Functional encryption (FE) is a generalization of public-key encryption in which possessing a secret key allows one to learn a function of what the ciphertext is encrypting.

More precisely, a functional encryption scheme for a given functionality ${\displaystyle f}$ consists of the following four algorithms:

• ${\displaystyle ({\text{pk}},{\text{msk}})\leftarrow {\textsf {Setup}}(1^{\lambda })}$: creates a public key ${\displaystyle {\text{pk}}}$ and a master secret key ${\displaystyle {\text{msk}}}$.
• ${\displaystyle {\text{sk}}\leftarrow {\textsf {Keygen}}({\text{msk}},f)}$: uses the master secret key to generate a new secret key ${\displaystyle {\text{sk}}}$ for the function ${\displaystyle f}$.
• ${\displaystyle c\leftarrow {\textsf {Enc}}({\text{pk}},x)}$: uses the public key to encrypt a message ${\displaystyle x}$.
• ${\displaystyle y\leftarrow {\textsf {Dec}}({\text{sk}},c)}$: uses secret key to calculate ${\displaystyle y=f(x)}$ where ${\displaystyle x}$ is the value that ${\displaystyle c}$ encrypts.

The security of FE requires that any information an adversary learns from an encryption of ${\displaystyle x}$ is revealed by ${\displaystyle f(x)}$. Formally, this is defined by simulation.^[1]

Functional encryption generalizes several existing primitives including Identity-based encryption (IBE) and attribute-based encryption (ABE). In the IBE case, define ${\displaystyle F(k,x)}$ to be equal to ${\displaystyle x}$ when ${\displaystyle k}$ corresponds to an identity that is allowed to decrypt, and ${\displaystyle \perp }$ otherwise. Similarly, in the ABE case, define ${\displaystyle F(k,x)=x}$ when ${\displaystyle k}$ encodes attributes with permission to decrypt and ${\displaystyle \perp }$ otherwise.

Functional encryption was proposed by Amit Sahai and Brent Waters in 2005^[2] and formalized by Dan Boneh, Amit Sahai and Brent Waters in 2010.^[3] Until recently, however, most instantiations of Functional Encryption supported only limited function classes such as boolean formulae. In 2012, several researchers developed Functional Encryption schemes that support arbitrary functions.^[1][4][5][6]