There is no shortage of minds working on to create backdoors, or develop cryptographic methods that have backdoors, just look at Dual_EC_DRBG. It was a backdoor for the "good guys", but now its backdoor for everyone - eventually people will study the code and see the backdoor exists.
The crux of the issue is mathematics has no concept of good guys or bad guys, so as far as mathematics is concerned a back door for anyone is a backdoor for everyone.
Also can't the role of the good guy be split up among a group? Similar to the two man rule to prevent rogue agents from launching missiles, can't we have some sort of process that requires agreement among a majority of a few parties including the end user, the company who owns the software, law enforcement, and the (public) judicial system. If all it takes to break down the door to my home are a judge and law enforcement to agree, why can't we accept similar when it comes to data?
I can't comment on the mathematics involved but let us assume it's mathematically possible. You engineer this mythical nearly foolproof backdoor. You can decrypt this text with any of two keys. (It's my understanding that such algorithms actually exist already.) Congratulations you have achieved your goal. You have a working algorithm.
Now let's examine the results of actually using this algorithm:
You now have twice the opsec problem you had before. You have transmit this second key to a Government agency securely. You have to trust that Government agency to securely store, use, and dispose? of this key when they obtain it.
And what is the number one threat to secure systems? Operational Security. In fact many security professionals will tell you that the hardest part of security isn't the math behind the encryption. It's the opsec. In one fell swoop you double the threat in the most fragile part of your security.
I would also love a more detailed description of just "it is impossible because math" that everyone seems to be giving.
Really, it's inevitable. Someone doesn't even need to crack it, you just need a single careless or corrupt government employee to compromise the whole system for everyone for all time. People are proposing adding a single point of failure to systems whose usefulness is currently defined by their lack of such a single point of failure. Put that in there and we may as well all go back to using DES for everything.
Nothing anyone posted here has said why there can't be a multikey solution that allows access to data in a reliable way that would not be susceptible to a single point of failure or abuse. That sounds like a very hard problem, but I'm not convinced it is an impossible problem.
I'm not a cryptographer, but lets assume a multikey solution is 100% possible.
The very notion that you can trust the government with a global key to all encryption is the crux of the issue. How do you know that Donald Trump won't wake up tomorrow and sell that key China? What do you then do if Germany then demands that key? What if Congress decides that giving Israel he private key is important to stability in the middle east? Then what do you do is some nationstate sells this key to a blackhat organization? Welp, all of Google's encryption is now worthless because this "multikey" that was supposed to be for the USG ended up in the hands of a blackhat - and now we have another Fappening 3.0 on our hands.
Great now the whole world has this "multikey" making it virtually worthless because the entire world can decrypt it. If you as an end user cannot control who can and cannot decrypt your messages, then its worthless as an encryption scheme.
Its not a technical issue, and the solution isn't limited because we aren't smart enough. The fundamental problem is that you cannot trust any third party with such a multikey.
You could split the second key so no single party has the whole key which would mitigate but you still have the same problem where you have effectively doubled your opsec problem.
Additionally if half the key is compromised that still greatly reduces the work required to decrypt the text.
To make a car analogy, we can make a submarines that are waterproof, and we can make cars which looks like a car and you can drive on the road. However, to make a car that is also a submarine is quite hard, and close to impossible if you also had to make it look like a normal car. It would even be harder if it need perfect obscurity so that you couldn't even tell if you opened up the hood or started to disassemble the car.
Let's pause to consider this. Math works, that's why even the NSA can't break encryption. I wouldn't want to tell you wrong and say that it's impossible, it's not, but it would take something like ten billion years to crack. Needless to say, there's a reason why they need a backdoor, and that's because math works.
However, if a backdoor were put into all electronic products, the strength of the encryption is now meaningless as any would-be attacker (government or otherwise) would just target the backdoor instead of trying to break the encryption. Why wait ten billion years for a computer to brute force the message when you could just find a flaw in something designed by the government?
"The problem with cryptographic backdoors isn't that they're the only way that an attacker can break into our cryptographic systems. It's merely that they're one of the best. They take care of the hard work, the laying of plumbing and electrical wiring, so attackers can simply walk in and change the drapes."
It's perfectly possible and trivial to put in a backdoor that only works for people who have access to a specific private key.
Obviously if that private key gets stolen anyone can then access the backdoor, but that's true for anything, and you can mitigate it by storing the key in self-destructing immovable hardware with access limitations, as well as periodically changing the keypair (with signed updates).
The real problem is that there is no single "good guy" to entrust with that private key: in particular humans are inherently not fully trustable or good and both individual consumers and other governments have no interest in using or allowing backdoored products.