Generalized Implicit Factorization Problem
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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.
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