Elliptic Curves, Base Points and Scalars

I discovered the Diffie-Hellman method, and I loved it. Another great discovery was how RSA worked. But the real eye-opener was the discovery of how elliptic curves work in elliptic curve cryptography (ECC). It is a truly wonderful technique and protects your online safety like no other method. We see it in key exchanges with ECDH (Elliptic Curve Diffie Hellman), and in digital signatures with ECDSA (Elliptic Curve Digital Signature Algorithm).

Within elliptic curves, we can have a scalar value (a) and a base point on the curve (G). We then perform an add operation to produce another point on the curve as aG. This is the equivalent operation of adding the point G for a times. Normally we would define aG as a public key point, and a as a private key scalar value. Typically these operations are done within a finite field defined by a prime number (p).

In the case of Curve 25519, this prime number is 2²⁵⁵−19. With the Ed25519 curve, we only need one of the co-ordinate points, and so the public key point is the same length as the private key scalar value. With other curves (such as…