How is NSA breaking so much crypto?

How is NSA breaking so much crypto? "Based on the evidence we have, we can’t prove for certain that NSA is doing this. However, our proposed Diffie-Hellman break fits the known technical details about their large-scale decryption capabilities better than any competing explanation."

For the nerds in the audience, here’s what’s wrong: If a client and server are speaking Diffie-Hellman, they first need to agree on a large prime number with a particular form. There seemed to be no reason why everyone couldn’t just use the same prime, and, in fact, many applications tend to use standardized or hard-coded primes. But there was a very important detail that got lost in translation between the mathematicians and the practitioners: an adversary can perform a single enormous computation to ‘crack’ a particular prime, then easily break any individual connection that uses that prime.

Since a handful of primes are so widely reused, the payoff, in terms of connections they could decrypt, would be enormous. Breaking a single, common 1024-bit prime would allow NSA to passively decrypt connections to two-thirds of VPNs and a quarter of all SSH servers globally. Breaking a second 1024-bit prime would allow passive eavesdropping on connections to nearly 20% of the top million HTTPS websites. In other words, a one-time investment in massive computation would make it possible to eavesdrop on trillions of encrypted connections.

Update: Bruce Schneier's comments.