Functional encryption is a cryptographic concept that allows a key holder to learn a specific function of encrypted data without learning anything else about the data. The paper introduces formal definitions and security challenges for functional encryption. It shows that defining security for functional encryption is non-trivial, as natural game-based definitions may be inadequate for certain functionalities. A simulation-based definition is proposed, which cannot be satisfied in the standard model but can be satisfied in the random oracle model. The paper also discusses how existing encryption concepts, such as attribute-based encryption and predicate encryption, can be mapped to functional encryption. It highlights several open problems in the field, including the construction of functional encryption for more general functionalities and understanding the relative power of different functionalities. The paper also presents a formal syntax for functional encryption and discusses various sub-classes, including predicate encryption with public index, anonymous identity-based encryption, hidden vector encryption, and inner product predicate encryption. It concludes with security definitions for functional encryption, including game-based and simulation-based definitions, and shows that certain functionalities are insufficiently secure under game-based definitions. The paper also presents a simulation-secure brute-force construction for functional encryption and shows that public index schemes secure under game-based definitions are also secure under simulation-based definitions in the random oracle model. The paper concludes with discussions on extending functional encryption to multiple parameters, multiple systems, and hiding information about key capabilities.Functional encryption is a cryptographic concept that allows a key holder to learn a specific function of encrypted data without learning anything else about the data. The paper introduces formal definitions and security challenges for functional encryption. It shows that defining security for functional encryption is non-trivial, as natural game-based definitions may be inadequate for certain functionalities. A simulation-based definition is proposed, which cannot be satisfied in the standard model but can be satisfied in the random oracle model. The paper also discusses how existing encryption concepts, such as attribute-based encryption and predicate encryption, can be mapped to functional encryption. It highlights several open problems in the field, including the construction of functional encryption for more general functionalities and understanding the relative power of different functionalities. The paper also presents a formal syntax for functional encryption and discusses various sub-classes, including predicate encryption with public index, anonymous identity-based encryption, hidden vector encryption, and inner product predicate encryption. It concludes with security definitions for functional encryption, including game-based and simulation-based definitions, and shows that certain functionalities are insufficiently secure under game-based definitions. The paper also presents a simulation-secure brute-force construction for functional encryption and shows that public index schemes secure under game-based definitions are also secure under simulation-based definitions in the random oracle model. The paper concludes with discussions on extending functional encryption to multiple parameters, multiple systems, and hiding information about key capabilities.