Any otherwise well-designed system with a built in guarantee of access by a third party is only secure if you can trust that third party. You don't need to prove any theorems to know that. I don't think anyone is arguing that such a system is impossible to build, they're arguing that no such trusted third party exists, and no one who actually cares about keeping a secret would ever willingly use such a system when alternatives exist.
I'm interested in the argument line that would be "Ok, let's assume you are impeccably honest, infallible, and have unwavering integrity and unstealable secrets ... even under those laughable, impossible conditions, it's still a terrible idea because of the following..." and go from there.
The perfect recipient would still have to be sent backdoors from many creators. Many more people would be handling the secret, each one a target to be social engineered into handing it over. I imagine when a more typical backdoor is made, very few people even know it exists let alone know the key to open it. Mandate backdoors and everyone knows they exist so more people will work to find and crack them. They would be very high value targets. Once opened, a backdoor would take a lot of work and expense to be closed, if you even know it had been opened.
Yes, the whole idea is about as plausible as this april fools joke: https://en.wikipedia.org/wiki/Evil_bit ... but it's been put forth and implemented too many times for comfort.
Each time these silly systems like DVD-CSS broke down and became worthless or like DIVX, were widely panned and rejected by the consumer.*
Showing how this will always and forever be the case at a more fundamental level to stop trying this deadbeat idea with different gift-wrapping would be great.
* Even in MP3, you have bits 29 and 30 which are for copyright. What were they thinking? people would re-implement /bin/cp to look for that and fail if the bit is set? Really? AAC has something similar. silly.
Even with absolutely perfect systems, which do not exist in practice even when they may exist in math, the humans with the keys are the weakest link. Stolen credentials are the biggest cause of data breaches in enterprise web applications (according the the Verizon DBIR). And even for absolutely perfect humans with impeccable morals, which do not exist as no one is perfect, there is coercion in the form of rubber hose cryptography.
Even if the system were perfect and the users with legit access had impeccable credential management and physical security, the escrow will be used a lot to unlock devices and communications. It becomes routine.
Processes which become routine are prone to being subverted, as Mrs. Landau stated in her recent congressional testimony that I linked to in yesterday's article on the FBI and key escrow - https://www.youtube.com/watch?v=g1GgnbN9oNw&t=3h35m50s.
Even if you do trust the third party, you've created another attack vector, and worse, you've created one that is far more valuable to attack than any other because you know they hold the keys to the kingdom.
