I think it's less of an "NSA directly did this" and more of a "NSA invented cryptography that absolutely no one thought was a good idea."
The juicy bit, and the piece that really is obnoxiously bad, is that NIST, who decides cryptographic standards, contracted out to two agencies when they were looking at introducing new cryptography in 2006. Those two agencies? RSA and NSA. They paid both of these organizations for the privilege of getting insight.
NSA had been pushing for Dual_EC for a couple years at this point, and wanting people to take the bait, they secretly gave $10 million to RSA for them to start using it in some of their products and for them to tell NIST that they thought it was cryptographically sound. All completely behind the back of NIST and the public.
So NIST is consulting out with two of the biggest names in cryptography, one of which had been championing Dual_EC for years (NSA), and one of which started using it in their products (which are primarily sold to government agencies that MUST use NIST-approved crypto, and presumably stood to lose a lot of money by "betting" on Dual_EC).
NIST never saw it coming.
And that's the irony of the whole thing. NSA is supposed to make the US, especially the US government, more technologically secure... yet they directly undermined cryptography that was predominately used by government agencies. Meanwhile the rest of the tech world saw it for the bullshit it was. The only people who were hurt by it was our government.
Thomas Massie gave a great speech to congress about this earlier this year, and pushed through a vote that now prevents NIST from contracting out with NSA. I wish I could find it.
(Aside: Thomas Massie, a congressman from Kentucky, graduated from MIT with a BS in EE and a MS in ME, founded a successful tech startup, and now serves on the Committee on Science, Space and Technology. One of the few cases where I feel someone in our government is adequately educated in what they rule over.)