We extend the CDPR's quantum attack from ideal lattices to module lattices over $2^k$-th cyclotomic rings. Using trace orthogonality of the power basis, we decompose a rank-$d$ module into mutually orthogonal rank-$1$ submodules, and apply CDPR's analysis to each independently and return the shortest candidate. The Hermite factor $\exp(\tilde{O}(\sqrt{n}))$ matches the ideal case, with a module reduction factor $α_d=O(1)$ independent of the rank, under a balance hypothesis (proved for Gaussian distribution) automatic for MLWE-distributed bases. To enable a bounded-precision implementation, we replace coordinate-wise rounding with Chinese Remainder Theorem-scaled rounding at totally split primes, reducing the Gram-Schmidt rounding radius from $n/2$ to $\le 1$ at cost $O(d^2 r n \log n)$. Finally, we reformulate the CDPR's sign-selection step as a mixed-integer linear program and prove its optimum is no more than 1/2 for all $k$ ($\approx 0.4407$ for all tested $k\le 12$, conjecturally universal). This replaces the previous heuristic discrepancy $Θ(\sqrt{nk})$. All results build on the class number condition $h_k^+=1$ established in Part I of this series.

翻译：我们将CDPR的量子攻击从理想格拓展至$2^k$次分圆环上的模格。利用幂基的迹正交性，将秩为$d$的模分解为相互正交的秩$1$子模，独立地对每个子模应用CDPR分析并返回最短候选解。在平衡假设（对高斯分布已验证）且MLWE分布基自动满足的条件下，埃尔米特因子$\exp(\tilde{O}(\sqrt{n}))$与理想情形一致，模约简因子$\alpha_d=O(1)$与秩无关。为实现有界精度计算，我们以完全分裂素数的中国剩余定理缩放舍入替代逐坐标舍入，将格莱姆-施密特舍入半径从$n/2$降至$\le 1$，代价为$O(d^2 r n \log n)$。最后，将CDPR的符号选择步骤重新表述为混合整数线性规划，并证明对所有$k$其最优值不超过$1/2$（对测试的所有$k\le 12$约为$0.4407$，推测具有普适性）。这替代了此前的启发式偏差$\Theta(\sqrt{nk})$。所有结果均建立在该系列第一部分确立的类数条件$h_k^+=1$之上。