This article describes a lightweight additive homomorphic algorithm with the same encryption and decryption keys. Compared to standard additive homomorphic algorithms like Paillier, this algorithm reduces the computational cost of encryption and decryption from modular exponentiation to modular multiplication, and reduces the computational cost of ciphertext addition from modular multiplication to modular addition. This algorithm is based on a new mathematical problem: in two division operations, whether it is possible to infer the remainder or divisor based on the dividend when two remainders are related. Currently, it is not obvious how to break this problem, but further exploration is needed to determine if it is sufficiently difficult. In addition to this mathematical problem, we have also designed two interesting mathematical structures for decryption, which are used in the two algorithms mentioned in the main text. It is possible that the decryption structure of Algorithm 2 introduces new security vulnerabilities, but we have not investigated this issue thoroughly.

翻译：本文描述了一种加密密钥与解密密钥相同的轻量级加法同态算法。与Paillier等标准加法同态算法相比，该算法将加密和解密的计算开销从模幂运算降低为模乘运算，并将密文加法的计算开销从模乘运算降低为模加法运算。该算法基于一个新的数学问题：在两个除法运算中，当两个余数存在关联时，是否可能根据被除数推断出余数或除数？目前尚不清楚如何破解该问题，但需进一步探索以确定其是否具有足够难度。除该数学问题外，我们还为解密设计了两种有趣的数学结构，这两种结构分别用于正文所述的两个算法中。算法2的解密结构可能会引入新的安全漏洞，但我们尚未对此问题开展深入研究。