# RareSkills Authors - Elliptic Curve Point Addition (Highlights) ![rw-book-cover|256](https://static.wixstatic.com/media/935a00_47a61dc12ed54c2b9a36415cceea5b54~mv2.png/v1/fill/w_1000,h_978,al_c,q_90,usm_0.66_1.00_0.01/935a00_47a61dc12ed54c2b9a36415cceea5b54~mv2.png) ## Metadata **Review**:: [readwise.io](https://readwise.io/bookreview/38631570) **Source**:: #from/readwise #from/reader **Zettel**:: #zettel/fleeting **Status**:: #x **Authors**:: [[RareSkills Authors]] **Full Title**:: Elliptic Curve Point Addition **Category**:: #articles #readwise/articles **Category Icon**:: 📰 **URL**:: [www.rareskills.io](https://www.rareskills.io/post/elliptic-curve-addition) **Host**:: [[www.rareskills.io]] **Highlighted**:: [[2024-03-13]] **Created**:: [[2024-03-16]] ## Highlights - I like to think of the identity element as “the point that is nowhere” because if you combine nowhere with any real point, nothing changes. Annoyingly, mathematicians call this point, the identity element “the point at infinity.” ([View Highlight](https://read.readwise.io/read/01hrvk5zdmps89pakayhgc7sn2)) ^692156313 - Ahh, but remember, we can define sets however we like! We define the set that makes up the elliptic curve as points on the elliptic curve and the nowhere point. ([View Highlight](https://read.readwise.io/read/01hrvk67nevat85a5sk93n43tr)) ^692156327 - As long as x is not "perfectly vertical" it will always intersect with a third point eventually. ([View Highlight](https://read.readwise.io/read/01hrvk8qk5zcwj3ymvfg2emr9c)) ^692161577 - So what we have is an important lemma here: elliptic curves eventually “rise faster” than straight lines as x gets larger. This means that eventually, the blue curve will “catch up” to the purple line and they will intersect… ([View Highlight](https://read.readwise.io/read/01hrvkb4tck0qazq6dhbercfqx)) ^692163990 - If a straight line crosses an elliptic curve at exactly two points, then it must be perfectly vertical. ([View Highlight](https://read.readwise.io/read/01hrvkc1dkn3pwcbydsasr5fnh)) ^692165490 - The identity element is the “point at infinity” we alluded to earlier is simply the point “way up there” when we draw a vertical line. ([View Highlight](https://read.readwise.io/read/01hrvkdkm05g1afmn1vj11jnpx)) ^692167881 The third point intersects at infinity. - In it’s current form, it has a bug if we add two points where the intersection happens in the middle. ([View Highlight](https://read.readwise.io/read/01hrvkg313gq4cjynh1pn7jsf4)) ^692171689 - Adding a point to itself is like bringing two points infintesimally close to each other until they become the same point. When this convergence happens, the slope of the line will lie tangent to the curve. So adding a point to itself is simply taking the derivative at that point, getting the intersection, then flipping the y axis. ([View Highlight](https://read.readwise.io/read/01hrvqs3qnsq01kw3qh2qyg6dq)) ^692194173 - 1000A = 512A ⊕ 256A ⊕ 128A ⊕ 64A ⊕ 32A ⊕ 8A ([View Highlight](https://read.readwise.io/read/01hrvqtn38rxkzjq2t4wxqtwpv)) ^692194220 - Scalar “multiplication” isn’t “distributive” the way we would think about normal algebra. It’s just shorthand for rearranging the order in which we add P to itself. Under the hood, we just added P to itself (a + b + c) times. The order we do it in doesn’t matter because of associativity. ([View Highlight](https://read.readwise.io/read/01hrvqxfqmbhhshtnpeay9zt5g)) ^692194332