I've just reviewed the Wikipedia entries for RSA and Elliptic Curve, and like most "Scientific American" level articles it tells you WHAT but not WHY. For example, it mentions the totient function, but doesn't explicitly say that the totient function counts the number of remaining group members after all the "zero divisors" have been cast out. Similarly, there is a level of detail in the ECC stuff that is missing, although they did make the major distinction between n>3 systems (which is the example usually used in "Scientific American" level descriptions) and the n=2 systems that are actually in production use.
It's important to realize that given a number k there are exactly 0 or 1 (distinct) finite fields of size k (as opposed to groups, where there may be many different groups of the same size). Furthermore, whether there is none at all or exactly one can be determined by the prime factorization of k! If k is of the form P ^ j where P is a prime number and j is an integer, there is one unique field of size k. If k does not fit this pattern, there is NO such field. So, there is a finite field of size 25 (5 ^ 2) but there is no such field of size 6! And it's not just as simple as using modulo arithmetic. If you just tried to do the math mod 25, you are shot down by zero-divisors 5, 10, 15, and 20. Instead, you have to do a 2-vector in which each of the two elements are a number modulo 5. Similarly (and this is key) to do a binary elliptic curve system you have a large number (perhaps 192) of vector members each of which is one bit (an integer either 0 or 1). That's what the n=2 means. Also note that the addition modulo 2 used in this system is the XOR operator, so you can use word-wide XOR operations to do addition quickly. The usual pedantry is to consider these vectors to be coefficients in a set of polynomials, where for addition you just add corresponding coefficients modulo the base field size, and then you have to have some prime polynomial to reduce the order of the resulting equation when you multiply. However, the fact that the field is unique (see above) means that even though there may be other ways of looking at it, the results are always going to be the same simply because the base field IS unique. Down to a renaming of elements. And this is a good insight - choice of a different prime polynomial corresponds simply to a renaming of elements (since the base field is unique). Sorry to pontificobloviate. Anybody who is interested in the why might be interested in at least looking at some of the books I've been able to assemble. Anybody else feel free to ignore this.
