If you're interested in reducing the dimension of your explanatory variables (20 IVs) then Principal Component regression (based off Principal Component Analysis) would be a better choice. That is unless you have some reason to believe the 20IVs are manifest variables relating some type of latent factor.

Remember, Factor Analysis assumes a statistical model that has a different purpose from Principal Components and, from the way you describe the problem, it seems like PCA is what you want.

I have not heard before about Principal Component regression within the context of logistic regression but it seems like a common-enough problem that someone surely has come up with something somewhere.