or mathematical identity , it goes like:

**R** would be the third point in question , the result of “adding” **P** and **Q:**

```
RX = S ** 2 - QX - PX
RY = -( QY + S * ( RX - QX))
```

If you’re paying attention , you noticed in the graph above that the resultant of the addition it’s called **-R ,** and that is cause **R**, the result we want , is the reflection over the **X** axis of **-R ,** in such way:

Below is a small code sample of how this should work :

That’s all for today , there’s still one very important operation called point doubling that we need to look at , probably in the next article.

There has been a few **questions** about **exceptions** , situations where a line would cross two points but not a third one , defeating the rule we spoke in the first article:

You can see that neither of these possibilities will cross a third point , we will cover all these in the next article , but if you can’t wait check “**the group law**” in wikipedia :)

Thanks a lot for the tons of reads this is getting and also the comments!