Looking at elliptic curves over the reals makes for pretty pictures, but really fails to convey why ECC should be secure at all. As an example, the function m

Looking at elliptic curves over the reals makes for pretty pictures, but really fails to convey why ECC should be secure at all. As an example, the function m => m^65537 is a completely useless encryption function over the reals because it's trivial to take 65537-th roots over R (even if the numbers are unwieldy in size), but it's a surprisingly strong encryption function when computed over certain finite groups (i.e. RSA). Similarly, any ECC scheme defined over real numbers is pretty useless. It only gets interesting when you use a finite field. Unfortunately, the math is less simple that way.