A New LSB Attack on Special-Structured RSA Primes

Asymmetric key cryptosystem is a vital element in securing our communication in cyberspace. It encrypts our transmitting data and authenticates the originality and integrity of the data. The Rivest-Shamir-Adleman (RSA) cryptosystem is highly regarded as one of the most deployed public-key cryptosystem today. Previous attacks on the cryptosystem focus on the effort to weaken the hardness of integer factorization problem, embedded in the RSA modulus, N=pq. The adversary used several assumptions to enable the attacks. For examples, p and q which satisfy Pollard's weak primes structures and partial knowledge of least significant bits (LSBs) of p and q can cause N to be factored in polynomial time, thus breaking the security of RSA. In this paper, we heavily utilized both assumptions. First, we assume that p and q satisfy specific structures where p=am+rp and q=bm+rq for a,b are positive integers and m is a positive even number. Second, we assume that the bits of rp and rq are the known LSBs of p and q respectively. In our analysis, we have successfully factored N in polynomial time using both assumptions. We also counted the number of primes that are affected by our attack. Based on the result, it may poses a great danger to the users of RSA if no countermeasure being developed to resist our attack.