Matthew Green has a nice post recounting the history of government mandated backdoors. If you're too young to remember the first round of the Crypto Wars from the 1990s, you'll find Green's history fascinating. Back then it was about the NSA designed Clipper chip that would allow secure communications while at the same time ensuring government access to those communications when needed.
As Green points out, the system was extremely fragile and Matt Blaze wrote a famous paper showing how the system could be defeated. For that and various other reasons the Clipper chip was abandoned and the government gave up on mandating backdoors.
More recently, we've had the FREAK and LogJam exploits both of which were enabled by the government's now abandoned crypto export regulations. The point of these episodes is that trying to insert backdoors into crypto systems always has unintended consequences and introduces weaknesses.
Nonetheless, the government is back for another round and insisting loudly that the fate of the nation depends on their being able to read all our communications. The best way to read Green's history is as a cautionary tale of what's apt to happen if they succeed. If you have any interest in this area, be sure to read the post.