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Michael Zhang 2022-03-06 20:00:00 -06:00
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content/posts/2022-03-03-learn-by-implementing-elliptic-curve-crypto.md → content/posts/2022-03-04-learn-by-implementing-elliptic-curve-crypto.md
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@ -35,10 +35,9 @@ production, but it's not great for understanding the algorithm.
## Basic Ideas ## Basic Ideas
ECC starts with the idea that starting with an elliptic curve formula like $y^2 ECC starts with the idea that starting with an elliptic curve formula like $y^2
= x^3 + ax + b$ that operates over a finite field $\mathbb{F}_p$, and defining a = x^3 + ax + b$ that operates over a finite field $\mathbb{F}_p$, and defining
_custom_ addition operation over two points, you can form a cyclic structure an addition operation over two points, you can form a cyclic structure where
where adding a point to itself some number of times gets you back where you adding a point to itself some number of times gets you back where you started.
started.
The interesting thing about this cyclic structure is that given the starting The interesting thing about this cyclic structure is that given the starting
point $G$, also called the **generator** and some number $n$, you can find the point $G$, also called the **generator** and some number $n$, you can find the
@ -80,12 +79,13 @@ import (
) )
``` ```
> You can run this blog post using [Markout]: > This is a [literate document][literate]. You can run this blog post using [Markout]:
> ``` > ```
> markout -l go content/posts/2022-03-03-learn-by-implementing-elliptic-curve-crypto.md > /tmp/ecc.go > TODO:
> go run /tmp/ecc.go
> ``` > ```
[literate]: https://en.wikipedia.org/wiki/Literate_programming
### Math primitives ### Math primitives
```go ```go
@ -96,8 +96,18 @@ type Point struct {
} }
``` ```
Addition on $P$ and $Q$ is defined by first finding the line $PQ$, determining
the point $-R$ where it intersects the curve again, and then returning $R$. We
can find the line $PQ$ by using high school geometry:
$$\begin{aligned}
(y - y_0) = m(x - x_0)
\end{aligned}$$
```go ```go
func (a Point) Add(b Point) Point { func (A Point) Add(B Point) Point {
// Find the slope between points A and B.
slope := big.NewRat(A.y - B.y, A.x - B.x)
return Point{} return Point{}
} }
``` ```