From Single-Key to Collusion-Resistant Secret-Key Functional Encryption by Leveraging Succinctness

We show how to construct secret-key functional encryption (SKFE) supporting unbounded polynomially many functional decryption keys, that is, collusion-resistant SKFE solely from SKFE supporting only one functional decryption key. The underlying single-key SKFE needs to be weakly succinct, that is, the size of its encryption circuit is sub-linear in the size of functions. We show we can transform any quasi-polynomially secure single-key weakly-succinct SKFE into quasi-polynomially secure collusion-resistant one. In addition, if the underlying single-key SKFE is sub-exponentially secure, then so does the resulting scheme in our construction. Some recent results show the power and usefulness of collusion-resistant SKFE. From our result, we see that succinct SKFE is also a powerful and useful primitive. In particular, by combining our result and the result by Kitagawa, Nishimaki, and Tanaka (ePrint 2017), we can obtain indistinguishability obfuscation from sub-exponentially secure weakly succinct SKFE that supports only a single functional decryption key.