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In this paper we consider the description of homomorphic and fullyhomomorphic ciphers in the $p$-adic model of encryption. This model describes awide class of ciphers, but certainly not all. Homomorphic and fully homomorphicciphers are used to ensure the credibility of remote computing, including cloudtechnology. The model describes all homomorphic ciphers with respect toarithmetic and coordinate-wise logical operations in the ring of $p$-adicintegers $Z_p$. We show that there are no fully homomorphic ciphers for eachpair of the considered set of arithmetic and coordinate-wise logical operationson $Z_p$. We formulate the problem of constructing a fully homomorphic cipheras follows. We consider a homomorphic cipher with respect to operation "$*$" on$Z_p$. Then, we describe the complete set of operations "$G$", for which thecipher is homomorphic. As a result, we construct a fully homomorphic cipherwith respect to the operations "$*$" and "$G$". We give a description of alloperations "$G$", for which we obtain fully homomorphic ciphers with respect tothe operations "$+$" and "$G$" from the homomorphic cipher constructed withrespect to the operation "$+$". We also present examples of such "new"operations.

Secure cloud computations: Description of (fully)homomorphic ciphers within the P-adic model of encryption