I'm looking to do a PhD in probability, with my project(s) having applications to for example the following (non-exhaustive).
Coding theory: eg algorithms for sending data quickly which can be decoded rapidly with a high probability of correct decoding, including error correction (with high probability) codes; also more general information theory
Cryptography: similar to above, but ways of securely encrypting given a key and then decrypting with high probability of success given the key but with very low probability of success without the key
Computer science more generally: fast algorithms for sampling (eg Markov mixing)
I'm after some advice as to possible topics/projects that I could consider in this field. Most of what I've looked at only uses very basic probability (eg discrete conditioning on events); this isn't what I'm lookin for. I'm interested in the PhD being in probability, but with above applications. I did probability for Part III, so this is the level of probability I want to use. (Yes, I realise real research isn't quoting/proving theorems that other people have used, but hopefully this is clear what I'm meaning.) It should include, for example, hitting/mixing times of random walks, maybe the odd martingale, even maybe some percolation. Another example might be to do with showing concentration results.
If people have any suggestions, then I'd very very interested in hearing about them. Thank you.
Lastly, hopefully this is on topic. If you have any advice how to make it more on topic, then please let me know... preferably without being too rude ;)