Below are some problems related to computer security (specifically poorly implemented security). Do Them. You are free to use any language and environment you like to complete them. The problems require familiarity with programming, but not necessarily with applied cryptography or computer security in general. An understanding of how some of the algorithms and tools in the problems work will be of use, so as a starting point you can find some resources to help you begin here. If this topic is new to you, you may want to start with the Tutorial Problems.
ECDSA Nonce Recovery
Nonce reuse leading to private key recovery, this time with elliptic curves
Elliptic Curve Private Key Recovery
Given the public parameters of an elliptic curve, recover the poorly chosen private key