Secure cloud computations: Description of (fully)homomorphic ciphers within the P-adic model of encryption
This addresses secure cloud computations for cryptography researchers, but appears incremental as it builds on existing p-adic models.
The paper tackles the problem of constructing fully homomorphic ciphers within the p-adic encryption model, showing that no such ciphers exist for certain operation pairs but successfully constructing them for specific operations like '+' and derived sets.
In this paper we consider the description of homomorphic and fully homomorphic ciphers in the $p$-adic model of encryption. This model describes a wide class of ciphers, but certainly not all. Homomorphic and fully homomorphic ciphers are used to ensure the credibility of remote computing, including cloud technology. The model describes all homomorphic ciphers with respect to arithmetic and coordinate-wise logical operations in the ring of $p$-adic integers $Z_p$. We show that there are no fully homomorphic ciphers for each pair of the considered set of arithmetic and coordinate-wise logical operations on $Z_p$. We formulate the problem of constructing a fully homomorphic cipher as follows. We consider a homomorphic cipher with respect to operation "$*$" on $Z_p$. Then, we describe the complete set of operations "$G$", for which the cipher is homomorphic. As a result, we construct a fully homomorphic cipher with respect to the operations "$*$" and "$G$". We give a description of all operations "$G$", for which we obtain fully homomorphic ciphers with respect to the operations "$+$" and "$G$" from the homomorphic cipher constructed with respect to the operation "$+$". We also present examples of such "new" operations.