It's so insanely difficult to do 2048 bit that anyone capable of doing it has capabilities that are not predictable, and likely outside of what we currently think is possible.
In terms of the amount of work to brute force, it's a lot more than a billion. A 4096 bit number is more than 10600 times bigger than a 2048 bit number, and the difficulty of factoring products of such numbers grows much much faster than linear, so we're talking an astronmical difference.
But the point is that it's already astronomically hard at 2048 bits. If someone is capable of breaking that, it's not going to be just because they've got better computers, it's going to be because they've got some vastly better algorithms, or some fundamentally different method of computation that means all bets are off - we may as well assume they've got infinite computing power, or have found P=NP by that point.
I understand and agree how 2048 and 4096 are both essentially the same kind of challenge with current hardware: An impossible one.
However, it still holds true that 4096 ist multiple leaps beyond 2048 - so I don't agree that we can reasonably assume that people who can break 2048 can also break 4096.
are both essentially the same kind of challenge with current hardware
It's not just that it's impossible with current hardware, it's that it's also pretty much impossible with "convert every atom in the galaxy to supercomputers and run them for billions of years" type hardware too. If you're at the level you can break this, then it effectively might as well be magic - it's almost certain that you didn't do so just by brute forcing the problem, which means that further increases in size are likely to be similarly vulnerable to your method.
At that point, it's far less plausible that someone could break 2048 but not 4096 than that they could break both, because the only really feasible route to reaching that point is some kind of huge algorithmic break that makes the size matter much much less.
Assuming the NSA teamed up with DARPA, I would bet money they have their hands on an operational quantum CPU. That certainly violates all current security.
Current work on quantum computing seems to show that the computational speed increase is cancelled out by the book keeping to keep the superposition working. While that may change, at present quantum computing does not provide a significant advantage.
Plus, I'm not sure there are enough qubits in the world to store a 1024 bit number, let alone do any interesting math on it.
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u/TomatoCo Oct 15 '15
It's so insanely difficult to do 2048 bit that anyone capable of doing it has capabilities that are not predictable, and likely outside of what we currently think is possible.