Why do we use groups, rings and fields in cryptography? I'm a student of Masters in Cyber Security. I have a habit to understand things from their first principles (at the very beginning). Kindly use any simple mathematical example to answer because I have read plenty of technical answers but they aren't clear to me. For example, we use primes because we know factoring them is extremely hard. Furthermore, we use injectives because we don't want one to many relations in encryption because it will be a disaster on the decryption side. Same like my examples can you provide me an example of using groups that makes sense as primes and injective do?Kindly, let me know what was the actual problem which leads us to use groups in cryptography?
I know what a group is and its properties, I know about abstract algebra, but I don't know why we use them and what happens if we don't use them.
Kindly, let me know what was the actual problem which leads us to use groups in cyptogrpahy?

Well, we use groups and other similar mathematical constructs because:

We found there are problems that appeared to be difficult to solve with those groups

We found ways to translate the difficulty of solving those problems into the cryptographical strength of some object (such as an RSA public key encryption, or a signature), that is, you can't break the cryptographical object (e.g. decrypt the encrypted ciphertext without the key) unless you can solve the hard problem.


The second part is crucial; we know problems that are much harder to solve than the ones we use in cryptography; however we don't know how to exploit those.
That's pretty much the answer; the answer to &quot;why do we use groups&quot; is pretty much the same as Willy Sutton supposedly gave to the question &quot;why do you rob banks&quot; - &quot;because that's where the money is&quot; - that is, we do it because it provides the cryptographical security we're looking for.

I believe if I don't know &quot;why&quot; then everything is useless.

Then you're in trouble with cryptography - one of the things we do not know is why these hard problems are actually hard (if, in fact, they are) - the most we know is that a number of fairly bright people have tried to do a number of rather clever tricks to solve them, and none of them worked.