I am somewhat new to cryptography.
Repeating the basics of RSA from How are the primes used to generate RSA keys?:
Textbooks say the one-way function is merely two primes (with some critical constraints) ...
may be a good base of this question, as well as What prime lengths are used for RSA?. Also, Big data and modern crypto systems already asks the same thing as the question here, but in a more abstract way.
Why not calculating in advance, on many computers, a vast amount of multiplications of primes that you loop over systematically, in the constraints of RSA? And then you would save the outcome as keys in a Big Data system that you can quickly search through? You could use a Bloom filter, a key coordinator or other Big Data tricks to quickly find the computers that might have the multiplied value at hand, and then quickly find a value's underlying primes.
You could have 10000 computers, with chosen different starting prime ranges. The first and second primes could be within a likely range that is needed to meet RSA standards of 4096 bit. You may also guess a bit, meaning to take a range that is rather around the square root of a chosen outcome, or take other tricks to guess a bit.
Then it just depends on how many computers you use. Even if you hit just 0.001 % of the possible combinations you could already get the private key from the public key in 0.001 % of the cases. Or do I oversee something here.
It cannot be that easy, can it? Can Big Data attack RSA by just calculating many prime multiplications in advance?