Abstract: In a traditional public key encryption system, data is encrypted to a particular individual who is fixed by a public key. Functional encryption (FE) gives a new way of sharing encrypted data, and removes this restriction on receivers, where a sender can specify a wide variety of receiver sets, and has fine-grained control over revealed information on encrypted data, under the common public key. Several specific encryption systems fall into this general framework, e.g. identity-based encryption (IBE), inner product encryption (IPE), and attribute-based encryption (ABE), which realize flexible private data sharing. This presentation will introduce the notion of FE, and describe recent progress on security enhancement techniques on FE. The cryptosystems are implemented by using a pairing operation on an elliptic curve, which has achieved many new applications including FE in recent years. I also give a brief summary of the attractive mathematical mechanism, e.g., the algorithm, parameter selection, and underlying security.