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"I can state flatly that heavier than air flying machines are impossible."

Lord Kelvin, 1895

No source?

Source: New Scientist 10 Impossibilities Conquered by Science

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"Brute force attacks against 256-bit keys will be infeasible until computers are built from something other than matter and occupy something other than space".

-Bruce Schneier

"I can state flatly that heavier than air flying machines are impossible."

Lord Kelvin, 1895

Brute force, so checking every combination, makes some pretty easily verifiable claims.

If you build a supercomputer that occupies a space of only 1 cubic centimetre, that can brute force 1 trillion keys a second, and cover the entire earths surface with these computers, the Sun will swallow the Earth, before you have time to search the whole 256 bit key space.

I have also seen other calculations that information represented as its absolute minimum energy in the laws of physics, that there is not enough energy in the Sun to search a 256 bit key space either.

These numbers have nothing to do with the technology of the devices; they are the maximums that thermodynamics will allow. So these Laws of Thermodynamics will have to be broken first.

I would very much like to see the proof regarding the necessity of the Laws of Thermodynamics to be broken in order to brute force a 256-bit key. Is there any link of such proof? I'm a bit concerned about the use of energy to do the calculations.

Applied Cryptography (1996) by Bruce Schneier page 157 (I am looking at my 2nd edition here):

Quote from: Applied Cryptography - p157

One of the consequences of the second law of thermodynamics is that a certain amount of energy is necessary to represent information. To record a single bit by changing the state of a system requires an amount of energy no less than kT, where T is the absolute temperature of the system and k is the Boltzman constant. (Stick with me; the physics lesson is almost over.)

Given that k = 1.38×10-16 erg/°Kelvin, and that the ambient temperature of the universe is 3.2°Kelvin, an ideal computer running at 3.2°K would consume 4.4×10-16 ergs every time it set or cleared a bit. To run a computer any colder than the cosmic background radiation would require extra energy to run a heat pump.

Now, the annual energy output of our sun is about 1.21×1041 ergs. This is enough to power about 2.7×1056 single bit changes on our ideal computer; enough state changes to put a 187-bit counter through all its values. If we built a Dyson sphere around the sun and captured all its energy for 32 years, without any loss, we could power a computer to count up to 2192. Of course, it wouldn't have the energy left over to perform any useful calculations with this counter.

But that's just one star, and a measly one at that. A typical supernova releases something like 1051 ergs. (About a hundred times as much energy would be released in the form of neutrinos, but let them go for now.) If all of this energy could be channeled into a single orgy of computation, a 219-bit counter could be cycled through all of its states.

These numbers have nothing to do with the technology of the devices; they are the maximums that thermodynamics will allow. And they strongly imply that brute-force attacks against 256-bit keys will be infeasible until computers are built from something other than matter and occupy something other than space.

He does quote this paragraph quite often, for example here: https://www.schneier.com/blog/archives/2009/09/the_doghouse_cr.html

Many thanks for this. Very interesting and reassuring for our cryptography. The problem comes when you remove the words "brute force", in which case the situation is far from reassuring. In any case, and no matter how certain we may think we are about anything, we must keep an open mind about what may be possible in the future.