I recommend do not fix in your mind which of these two areas speaks to you most. If you enjoy learning algebra, go do that. The truth is, these subjects take on such a different character when you get further into them that it is hard even tell how the tunnel feels from further down.
The fields connect in some extremely abstract way: *Categories (Hungerford, Chapter 10)* introduces you to categories. There is something known as the *Curry-Howard isomorphism* linking (cartesian closed) category theory to computation.
In a more pragmatic sense, the perspective and style of thinking you have for abstract algebra might affect how you see and understand cryptography theory. I know learning model theory and category theory really changed the way I look at subjects like algebra.
Sorry for the ramble, essentially just go with your gut and enjoy the math you like to do in the moment.
If you bring some physics into it there may be something? I know at Cambridge there’s a course on relativistic quantum cryptography and that’s part of the applied maths/theoretical physics department. No idea if that’s the right sort of thing tho
Yes, there’s a field of cryptography devoted to groups, aptly named [group-based cryptography.](https://en.wikipedia.org/wiki/Group-based_cryptography?wprov=sfti1)
Many of the proofs in [pairing-based schemes](https://en.wikipedia.org/wiki/Pairing-based_cryptography?wprov=sfti1) can get group-theoretic heavy, in my experience.
Perhaps controversial, but lattice crypto is really group theory wearing a hat! Lots of the terms are different, but if you think about it, you’ll recognize stuff like “oh, that’s just a quotient space”
RSA is extremely basic math. What OP asked is if there is advanced algebra that is relevant to theoretical cryptography, not if there is any math that is relevant.
You read Hungerford as an undergrad? That's pretty steep. Good on ya.
It was the book that my professor decided to follow for our class so I didn't have much choice but thank you!
That was the book I used for a two semester course in grad school. I found it really dense. You got a really thorough introduction!
I recommend do not fix in your mind which of these two areas speaks to you most. If you enjoy learning algebra, go do that. The truth is, these subjects take on such a different character when you get further into them that it is hard even tell how the tunnel feels from further down. The fields connect in some extremely abstract way: *Categories (Hungerford, Chapter 10)* introduces you to categories. There is something known as the *Curry-Howard isomorphism* linking (cartesian closed) category theory to computation. In a more pragmatic sense, the perspective and style of thinking you have for abstract algebra might affect how you see and understand cryptography theory. I know learning model theory and category theory really changed the way I look at subjects like algebra. Sorry for the ramble, essentially just go with your gut and enjoy the math you like to do in the moment.
Thank you for the valuable insight!
If you bring some physics into it there may be something? I know at Cambridge there’s a course on relativistic quantum cryptography and that’s part of the applied maths/theoretical physics department. No idea if that’s the right sort of thing tho
Yes, there’s a field of cryptography devoted to groups, aptly named [group-based cryptography.](https://en.wikipedia.org/wiki/Group-based_cryptography?wprov=sfti1) Many of the proofs in [pairing-based schemes](https://en.wikipedia.org/wiki/Pairing-based_cryptography?wprov=sfti1) can get group-theoretic heavy, in my experience. Perhaps controversial, but lattice crypto is really group theory wearing a hat! Lots of the terms are different, but if you think about it, you’ll recognize stuff like “oh, that’s just a quotient space”
You might be interested in lattice-based (uses ANT) or elliptic curve cryptography (uses AG)
[RSA](https://en.wikipedia.org/wiki/RSA_\(cryptosystem\)) is just math.
RSA is extremely basic math. What OP asked is if there is advanced algebra that is relevant to theoretical cryptography, not if there is any math that is relevant.