I am currently finishing my master's degree in theoretical maths (mostly courses around algebra, geometric algebra, a bit of Riemann geometry, K-theory etc) and I don't know which area to go into for a Phd. I was interested in cryptography but have never taken any advance classes in coding theory. Is knowledge in "pure math" enough to apply for a PhD in this area? I wanted to do something which uses my theoretical math but into a more applied subject ... any ideas would be very welcome !
A Math Master's is a good start for any Math Ph.D. program. You may need to spend a minute or two convincing an advisor that you know enough about the field to be interested in it, but that won't take specific coursework most of the time.
It's certainly possible to get into PhD programs in fields like computer science, applied math, or theoretical/computational physics or chemistry with a pure math MS. Of course, getting accepted still depends on your grades, letters, test scores, and any published research. You'll also want to use the personal statement to discuss how you will transition or how the knowledge you have will be useful in the new area.