The Mighty Base Point (G) in Elliptic Curve Cryptography — And What’s An Order?

Your online security is highly dependent on a simple point on a curve — the base point, G. Every time you make a connection to the Web, you and the server are using the magical point to create a secret key which only you and server know. This is the basic mechanics of the Elliptic Curve Diffie Hellman method (ECDH), and its usage of the base point (G).

You might think that the Diffie-Hellman method is cool, and that RSA has a neat backdoor method, but it is when you discover elliptic curve cryptography that you really see the beauty of cryptography. Much of our original public key and key exchange methods were based on the discrete logarithm method of:

Y=g^x (mod p)

and where we have a base of g and a prime number of p. The value of g needs to be selected carefully so that it maximises the number of possible mapping of x to Y, and where every one of these has a mapping back. If you are interested, when we pick the right value of g and p, the total number of mappings from x to Y will be p-1. This value is known as the order of the mapping. So that:

Y = 7^x (mod 997)

will have an order of 996 (as 7 is a safe generator for 997 [here]). But what about elliptic curves? Well, they have a base point G on the curve…