Homomorphic encryption is a powerful cryptographic tool that enables secure computations on the private data. It evaluates any function for any operation securely on the encrypted data without knowing its corresponding plaintext. For original data $p$, $c$ denotes the ciphertext of the original plaintext $p$, i.e. $c = Encrypt_k(p)$. This is crucial for any sensitive application running in the Cloud, because we must protect data privacy even in the case when the server has falled victim to a cyber attack. The encryption scheme $Encrypt_k$ is said to be homomorphic with respect to some set of operations $\mathcal{O}$, if for any operation $\circ \in \mathcal{O}$ one can compute $Encrypt_k(p_1 \circ p_2)$ from $Encrypt_k(p_1) \circ Encrypt_k(p_2)$. Those schemes come in three forms: somewhat, partially and fully homomorphic. In this survey, we present the state of art of the known homomorphic encryption schemes based on coding theory and polynomials.

翻译：同态加密是一种强大的密码学工具，能够对私有数据进行安全计算。它可以在不知道对应明文的情况下，安全地对加密数据计算任意函数或执行任意操作。对于原始数据 $p$，$c$ 表示原始明文 $p$ 的密文，即 $c = Encrypt_k(p)$。这对于任何运行在云端的敏感应用至关重要，因为即使在服务器遭受网络攻击的情况下，我们也必须保护数据隐私。若对于某一操作集合 $\mathcal{O}$ 中的任意操作 $\circ$，可以从 $Encrypt_k(p_1) \circ Encrypt_k(p_2)$ 计算出 $Encrypt_k(p_1 \circ p_2)$，则称加密方案 $Encrypt_k$ 对 $\mathcal{O}$ 具有同态性。这些方案有三种形式：近似同态、部分同态和全同态。在本综述中，我们介绍了现有基于编码理论和多项式的同态加密方案的最新进展。