I'm currently doing a PhD in probability theory, specifically (discrete space) Markov chains and their mixing properties. As well as my current main project, I'm looking to have a side project, eg to work on if I want a (temporary) change from my main project. What I'm doing is fairly applied in the sense that we have certain graphs and we're looking at mixing on them; it's fully rigorous though.
While it's not hugely related to Markov mixing, I'm also very interested in cryptography/cryptanalysis. I expect cryptanalysis is more related to my kind of probability theory, but I could be wrong. I'd like to be able to apply similar principles of my main project (Markov chains, in particular mixing) to this.
I'm interested in using high level probability theory. By this, I don't mean only having to know about discrete distrubutions and conditional probabilities, but more -- for example, material in the Aldous/Fill book. By this I mean that it would be a probability project on (/applied to) cryptography/cryptanalysis.
In a related field, I'd also be interested in similar applications to coding theory, eg codes that error-correct with high probability. I know coding theory is based on basic probability, but again I'd be looking for more than just having to know some basic rules.
After some searching it doesn't appear that anything like this has been done. It might be a nice opportunity to do develop some new tools. After all, Tim Gowers got his field medal for combining combinatorics and functional analysis. Not suggesting I'm on Tim Gowers' level, of course! Bollobas successfully used probabilistic methods to show lots of stuff about graphs (again, no claim to be on his level!). This could include things such as developing algorithms via Markov chains and proving fast (polynomial) mixing. I'd be happy to learn new things -- maybe some machine learning techniques would be helpful? -- in order to do this. But I would like it to be probability at heart.
Now, it might be that there isn't literature on this topic because the tools I want to use just aren't applicable in this situation, or it might just be that people haven't done it it. People have different interests, and maybe this is a less common crossover. Let's hope for the latter!
So I'm wondering if anyone has experience of, or knows about, any such questions or areas of research? I've looked at various literature, but maybe someone of a more cryptographical/cryptanalytical background would know of some such problems.
I'd certainly appreciate any comments. This is a cross-post from Maths Overflow (see here).
Note the highly related question "Current mathematics theory used in cryptography/coding theory". In this, however, there is no mention of probabilistic tools being used there; it's mostly number theoretic or algebraic. Note that there is the tag "probabilistic encryption".