Cryptography is the science of secure computation and communication. The scope of cryptosystems, nowadays, ranges from private-key encryption and authentication to more complex systems such as publickey encryption, electronic voting, secure multi-party computation on large data sets, etc. My specialization is in foundations of cryptography, a field that aims at designing cryptographic protocols with provable security, and computational complexity theory which provides the framework for such studies. My research has significantly improved our understanding of the assumptions behind the way modern cryptosystems are modeled and proved to be secure. Cryptographic assumptions exist in two forms: 1. Physical assumptions formalize the format of an attack against a cryptosystem. For example, is the adversary able to partially modify the internal state of an encryption device (e.g., by sending a computer virus), or are tamper-proof devices available to be used in cryptographic protocols? 2. Computational assumptions describe the computational tasks that are feasible for the adversary. The goal is to design a cryptosystem that is as hard to break for the adversary as performing an unfeasible computational task. Here, computational complexity theory comes to play a central role. The security of the modern computing systems are all based on assumptions and may completely break down if these assumptions are not true. This puts forward to the following fundamental question:

Predicate encryption is an advanced form of public-key encryption that yield high flexibility in terms of access control. In the literature, many predicate encryption schemes have been proposed such as fuzzy-IBE, KP-ABE, CP-ABE, (doubly) spatial encryption (DSE), and ABE for arithmetic span programs. In this paper, we study relations among them and show that some of them are in fact equivalent by giving conversions among them. More specifically, our main contributions are as follows: − We show that monotonic, small universe KP-ABE (CP-ABE) with bounds on the size of attribute sets and span programs (or linear secret sharing matrix) can be converted into DSE. Furthermore, we show that DSE implies non-monotonic CP-ABE (and KP-ABE) with the same bounds on pa-rameters. This implies that monotonic/non-monotonic KP/CP-ABE (with the bounds) and DSE are all equivalent in the sense that one implies another. − We also show that if we start from KP-ABE without bounds on the size of span programs (but bounds on the size of attribute sets), we can obtain ABE for arithmetic span programs. The other direction is also shown: ABE for arithmetic span programs can be converted into KP-ABE. These