Utilization of secure data requires data concealment and statistical processing. Homomorphic encryption can perform operations such as addition in the encrypted state, and can realize utilization of secure data. Homomorphic cryptography consists of a public key cryptosystem based on security that is mathematically difficult to calculate.
In this study, we focused on the Ring-LWE problem [1], which constitutes quantum-resistant homomorphic cryptography. The difficulty of the Ring-LWE problem depends on the field of the components. We proposed a homomorphic cipher using the Ring-LWE problem on a generalized decomposition field [2]. In this study, we performed a lattice attack and a χ2 attack on the Ring-LWE problem on a dichotomy and a decomposition field. As a result, we obtained an experimental result that the Ring-LWE problem on the decomposed field can be expected to have the same or higher security in the lattice attack as the Ring-LWE problem on the circular field. Proposed an extended attack.