You can't sign reality, but people can sign statements.
If you see a video at a date D1 where you say Statement1, and cryptographically sign it with your key K, then at a later date someone can verify that at least you said you've watched an backed your statement.
In a way all of security relies on the physical safety of some kind of secret data. So you have the deniability of key compromise in any case.
If everything a president states publicly is signed with his key Kp, then:
1) If something controversial is published without signature, the president can say it's not standard procedure and a plausible forgery;
2) If the president publishes officially without a signature, the public can demand one so there's no later equivocation;
3) Anything that has been said can be verified in the future by checking the presidential signature.
In this case, the worst case is really a compromised key (although key scheduling should mitigate it), but most forgery cases of statements (and potentially documents, mandates, etc.) are eliminated.
In practice, it would be difficult to get your public figures to sign everything they say (and difficult to get them to accept this kind of potential auto-incrimination for the public good).
That doesn't work for adversarial recordings. No one will sign an embarrassing or damning video of themselves, and those are the cases where authenticity is the most important.
Your solution is technologically cool, but I think the current system of "was this published by a domain controlled by the office of the press secretary" is probably accomplishing this case well enough.
For adversarial recordings, I wonder if we could have a camera that instantly uploads a timestamp and hash of each video taken to a blockchain. This way, we could have videos that we know for a fact were recorded at the latest at a certain time. It would still be vulnerable for a fake video to be post dated, but never pre dated.
If you see a video at a date D1 where you say Statement1, and cryptographically sign it with your key K, then at a later date someone can verify that at least you said you've watched an backed your statement.
In a way all of security relies on the physical safety of some kind of secret data. So you have the deniability of key compromise in any case.
If everything a president states publicly is signed with his key Kp, then:
1) If something controversial is published without signature, the president can say it's not standard procedure and a plausible forgery;
2) If the president publishes officially without a signature, the public can demand one so there's no later equivocation;
3) Anything that has been said can be verified in the future by checking the presidential signature.
In this case, the worst case is really a compromised key (although key scheduling should mitigate it), but most forgery cases of statements (and potentially documents, mandates, etc.) are eliminated.
In practice, it would be difficult to get your public figures to sign everything they say (and difficult to get them to accept this kind of potential auto-incrimination for the public good).