  In this paper, we propose the first homomorphic based proxy re-encryption
(HPRE) solution that allows different users to share data they outsourced
homomorphically encrypted using their respective public keys with the
possibility to process such data remotely. More clearly, this scheme makes
possible to switch the public encryption key to another one without the help of
a trusted third party. Its originality stands on a method we propose so as to
compute the difference between two encrypted data without decrypting them and
with no extra communications. Basically, in our HPRE scheme, the two users, the
delegator and the delegate, ask the cloud server to generate an encrypted noise
based on a secret key, both users previously agreed on. Based on our solution
for comparing encrypted data, the cloud computes in clear the differences
in-between the encrypted noise and the encrypted data of the delegator,
obtaining thus blinded data. By next the cloud encrypts these differences with
the public key of the delegate. As the noise is also known of the delegate,
this one just has to remove it to get access to the data encrypted with his
public key. This solution has been experimented in the case of the sharing of
images outsourced into a semihonest cloud server.
