Alogrithms for the ACD Problem and Cryptanalysis of ACD-based FHE schemes, Revisited

The security of several full homomorphic encryption (FHE) schemes depends on the hardness of the approximate common divisor (ACD) problem. The analysis of attack and defense against the system is one of the frontiers of cryptography research. In this paper, the performance of existing algorithms, including orthogonal lattice, simultaneous diophantine approximation, multivariate polynomial and sample pre-processing are reviewed and analyzed for solving the ACD problem. Orthogonal lattice (OL) algorithms are divided into two categories (OL-$\land$ and OL-$\vee$) for the first time. And an improved algorithm of OL-$\vee$ is presented to solve the GACD problem. This new algorithm works well in polynomial time if the parameter satisfies certain conditions. Compared with Ding and Tao's OL algorithm, the lattice reduction algorithm is used only once, and when the error vector $\mathbf{r}$ is recovered in Ding et al.'s OL algorithm, the possible difference between the restored and the true value of $p$ is given. It is helpful to expand the scope of OL attacks.