Comment by tialaramex

The algorithms don't care because it doesn't make sense to care, the same reason a "smallest non-interesting number" wouldn't make sense - now that number is interesting precisely because it was cited as the smallest non-interesting number.

However practical applications do care, and so e.g. the "Debian Weak Keys" are a relatively small set of keys which would have been generated if you used a particular patched OpenSSL to generate "random" key pairs after a mistake patched out most of the randomness. Many of them were used by Debian sysadmins particularly in the window between the patch and the discovery - and so all CAs are expected to check that you're not trying to get a certificate for the public half of one of those pairs.

There's nothing "bad" about those keys except that well, we know some people used them so bad guys will try the private halves of those keys just in case. Likewise there are a few apps which shipped with a baked in "default" private key, and all the paperwork to apply for a certificate but obviously a user may not realise that if they just fill it out with their name and details, they're now vulnerable to attack by literally any other user of that software. They were supposed to generate a random key pair and re-generate the paperwork for the resulting random values. If they don't realise we need the CA to say "Nope, that's a bad idea, not issuing" even though these keys aren't special in any other way than they're given to every single user of that software.

IIRC when I worked for a famous US corporation their middleboxes relied on such a well-known key and at some point I asked "Hey, this seems wrong. Why did you do this?" and as often has happened in my career I got no actual reply to my question but magically the issue was fixed in short order. The first clumsy attempt to fix it broke some things, but they gradually figured it out. Glad I could help, I guess.