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$N_2=p_2q_2$，当它们的素数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基的算法来解决该问题，并将其推广到$k>2$个RSA模数的情况。此后，隐式分解问题的多种变体被研究，包括$p_1$和$p_2$共享某些最高有效位(MSBs)、中间位、或同时共享相同位置的MSBs和LSBs的情况。本文探讨了隐式分解问题的一个更一般情形，其中不同素数共享的比特位于不同且未知的位置。我们提出了一种基于格基的算法，并在特定条件下分析了其效率。我们还给出了实验结果以支持我们的分析。