ECDSA

• Definition of an elliptic curve: $y^2 = ax^3 + bx + c$

• Elliptic curve that Ethereum uses: $y^2 = x^3 + 7$

• unique characteristic of an elliptic curve is that its possible to connect two points to produce a third point - kind of like a binary operation

• points on an elliptic curve form a group

• identity element - point at infinity (straight line; gradient is infinite)

• if A = (x, y), the inverse of A is A = (x, -y)

• ECDSA malleability attack

• every point has an image across the x-axis (how about the point on the x-axis where the tangent is vertical)

• points on an ECDSA also form an abelian group

• tl;dr of elliptic curve addition:

• we have the points

• in the curve

• which gives us a third point (x_r, y_r)

• ethereum has ECDSA precompiles (ecrecover) so we can do elliptic curve manipulation on chain

• bn128 curve is pairing friendly