It is perhaps unsurprising that many of these standards do indeed produce 'secure' RSA moduli however a closer inspection of some of the algorithms gives some surprising results. For example if we consider RSA integers generated from the GNU Crypto library it turns out the second prime q is only chosen after p, and the product must be of fixed length. Hence, if p is large there are far less choices for q. One might expect the loss in entropy to be large but in fact this is not the case and the output modulus pq is still suitably random.
This rather unusual and interesting work provides a theoretical framework for analysing practical cryptography and it would be good if other algorithms that have been standardised came under similar scrutiny. This could both prove the quality of some standards and expose weaknesses in others.
