Possible Cryptology Projects

This is a list of some project ideas you may find interesting. 
It is not meant to be exhaustive or exclusive, so you are welcome
to propose a project not on this list. 

Crypto Pedagogy
  Develop tools to help those learning crypto to understand, explore,
  and experiment with cryptographic methods. These can include 
  classic and modern symmetric ciphers and cryptanalysis, as well as
  mathematical tools/mechanisms important to crypto, and public key
  cryptography. The emphasis is on exposing clearly what is going on
  with each of the methods.

Secure Hash Function/Pseuo-random Number Generator
  Devise and/or test secure hash functions/PRNGs. This would involve
  researching the various tests for randomness and applying them to
  various hash functions, and possibly developing a new or tweaking
  an existing secure hash function or PRNG. In particular, I am 
  interested in the one based on cellular automata, and the student
  (Garrett Cotton) who started the project is available for help and
  guidance. 

Factorization/Primality Testing
  Research the current state of the art in prime factorization and/or
  primality testing. Write a term paper and/or develop code to 
  implement various approaches. 

Protocol Vulnerability Report
  Pick a particular system or protocol in use today and research its 
  vulnerabilities. Write a report and/or develop code to exploit these 
  vulnerabilities and report on your results. There are numerous systems 
  from SSL/TLS to WEP and many, many more that have identified problems 
  in them ranging from implementation to the protocol itself.

Cryptographic Function Vulnerability Report
  Pick a particular cryptographic function (cipher, hash function) in use 
  today and research its vulnerabilities. Write a report and/or develop 
  code to exploit these vulnerabilities. There are numerous ciphers from
  RC4 to RSA that have identified weaknesses in them. 

Back Doors
  In light of recent revelations concerning certain government agency
  influence in the systems that are widely used for cryptography (and
  security in general), pick a particular system or type of system and
  research how back door access or weaknesses could have been introduced
  into it (them). Then write a report and/or implement code to try to
  exercise or demonstrate the weakness/back door.

Challenges to Cryptography
  The world is changing in many ways, much of them driven by new and
  possibly disruptive technologies. Security is often an afterthought
  in the rush to get a new gizmo to market, which can leave people and
  systems unprotected or poorly protected. From the cloud to the 
  Internet of Things, from utilities to health care, pick a particular
  type of system and research the security issues it has, and how they
  are being addressed with cryptography (or not). 

Usability
  "If it ain't usable, it ain't secure"
  I say this and it is true. Where are there mis-matches between the
  technology we could have and the use of that technology, and why?
  For example, email has long had provisions for encryption, but nobody
  uses it. Why not? What are the barriers to adoption? How can these be
  overcome? Write a research paper or do a proof-of-concept demo of a
  system that improves usability. 

There are many other good and interesting projects you can do. 
Explore and see what catches your imagination and interest, then follow
up and determine what you can do with it.