How much computing resource is required to brute-force RSA?
It's not possible.

The number of primes smaller than $x$ is approximately $\frac{x}{\ln x}$. Therefore the number of $512$ bit primes (approximately the length you need for $1024$ bit modulus) is approximately:

$$\frac{2^{513}}{\ln 2^{513}}-\frac{2^{512}}{\ln 2^{512}} \approx 2.76×10^{151}$$

The number of RSA moduli (i.e. pair of two distinct primes) is therefore:

$$\frac{(2.76×10^{151})^2}{2}-2.76×10^{151}=1.88×10^{302}$$

Now consider that the observable universe contains about $10^{80}$ atoms. Assume that you could use each of those atoms as a CPU, and each of those CPUs could enumerate one modulus per millisecond. To enumerate all $1024$ bit RSA moduli you would need:

\begin{eqnarray*}
1.88×10^{302}ms / 10^{80}&amp;=&amp;1.88×10^{222}ms\\
&amp;=&amp;1.88×10^{219}s\\
&amp;=&amp;5.22×10^{215}h\\
&amp;=&amp;5.95×10^{211} \text{years}
\end{eqnarray*}

Just as a comparison: the universe is about $13.75×10^9$ years old.

It's not a question of resources, it's simply not possible.

Also, it would not make any sense to do that. There are much faster ways to find out a secret key. In fact there are algorithms with sub-exponential running time for factoring integers.