I could imagine a Manahattan-like project that does quantum crypto and breaks standard RSA/ECC keys. Or an extension to what NSA already does: a giant database of exploits to most systems on the earth.

It's a really, really stupid and irrelevant discussion. The problem the government has is not that terrorists are using encryption. Is that EVERYBODY is using encryption compromising the data-mining and passive surveillance they rely on to generate leads. Once you know who the suspects are, the encryption they might be using is the last concern because you can use active methods.

So what Hillary is really asking is for a way to tap and analyze the billions of conversation of the citizens of the world, some of which might be terrorists.

It would look like a magician's trick... Look at this shiny object (money for impossible algorithms (aka just a little bit pregnant)) while setting up the real goal.

Now that the phone company keeps the records, we have access to more than metadata. Now that Congress passed the Omnibus, we can legally share with everyone. Now that we have a campaign to claim encryption = terrorist, we can make strong encryption a criminal-seeming activity. Now that we looked like we broke Tor nobody will trust it. Now that companies can't be sued, we don't have to worry about the courts. Now that ....

LOOK OVER THERE! ... A presidential candidate waving tax dollars!

Given that we're talking about Silicon Valley, I'm guessing the solution would involve Shamir secret sharing + proof of stake, along with some sort of USB device that mysteriously costs more than the expected value.

First, let's remember there's a perfect encryption scheme: one-time-pad.

Give this, a manhattan-like project to give (random evil entity) a backdoor into encrypted communications would have to be able to have (easy) access basically to the cleartext. If you have access to one-time-pad keys at the right time, they you also have access to the cleartext.

So you could imagine a system where the whole planet surface is monitored or monitorable and possibly recorded. But it's actually much cheaper, since while it would be quite interesting to be able to get access to all the data you can find about any plant, animal or grain of sand on the planet, Hillary is actually interested about YOU. That is, about human beings, and let's add for good measure, any other computer system and robot too (but of course, not the computers involved in the tracking). The order of magnitude is to track about a hundred billion or a trillion information sources.

So, imagine a system of nanobots that would duplicate and maintain themselves autonomously (for ease of manufacturing and distribution), and that would spread all over the planet, attaching itself to any human being, and any computer system or robot, collecting data about the attached entity, notably the information exchanges, but for version 2 you can also imagine it to be able to retrieve information stored in memories, be they wetware or hardware), and to filter it and forward it to the controlling system.

Now the second technological element you need of course is a massive wireless communication network, to be able to transmit all the collect data. The current internet/phone network is already near the order of magnitude the size needed, but you could build such a network using the nanobots spread all over as nodes. For isolated people, you would need to have intermediary nodes to connect them to the population centers where more bandwidth would be available.

One problem however would be that if you used electromagnetics for this communication network, it would be easily detected and neutralized (a mere Faraday cage would be enough, such as a aluminium paper hat ;-) (No, but really, you don't need to enclose eg. the RFID devices in metal, just placing a coin or a piece of aluminium over one side of the antena is enough to disable their communication.)). Anyways, let's assume instead some quantum communication that would be invisible to normal electromagnetics.

Now if you don't want to collect all the information about everybody all the time, that can simplify a little the system. You need to collect information that you don't have access to. But if the system can give you access to any information in any place, then you don't need to collect it, since you can always access it. (Ok, you may want to make some backup or snapshot for legal reasons).

If your nanobots once reaching a human can spread in his brain to read any memory or thought, and can receive commands from the nanobot network and perform some processing, you could just use this network to make a request such as: give me the list of all the people who thought that "Hillary is an evil cow" in the last hour. Perhaps you could even instruct the nanobots to make a red target appear on the forehead of any such person. Or just burn out the brain neurons where such thoughts are stored or initied.

Oh, by the way, if you have such nanobot technology avaialble, you definitely won't use a keyboard to make those requests, you will have your own nanobots in your brain.

Yep, this is the Borg. Only much nicer, since totally invisible. Until you start thinking what the hell, those slaves building missiles and drones, let's have nanobots graft tools directly on their limbs, that'll make more profit for me.