An Elliptic Curve and its rational points
Adding a Group Law to an elliptic curve
|Elliptic Curve (blue) with two points (P,Q) and their sum (P+Q) plotted, along with construction lines (red)|
Is that all there is to it?
What's that got to do with Cryptography?
- Specifically, fields of characteristic not equal to 2,3. That is, fields where $2\neq0$ and $3\neq 0$. Unfortunately, this obviously means that the results we discuss won't hold in binary fields, but that is rather beyond the scope of this talk.
- Justification for this comes from considering the elliptic curve as a curve in projective space, but for now it suffices that such a point exists.
- Associativity is by far the most complicated to show. This diagram on wikipedia explains the concept behind the proof, although the details are rather involved.
- Even as I write this, I'm sure someone will question the validity of this claim, but it is true that compared to many groups that one could construct in which the required problems are sufficiently hard, point arithmetic on an elliptic curve is comparatively tractable.