[enigma83]

The "cloud keyset" is another name for "public key". Usually, public key enables encryption but no decryption, here, the cloud key enables computations on ciphertexts, but not decryption.

Like in public key cryptography, you can give a RSA public key to an adversary, he can use it as much as he want, he will not be able to get any information on the private key (in any practical time). Most importantly, the cloud key is not secret, nor obfuscated: the cloud already knows each bits of it (side channel attacks are not relevant).

The "cloud key" is the public part of the key given to the cloud so that he can perform his operations (circuit, bootstrapping). There are strong security proofs that show that it is impossible to recover the secret key from the cloud key.

In the case of TFHE, if there was a polynomial algorithm that would recover even a single bit of the secret key given the cloud key, this algorithm could be used to break the LWE problem, and also worst case instances of lattice problems (which are much harder than factorization or discrete log for instance).