Why are finite groups used in cryptography? Most of the cryptographic schemes I know are all based on group theory, e.g. they use finite groups.
Can someone explain why is that the case? And why not base the schemes on elements and operations say from natural numbers or some other infinite sets? E.g. $\mathbb{Q, Z, R}$?
Can someone explain why is that the case?


Cryptosystems based on finite sets have two very nice properties:


There is an upper bound to the size of all involved mathematical objects. This also allows one to predict things like memory usage rather well. This also means that the precision / memory you need can't grow arbitrarily / infinitely.
You can actually draw numbers at random according to some distribution. You can't draw a number uniformly at random from the natural numbers because you'd have probability $0$ of drawing any single specific one. Also in theory if you wanted to sample a normal distribution, you could get $10^{10^{100}}$ drawn (even though that's unlikely) which also brings back the first point.