Last month I talked about Landauer's principle in the context of brute-forcing encryption keys. As a quick review, Landauer's principle defines a minimum necessary energy to flip a single bit as a function of the system's temperature. This is tied to the laws of thermodynamics and sets a firm lower bound on the amount of energy needed to brute-force an encryption key of a specified length.

The conclusion—after some miscalculation—was that current global energy production could fuel a 100%-efficient brute force computation, but that it would certainly be large enough to be noticeable and might not even produce proper results. This was based on a very generous calculation: that enumerating a key cost exactly one bit-flip operation. It also ignored the cost of performing the decryption attempt with each key—a cost that would necessarily be at least the same as generating the key itself.

But technology is moving faster than ever these days, and yesterday's supercomputers are tomorrow's cell phones. What sort of encryption keys are needed to protect information well into the future, even against ever-rising technological development? In short, what does key-cracking look like with more energy?

The Kardashev Scale

I'm going to summarize this briefly, but you can lose an afternoon on the Wikipedia page if you are so inclined. The Kardashev scale is a three-point scale which can describe advanced civilizations based on their energy production capabilities. It was originally described about fifty years ago and has since been re-described, extended, pre-pended, sub-divided, and otherwise roundly discussed. But here are the basics:

A Kardashev type I civilization can capture energy equivalent to the total energy being input to Earth by the sun. This shakes out to about $10^{16}$ watts.

A Kardashev II civilization can capture energy equivalent to the combined output of our sun. This is about $3.846 \times 10^{26}$ watts.

Finally, a Kardashev III civilization can capture energy on a galactic scale, or approximately $4 \times 10^{37}$ watts.

You are here: humans are currently a type 0, as we have a few centuries to go before reaching type I assuming consistent future growth. So anything too difficult for a Kardashev I civilization to break can stay safe for a long time.

Adjusting the Formula

Previously, we adapted the Landauer formula to solve for the number of bits needed in an encryption key. However, since the Kardashev scale is described in terms of wattage (which is energy per time), a better approach would be to see how much time it would take to break a key of a given size.

Since each step of the Kardashev scale also depends on successful space travel and infrastructure, it makes sense to place the super-efficient computer in cold vacuum rather than room temperature. So let's use 2.725K as the system temperature, as measured from the Cosmic Microwave Background. (Getting cooler than this would require refrigeration efforts, which cost more energy that could be used for key-cracking efforts.)

The new formula looks something like this:

$$s = \frac{2^{n} \times 2.725 \times k \ln{2}}{W}$$

Where s is the time in seconds, n is the number of bits in the key, k is the Boltzmann constant, and W is the available wattage.

128-bit Key

Typical key size for symmetric block ciphers is 128 bits. This is the default for AES which is widely used and even has built-in hardware support on many chipsets. Let's see how it holds up to advanced civilizations.

Kardashev Level	Wattage	Time
I	$10^{17}$	0.887 sec
II	$3.846 \times 10^{26}$	$2.308 \times 10^{-11}$ sec
III	$4 \times 10^{37}$	$2.219 \times 10^{-22}$ sec

Hrm, that doesn't look very promising. While still a colossal energy expenditure, it would be almost trivial for an advanced civilization to brute-for a 128-bit key.

192-bit Key

This key size is a bit of an odd duck, as it isn't a power of two. Still, AES supports it, so let's give it a try:

Kardashev Level	Wattage	Time
I	$10^{17}$	$5.188 \times 10^{11}$ years
II	$3.846 \times 10^{26}$	13 years, 179 days, 5 hours
III	$4 \times 10^{37}$	0.004 sec

That's quite a bit better. It's still breakable by a super-advanced civilization, but it's well out of reach of anything in humanity's foreseeable future.

256-bit Key

Just for completeness' sake, let's try a 256-bit key. This is the largest key size that AES supports:

Kardashev Level	Wattage	Time
I	$10^{17}$	$9.571 \times 10^{30}$ years
II	$3.846 \times 10^{26}$	$2.489 \times 10^{20}$ years
III	$4 \times 10^{37}$	2.392 million years

Ha! With a 256-bit key, brute-forcing requires so much energy that even the bug-eyed Andromedans can't manage it in time! Although you probably have bigger things to worry about if bug-eyed Andromedans are invading.

Really this all goes to show that brute-forcing an encryption key is the least effective way to break security.