The Implicit Factorization Problem was first introduced by May and Ritzenhofen at PKC'09. This problem aims to factorize two RSA moduli $N_1=p_1q_1$ and $N_2=p_2q_2$ when their prime factors share a certain number of least significant bits (LSBs). They proposed a lattice-based algorithm to tackle this problem and extended it to cover $k>2$ RSA moduli. Since then, several variations of the Implicit Factorization Problem have been studied, including the cases where $p_1$ and $p_2$ share some most significant bits (MSBs), middle bits, or both MSBs and LSBs at the same position. In this paper, we explore a more general case of the Implicit Factorization Problem, where the shared bits are located at different and unknown positions for different primes. We propose a lattice-based algorithm and analyze its efficiency under certain conditions. We also present experimental results to support our analysis.

翻译：隐式分解问题最早由 May 和 Ritzenhofen 在 PKC'09 会议上提出。该问题旨在分解两个 RSA 模数 $N_1=p_1q_1$ 和 $N_2=p_2q_2$，当它们的素因子共享一定数量的最低有效位（LSBs）时。他们提出了一种基于格（lattice）的算法来解决该问题，并将其推广至 $k>2$ 个 RSA 模数的情况。此后，隐式分解问题的多种变体得到了研究，包括 $p_1$ 与 $p_2$ 共享部分最高有效位（MSBs）、中间位、或同一位置同时共享 MSBs 与 LSBs 的情形。本文探讨了隐式分解问题的一种更一般情况，其中不同素数的共享位位于不同且未知的位置。我们提出了一种基于格的算法，并在特定条件下分析了其效率。同时，我们给出了支持理论分析的实验结果。