One example of a crypto backdoor: NSA's backdoor in Lotus-Notes January 2015
Excellent finding from Adam Back.
If I understand the article correctly, when exporting encrypted content with Lotus-Notes, 24 bits of the 64 bits key would be encrypted under one of the NSA's public key and then appended to the encrypted content (I guess). This would allow NSA to decrypt those 24 bits of key with their corresponding private key and they would then have to brute force only 40 bits instead of 64 bits.
This shouldn't allow any bad attacker to get any advantage if they don't know the NSA's private key to decrypt those bits. And if they do acquire it, and they do decrypt 24bits of key, they would still have to have the computing power to brute force 40 bits of key. I have no idea what I'm talking about but I have the feeling the NSA might be the most powerful computing power when it comes to brute forcing ciphers.