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====== Elliptic Curve Cryptography ======

Mimblewimble relies entirely on Elliptic-curve cryptography (ECC), an approach to public-key cryptography. Put simply, given an algebraic curve of the form y^2 = x^3 + ax + b, pairs of private and public keys can be derived. Picking a private key and computing its corresponding public key is trivial, but the reverse operation public key -> private key is called the discrete logarithm problem, and is considered to be computationally infeasible.

Let's review the basics.

==== Operations ====
These are the relevant mathematical operations we can do on Elliptic-curve points.

=== Addition ===
Given two points, we can add them to one another (or subtract) and the result would be a new point on the curve.
Multiplication - Given a point, we can multiply it any number of times.
Addition
Given three aligned points P, Q and R, their sum is always 0. We treat this as an inherent property of elliptic curves.

''P + Q + R = 0''

We can then write it as:

''P + Q = -R''

So that adding the two points P and Q results in -R, the inverse of R.

If we draw a line passing through P and Q, this line will cross a third point on the curve, R (so that P, Q and R are aligned). If we take the inverse of this point, which is simply the one symmetric to it about the x-axis, we have found the result of adding two curve points, P + Q. Let’s illustrate:

In other words, addition of points is basically hopping around on the curve to a different, seemingly random point; It looks random unless you know the exact operation performed to reach it.

=== Multiplication ===
We can’t multiply a point by another point, but we can multiply a point by a number (scalar). Multiplying point P by scalar k would simply require adding point P onto it self k times. This operation is easily demonstrated by assigning k=2 so that k*P = P+P. To illustrate how it would look like on the curve, we draw a tangent line. You can imagine that the line intersects three points, whereas two of them are P, such that:

''P + P = -R''

To calculate 8*P for e.g. wouldn’t take 8 operations, but only 3; you can find 2*P, then add it onto itself, and then add 4*P onto itself, for the final result of 8*P.

==== Key Pairs ====
An ECC system defines a publicly known constant curve point called the generator point, G. The generator point is used to compute any public key. A key pair consists of:

Private key k – A randomly chosen 256-bit integer (scalar).
Public key P – An Elliptic-curve point derived by multiplying generator point G by the private key.
And more clearly, a public key (of private key k) is as follows:

''P = k*G''

This is easy to compute.

But, if everybody knows points P and G, can they find out what k is? The answer is no; The difficulty of getting from one point to another is precisely the definition of the Elliptic curve discrete logarithm problem.

=== Secp256k1 ===

The specific Elliptic curve that Grin employs is rust-secp256k1 (y2 = x3 + 7) using Schnorr signature scheme.
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