Implicit factorization of unbalanced RSA moduli

Let $$N_{1} = p_{1}q_{1}$$N1=p1q1 and $$N_{2} = p_{2}q_{2}$$N2=p2q2 be two RSA moduli, not necessarily of the same bit-size. In 2009, May and Ritzenhofen proposed a method to factor $$N_{1}$$N1 and $$N_{2}$$N2 given the implicit information that $$p_{1}$$p1 and $$p_{2}$$p2 share an amount of least significant bits. In this paper, we propose a generalization of their attack as follows: suppose that some unknown multiples $$a_{1}p_{1}$$a1p1 and $$a_{2}p_{2}$$a2p2 of the prime factors $$p_{1}$$p1 and $$p_{2}$$p2 share an amount of their Most Significant Bits (MSBs) or an amount of their Least Significant Bits (LSBs). Using a method based on the continued fraction algorithm, we propose a method that leads to the factorization of $$N_{1}$$N1 and $$N_{2}$$N2. Using simultaneous diophantine approximations and lattice reduction, we extend the method to factor $$k\ge 3$$k≥3 RSA moduli $$N_{i}=p_{i}q_{i}, i=1,\ldots ,k$$Ni=piqi,i=1,…,k given the implicit information that there exist unknown multiples $$a_{1}p_{1}, \ldots , a_kp_k$$a1p1,…,akpk sharing an amount of their MSBs or their LSBs. Also, this paper extends many previous works where similar results were obtained when the $$p_{i}$$pi’s share their MSBs or their LSBs.