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Formal definition

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

• ( pk , msk ) ← Setup ( 1 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.¹

Applications

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.

History

Functional encryption was proposed by Amit Sahai and Brent Waters in 2005² 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.⁴⁵⁶⁷

References

1. 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. 978-1-4503-2029-0 ↩

2. Amit 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. 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. http://eprint.iacr.org/2010/543.pdf ↩

4. 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. 978-1-4503-2029-0 ↩

5. Gorbunov, Sergey; Hoeteck Wee; Vinod Vaikuntanathan (2013). "Attribute-Based Encryption for Circuits". Proceedings of STOC. ↩

6. Sahai, Amit; Brent Waters (2012). "Attribute-Based Encryption for Circuits from Multilinear Maps" (PDF). arXiv:1210.5287. http://eprint.iacr.org/2012/592.pdf ↩

7. 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. 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. 978-3-642-40083-4 ↩