Your friend is not going to get PhD thesis topics by asking his friend to post on HN.
Regardless, here's what I think are interesting areas in recent crypto:
- Performance improvements in fully homomorphic encryption, starting with Gentry's work in 2009.
- Practical applications of secure multiparty computation, e.g. Dyadic Security and Google's SMC work.
- Non-NIST standards with actual adoption like Curve25519 and Chacha20-Poly1305
- Functional Encryption: http://eprint.iacr.org/2010/543
- Post-quantum crypto like New Hope (https://eprint.iacr.org/2015/1092) and Supersingular Isogenies (http://eprint.iacr.org/2011/506)
- Candidate functions for Multilinear Maps, e.g. https://eprint.iacr.org/2012/610
- Hardware-based secure enclaves like SGX