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Suppose that an $n$-bit integer $c$ has been randomly drawn from a distribution $\chi$, whose description is known. Is there a general method to check if this particular sampling helps factorize $n$? Are there any results in this direction (i.e. results like "don't choose an RSA modulus whose factors were sampled with this or that distribution")?

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    $\begingroup$ how would you specify such a distribution, just wondering... $\endgroup$ – kodlu Apr 22 '16 at 10:51
  • $\begingroup$ Say, for instance, that you know that $p,q$ are sampled according to prime search near a Gaussian distribution... Or in my case, $p$ and $q$ bits are sampled using Bernoulli trials with low probability of being 1. $\endgroup$ – Tal-Botvinnik Apr 22 '16 at 11:18
  • $\begingroup$ You can check the following paper: J. Brandt and I. Damgård. On generation of probable primes by incremental search. In Advances in Cryptology − CRYPTO ’92, vol. 740 of Lecture Notes in Computer Science, pp. 358–370, Springer, 1993. dx.doi.org/10.1007/3-540-48071-4_26 (DOI:10.1007/3-540-48071-4_26). $\endgroup$ – user94293 May 22 '16 at 22:45
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On trail to follow, from Handbook of Applied Cryptography fact 3.7:

Let $n$ be chosen uniformly at random form the interval $[1, x]$.

  1. if $1/2 \leq \alpha \leq 1$, then the probability that the largest prime factor of $n$ is $\leq x^{\alpha}$ is approximately $1+ ln(\alpha)$. Thus, for example, the probability than $n$ has a prime factor $> \sqrt(x)$ is $ln(2) \approx 0.69$

  2. The probability that the second-largest prime factor of $n$ is $\leq x^{0.2117}$ is about $1/2$.

  3. The expected total number of prime factors of $n$ is $ln ln x + \mathbb{O}(1)$. (If $n = \prod p_i^{e_i}$, the total number of prime factors of n is $\sum e_i$.)

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