With the development of Shor's algorithm, some nondeterministic polynomial (NP) time problems (e.g. prime factorization problems and discrete logarithm problems) may be solved in polynomial time. In recent years, although some homomorphic encryption algorithms have been proposed based on prime factorization problems, the algorithms may be cracked by quantum computing attacks. Therefore, this study proposes a post-quantum cryptography (PQC)-based homomorphic encryption method which includes the homomorphic encryption function based on a code-based cryptography method for avoiding quantum computing attacks. Subsection 3.2 proposes mathematical models to prove the feasibility of the proposed method, and Subsection 3.3 gives calculation examples to present the detailed steps of the proposed method. In experimental environments, the mainstream cryptography methods (i.e. RSA cryptography and elliptic curve cryptography (ECC)) have been compared, and the results show that the encryption time and decryption time of the proposed method are shorter than other cryptography methods. Furthermore, the proposed method is designed based on a non-negative matrix factorization problem (i.e. a NP problem) for resisting quantum computing attacks.

翻译：随着Shor算法的发展，某些非确定性多项式（NP）时间问题（如质因数分解问题和离散对数问题）可能在多项式时间内解决。近年来，虽然已有基于质因数分解问题的同态加密算法被提出，但这些算法可能受到量子计算攻击的破解。因此，本研究提出了一种基于后量子密码（PQC）的同态加密方法，其中包含基于编码密码学的同态加密函数，用以规避量子计算攻击。第3.2节提出了数学模型以证明所提方法的可行性，第3.3节通过计算示例展示了所提方法的详细步骤。在实验环境中，与主流密码学方法（即RSA密码学和椭圆曲线密码学（ECC））进行了比较，结果表明所提方法的加密时间和解密时间均短于其他密码学方法。此外，所提方法基于非负矩阵分解问题（即NP问题）设计，以抵抗量子计算攻击。