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

TL;DR

This paper explores the limitations and possibilities of homomorphic and fully homomorphic encryption within the p-adic model, showing that not all operation pairs admit fully homomorphic ciphers and constructing specific examples where they do.

Contribution

It characterizes the set of operations allowing fully homomorphic encryption in the p-adic model and provides explicit constructions and examples of such operations.

Findings

No fully homomorphic ciphers exist for all pairs of operations on Z_p.

Constructed a fully homomorphic cipher for specific operation pairs.

Identified all operations G that enable fully homomorphic encryption with addition.

Abstract

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.