Threshold Fully Homomorphic Cryptosystems recently, Craig Gentry published the first public key encryption scheme (over plaintext space {0,1}) which is fully homomorphic, meaning that one can efficiently and compactly evaluate AND and XOR on encrypted plaintexts without knowledge of the secret decryption key.

I am wondering if there is any obvious way to turn this public key cryptosystem into a threshold public key cryptosystem such that everybody can encrypt, AND and XOR, but decryption is only possible if some (all) people sharing the key team up.

I would be interested in any ideas about that subject.

Thanks in advance

fw
A new paper by Steven Myers, Mona Sergi, and Abhi Shelat on eprint, &quot;Threshold Fully Homomorphic Encryption and Secure Computation&quot;, claims a threshold fully homomorphic encryption scheme.
From their abstract:

...
Gentry [Gen09a] shows how to combine both ideas with fully homomorphic encryption in order to construct secure multi-party protocol that allows evaluation of a function $f$ using communication that is independent of the circuit description of $f$ and computation that is polynomial in $|f|$. This paper addresses the major drawback's of Gentry's approach: we eliminate the use of non-black box methods that are inherent in Naor and Nissim's compiler.
To do this we show how to modify the fully homomorphic encryption construction of van Dijk et al. [vDGHV10] to be threshold fully homomorphic encryption schemes.
...
Altogether, we construct the first black-box secure multi-party computation protocol that allows evaluation of a function $f$ using communication that is independent of the circuit description of $f$.