What are the relations between these number theorems and ECC?

Question

I'm currently working on a bigger assignment where I have to write about the connection between bitcoin/blockchain and maths. I've gotten far, but one of my tasks is to talk about principal rest, modulus, greatest common divisor, the fundamentals of arithmetic theorem, fermats theorem, eulers theorem and function.

The point of the assignment is to make everything as a flow, and not go out of context, so I'm wondering what all of these mathematical theories has to do with bitcoin and blockchain.

Any answers or sources to where I can read about it, would be highly appreciated!

Answer 1

Only on a high level (because I won't do your homework for you)

principal rest and modulus create groups in particular F_p which are fields arising from the the ring of integer modulo a prime. You can look at elliptic curves over such a field (which is a discrete set of points) and define a geometric point addition which (surprisingly?) is a group again. If you find cyclic subgroup with a generator of high (prime) order it will be useful for cryptography for two reasons:

1. The discrete log problem seems to be difficult
2. Related to 1.) factorizing the large generator into its primes is hard (which is related to the fact that computing the greatest common divisor) is also hard.

I hope you see the gist where this is going. I strongly recommend reading up the wikipedia articles of the topics you mention. But connecting the dots is obviously up to you.