By Jimmy Song, Principal Blockchain Architect
“…Finite fields are one thing and elliptic curves another. We can combine them by defining an elliptic curve over a finite field. All the equations for an elliptic curve work over a finite field. By “work”, we mean that we can do the same addition, subtraction, multiplication and division as defined in a particular finite field and all the equations stay true. If this sounds confusing, it is. Abstract algebra is abstract!
Of course, the elliptic curve graphed over a finite field looks very different than an actual elliptic curve graphed over the Reals. An elliptic curve over real numbers looks like this:
An elliptic curve over a finite field looks scattershot like this:
How to calculate Elliptic Curves over Finite Fields
Let’s look at how this works. We can confirm that (73, 128) is on the curve y2=x3+7 over the finite field F137. …”