# Functional encryption

General Amit Sahai Brent Waters, Dan Boneh, Shafi Goldwasser, Yael Kalai Public-key encryption Homomorphic encryption

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.

## Formal definition

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]

## Applications

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.

## History

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]

## References

1. ^ a b Goldwasser, Shafi; Kalai, Yael; Ada Popa, Raluca; Vaikuntanathan, Vinod; Zeldovich, Nickolai (2013). Reusable garbled circuits and succinct functional encryption - Stoc 13 Proceedings of the 2013 ACM Symposium on Theory of Computing. New York, NY, USA: ACM. pp. 555–564. ISBN 978-1-4503-2029-0.
2. ^ EUROCRYPT (2005-05-09). Fuzzy Identity-Based Encryption - Advances in Cryptology - EUROCRYPT 2005: 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings. Springer Science & Business Media. ISBN 978-3-540-25910-7.
3. ^ Boneh, Dan; Amit Sahai; Brent Waters (2011). "Functional Encryption: Definitions and Challenges" (PDF). Proceedings of Theory of Cryptography Conference (TCC) 2011.
4. ^ Gorbunov, Sergey; Hoeteck Wee; Vinod Vaikuntanathan (2013). "Attribute-Based Encryption for Circuits". Proceedings of STOC.
5. ^ Sahai, Amit; Brent Waters (2012). "Attribute-Based Encryption for Circuits from Multilinear Maps" (PDF). arXiv:1210.5287. {{cite journal}}: Cite journal requires |journal= (help)
6. ^ Goldwasser, Shafi; Yael Kalai; Raluca Ada Popa; Vinod Vaikuntanathan; Nickolai Zeldovich (2013). "How to Run Turing Machines on Encrypted Data" (PDF). Advances in Cryptology – CRYPTO 2013. pp. 536–553. doi:10.1007/978-3-642-40084-1_30. ISBN 978-3-642-40083-4. {{cite book}}: |journal= ignored (help)