In standard RSA, the modulus $n=p_1 p_2$ is a product of two primes $p_1,p_2$ of the same size.  Suppose we construct the modulus as a product of multiple primes $p_1,\dots,p_k$, i.e., $n=p_1 p_2 \cdots p_k$, where all the primes are of about the same size.  I'm wondering how much this reduces the security of RSA, for typical modulus sizes.

Let me be more concrete.  I want security comparable to that obtained with standard RSA with a 2048-bit modulus.  Can I use $k=3$ (three primes) without significant loss of security?  $k=4$?  What's the largest number of primes $k$ that I can use, without significant loss of security?  Assume that each prime is $2048/k$ bits long, so all the prime factors are of equal length.

Related: see also Who first published the interest of more than two prime factors in RSA?, which asks about the inventor of this technique.  I'm asking something slightly different; in this question, I'm not asking about its inventor; I'm asking about concrete security levels.


Question: How can I check USB drive contents safely?
Answer: One option is to reboot your Mac to a Linux installation CD/DVD.  

You can unmount your local hard disk once the Linux desktop loads.

You can then plug in the thumb drive and be isolated from your installation of Mac OS X as well as your local hard disk.