The Beauty of Elliptic Curves: Meet Twisted Edwards
I must admit, I loved discovering the Diffie-Hellman method, and then the beauty of RSA, but it was elliptic curves that really has me hooked. And basically the security that you are using to access this page is likely to be based an elliptic curves. The basic form is y² = x² + ax +b, but we can also have a Montgomery curve which is y²=x³+ax²+x. But the one I like most is the twisted Edwards curve, and which takes the form of
A plot of 10x²+y² = 1 + 6x²y² is [here]:
For Ed25519 — based on Curve 25519 — it has a finite field defined by a prime number of p=2²⁵⁵−19, a=-1, d=37095705934669439343138083508754565189542113879843219016388785533085940283555 and the base point is at x=5866666666666666666666666666666666666666666666666666666666666666.
The simplest operations we have is to take a base point G on this curve, and then perform point addition. We always end up with another point on the curve. So 2G is equal to G+G and where we get a new point on the elliptic curve. For 3G we can have G+2G, and so on. We can also perform a scalar multiplication, such as taking a scalar of 3 and finding 3G. In the following code we have three scalar values of…
