Applications of Probability Theory to Coding/Cryptography Theory 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 ;)
 A: The most important issue I see here is to get yourself ready for a PhD
program in probability. Depending on what is available in your current
program, presumably an undergraduate major in mathematics, you should
consider the following. Make sure you really know calculus through
multiple integration and and infinite series. Take a course in real
analysis. Learn something about measure theory. Take a mathematical
statistics course (or at least a statistics course with a calculus
prerequisite). Find out what Bayesian statistics is. And a course in linear algebra would certainly be useful.
Look at the web sites of departments to which you might feasibly be
admitted; some of them give specifics of the background they want
applicants to have.
You will be exposed to a lot of new ideas and possibilities in a PhD
program in probability. The CS related fields you mention are changing
so rapidly and progress in them is tightly held and not always public
information, that it might not be possible for you to prepare yourself
for those fields now. It is entirely possible that your objectives will
change as you progress towards your PhD in probability. So my suggestions
along lines of the CS topics you mention are more tentative and possibly not as important as
my suggestions on preparing for the probability PhD.
Of course, you should take obviously relevant courses in computer
science. Some less obvious possibilities are number theory, group theory,
and quantum mechanics. Learn to use a statistical programming package/language
such as R or Python (with Scripy)--data management, standard statistical analysis, simulation (e.g., permutation tests, bootstrapping, stochastic
processes, maybe even Gibbs sampling).
Outside of coursework, try to keep current in 
developments in various kinds of artificial intelligence, and proposals
for making the Internet more secure. You may find out more about these
things by reading the New York Times than from CS courses. (Generally speaking, it is a real
struggle for colleges and universities to keep current with developing
trends in CS because the non-academic job market for potential faculty members
is so active.) Much of the innovation in these areas is going on at 
places like Google and Facebook in the private sector and NSA in the government
sector, and you may only be able to get miscellaneous rumors, hints, and clues about the most important developments.
Personal note: All of that said, I went off to a PhD program in probability and statistics
out a first rate undergrad program, but with only a one-quarter course out of Feller's probability book to give me a clue
what I was getting into. I did survive, and over 50 years later I'm
still learning stuff. So I can't say planning is everything. Being at the
right place at the right time is important and unpredictable. (But never
having the curiosity to be anywhere interesting any of the time is clearly not
an optimal strategy.)
