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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

• Applications

• History

• References

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

• ( pk , msk ) ← ← Setup ( 1 λ λ ) {\displaystyle ({\text{pk}},{\text{msk}})\leftarrow {\textsf {Setup}}(1^{\lambda })} : creates a public key pk {\displaystyle {\text{pk}}} and a master secret key msk {\displaystyle {\text{msk}}} .

• sk ← ← Keygen ( msk , f ) {\displaystyle {\text{sk}}\leftarrow {\textsf {Keygen}}({\text{msk}},f)} : uses the master secret key to generate a new secret key sk {\displaystyle {\text{sk}}} for the function f {\displaystyle f} .

• c ← ← Enc ( pk , x ) {\displaystyle c\leftarrow {\textsf {Enc}}({\text{pk}},x)} : uses the public key to encrypt a message x {\displaystyle x} .

• y ← ← Dec ( sk , c ) {\displaystyle y\leftarrow {\textsf {Dec}}({\text{sk}},c)} : uses secret key to calculate y = f ( x ) {\displaystyle y=f(x)} where x {\displaystyle x} is the value that c {\displaystyle c} encrypts.

The security of FE requires that any information an adversary learns from an encryption of x {\displaystyle x} is revealed by f ( x ) {\displaystyle f(x)} . Formally, this is defined by simulation.cite-ref-0-1-0[1]

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

Functional encryption was proposed by Amit Sahai and Brent Waters in 2005cite-ref-2[2] and formalized by Dan Boneh, Amit Sahai and Brent Waters in 2010.cite-ref-3[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.cite-ref-0-1-1[1]cite-ref-4[4]cite-ref-5[5]cite-ref-6[6]

cite-note-0-11. ↑ citerefgoldwasserkalaiada-popavaikuntanathan2013Goldwasser, 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.

cite-note-22. ↑ citerefamit-sahaibrent-waters2005Amit Sahai; Brent Waters (2005). "Fuzzy Identity-Based Encryption". In Ronald Cramer (ed.). Advances in Cryptology. EUROCRYPT 2005: 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings. Springer. pp. 457–473. ISBN 978-3-540-25910-7. LCCN 2005926095.

cite-note-33. ↑ citerefbonehamit-sahaibrent-waters2011Boneh, Dan; Amit Sahai; Brent Waters (2011). "Functional Encryption: Definitions and Challenges" (PDF). Proceedings of Theory of Cryptography Conference (TCC) 2011.

cite-note-44. ↑ citerefgorbunovhoeteck-weevinod-vaikuntanathan2013Gorbunov, Sergey; Hoeteck Wee; Vinod Vaikuntanathan (2013). "Attribute-Based Encryption for Circuits". Proceedings of STOC.

cite-note-55. ↑ citerefsahaibrent-waters2012Sahai, Amit; Brent Waters (2012). "Attribute-Based Encryption for Circuits from Multilinear Maps" (PDF). arXiv:1210.5287.

cite-note-66. ↑ citerefgoldwasseryael-kalairaluca-ada-popavinod-vaikuntanathan2013Goldwasser, 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. Lecture Notes in Computer Science. Vol. 8043. pp. 536–553. doi:10.1007/978-3-642-40084-1_30. hdl:1721.1/91472. ISBN 978-3-642-40083-4.