Consider the following problem:
Factorize a $n$-bit integer $c$ knowing that it is the product of two integers with known Hamming weight $h$.
Is there a way to prove that this is still hard? I have parameters $n=1024$ and $h=80$, but am willing to augment $h$ if this turns out to be easy. I think this problem is current-factorization-algorithms-related, since I did not find an easy way to describe the distribution of these type of numbers (my try considered quadratic expressions of dependent Bernoulli trials). So the question may be reformulated: Does this hypothesis helps any of the actual factorization algorithms?