seirdy.one/content/posts/password-strength.md

---
date: "2021-01-12T00:03:10-08:00"
description: Using thermal physics, cosmology, and computer science to calculate
  password vulnerability to the biggest possible brute-force attack.
outputs:
- html
- gemtext
tags:
- security
- fun
title: Becoming physically immune to brute-force attacks
---

This is a tale of the intersection between thermal physics, cosmology, and a tiny
amount of computer science to answer a seemingly innocuous question: "How strong does
a password need to be for it to be physically impossible to brute-force, ever?"

*Note: this post contains equations. Since none of the equations were long or
complex, I decided to just write them out in code blocks instead of using images or
MathML the way Wikipedia does.*

Introduction
------------

I realize that advice on password strength can get outdated. As supercomputers grow
more powerful, password strength recommendations need to be updated to resist
stronger brute-force attacks. Passwords that are strong today might be weak in the
future. **How long should a password be in order for it to be physically impossible
to brute-force, ever?**

This question might not be especially practical, but it's fun to analyze and offers
interesting perspective regarding sane upper-limits on password strength.

Asking the right question
-------------------------

Instead of predicting what tomorrow's computers may be able to do, let's examine the
*biggest possible brute-force attack* that the laws of physics can allow.

A supercomputer is probably faster than your phone; however, given enough time, both
are capable of doing the same calculations. If time isn't the bottleneck, energy
usage is. More efficient computers can flip more bits with a finite amount of energy.

In other words, energy efficiency and energy availability are the two fundamental
bottlenecks of computing. What happens when a computer with the highest theoretical
energy efficiency is limited only by the mass-energy of *the entire [observable
universe](https://en.wikipedia.org/wiki/Observable_universe)?*

Let's call this absolute unit of an energy-efficient computer the MOAC (Mother of All
Computers). For all classical that are made of matter, do work to compute, and are
bound by the conservation of energy, the MOAC represents a finite yet unreachable
limit of computational power. And yes, it can play Solitaire with *amazing*
framerates.

How strong should your password be for it to be safe from a brute-force attack by the
MOAC?

### Quantifying password strength.

A good measure of password strength is **entropy bits.** The entropy bits in a
password is a base-2 logarithm of the number of guesses required to brute-force
it.[^1]

A brute-force attack that executes 2<sup>n</sup> guesses is certain to crack a
password with *n* entropy bits, and has a one-in-two chance of cracking a password
with *n*+1 entropy bits.

For scale, [AES-256](https://en.wikipedia.org/wiki/Advanced_Encryption_Standard)
encryption is currently the industry standard for strong symmetric encryption. As the
name suggests, its keys have 256 bits of entropy; if your password has more than 256
entropy bits, then the AES-256 encryption algorithm is the bottleneck.

To calculate the entropy of a password, I recommend using a tool such as
[zxcvbn](https://www.usenix.org/conference/usenixsecurity16/technical-sessions/presentation/wheeler)
or [KeePassXC](https://keepassxc.org/).

The Problem
-----------

Define a function `P`. `P` determines the probability that MOAC will correctly guess
a password with `n` bits of entropy after using `e` energy:

    P(n, e)

If `P(n, e) ≥ 1`, the MOAC will certainly guess your password before running out of
energy. The lower `P(n, e)` is, the less likely it is for the MOAC to guess your
password.

Caveats and estimates
---------------------

I don't have a strong physics background.

When estimating, we'll prefer higher estimates that increase the odds of it guessing
a password; after all, the point of this exercise is to establish an *upper* limit on
password strength. We'll also simplify: for instance, the MOAC will not waste any
heat, and the only way it can guess a password is through brute-forcing. Focusing on
too many details would defeat the point of this thought experiment.

I won't address any particular encryption algorithms; this is just a pure and simple
brute-force attack given precomputed password entropy. Furthermore, quantum computers
can use [Grover's algorithm](https://en.wikipedia.org/wiki/Grover%27s_algorithm) for
an exponential speed-up; to account for quantum computers using Grover's algorithm,
calculate `P(n/2, e)` instead.

Obviously, I'm not taking into account future mathematical advances; my crystal ball
broke after I asked it if humanity would ever develop the technology to make anime
real.

Computation
-----------

How much energy does MOAC use per guess during a brute-force attack? In the context
of this thought experiment, this number should be unrealistically low. I settled on
[`kT`](https://en.wikipedia.org/wiki/KT_(energy)). `k` represents the [Boltzmann
Constant](https://en.wikipedia.org/wiki/Boltzmann_constant) (about
1.381×10<sup>-23</sup> J/K) and `T` represents the temperature of the system. Their
product corresponds to the amount of heat required to create a 1 nat increase in a
system's entropy.

A more involved approach to picking a good value might utilize the [Plank-Einstein
relation](https://en.wikipedia.org/wiki/Planck%E2%80%93Einstein_relation).

It's also probably a better idea to make this value an estimate for flipping a single
bit, and to estimate the average number of bit-flips it takes to make a single
password guess. If that bothers you, pick a number `b` you believe to be a good
estimate for a bit-flip-count and calculate `P(n-b, e)` instead of `P(n, e)`.

What's the temperature of the system? Three pieces of information help us find out:

1.  The MOAC is located somewhere in the observable universe
2.  The MOAC will be consuming the entire observable universe
3.  The universe is mostly empty

A good value for `T` would be the average temperature of the entire observable
universe. The universe is mostly empty; `T` is around the temperature of cosmic
background radiation in space. The lowest reasonable estimate for this temperature is
2.7 degrees Kelvin.[^2] A lower temperature means less energy usage, less energy
usage allows more computations, and more computations raises the upper limit on
password strength.

Every guess, the MOAC expends `kT` energy. Let `E` = the total amount of energy the
MOAC can use; let `B` = the maximum number of guesses the MOAC can execute before
running out of energy.

    B = E/(kT)

Now, given the maximum number of passwords the MOAC can guess `B` and the bits of
entropy in our password `n`, we have an equation for the probability that the MOAC
will guess our password:

    P(n,B) = B/2ⁿ

Plug in our expression for `B`:

    P(n,E) = E/(2ⁿkT)

### Calculating the mass-energy of the observable universe

The MOAC can use the entire mass-energy of the observable universe.[^3] Simply stuff
the observable universe into the attached 100% efficient furnace, turn on the burner,
and generate power for the computer. You might need to ask a friend for help.

Just how much energy is that? The mass-energy equivalence formula is quite simple:

    E = mc²

We're trying to find `E` and we know `c`, the speed of light, is 299,792,458 m/s.
That leaves `m`. What's the mass of the observable universe?

### Calculating the critical density of the observable universe

Critical density is smallest average density of matter required to *almost* slow the
expansion of the universe to a stop. Any more dense, and expansion will stop; any
less, and expansion will never stop.

Let `D` = critical density of the observable universe and `V` = volume of the
observable universe. Mass is the product of density and volume:

    m = DV

We can derive the value of D by solving for it in the [Friedman
equations](https://en.wikipedia.org/wiki/Friedmann_equations):

    D = 3Hₒ²/(8πG)

Where `G` is the [Gravitational
Constant](https://en.wikipedia.org/wiki/Gravitational_constant) and `Hₒ` is the
[Hubble Constant](https://en.wikipedia.org/wiki/Hubble%27s_law). `Hₒd` is the rate of
expansion at proper distance `d`.

Let's assume the observable universe is a sphere, expanding at the speed of light
ever since the Big Bang.[^4] The volume `V` of our spherical universe when given its
radius `r` is:

    V = (4/3)πr³

To find the radius of the observable universe `r`, we can use the age of the universe
`t`:

    r = ct

Hubble's Law estimates the age of the universe to be around `1/Hₒ`

### Solving for E

Let's plug in all the derived values into our original equation for the mass of the
observable universe `m`:

    m = DV

Remember when I opened the article by saying that none of the equations would be long
or complex?

I lied.

    m = (3Hₒ²/(8πG))(4/3)π(ct)³
    m = c³/(2GHₒ)

    E = mc²
    E = c⁵/(2GHₒ)

Yay, we found an expression for the total energy the MOAC can consume!

Final Solution
--------------

    P(n,E) = E/(2ⁿkT)
    P(n, c⁵/(2GHₒ)) = c⁵/(2GHₒ*2ⁿkT)

Let's copy and paste the values for those constants from Wikipedia and Wolfram Alpha:

- c = 299,792,458 m/s
- G ≈ 6.67408×10<sup>-11</sup> m³/kg/s²
- Hₒ ≈ 2.2×10<sup>-18</sup> Hz (uncertain; look up the Hubble tension)
- T ≈ 2.7 K
- k ≈ 1.3806503×10<sup>-23</sup> J/K

Plugging those in and simplifying:

**P(n) ≈ 2.21×10<sup>92</sup> / 2<sup>n</sup>**

Here are some sample outputs:

- P(256) ≈ 1.9×10<sup>15</sup> (password certainly cracked after burning 1.9
  quadrillionth of the mass-energy of the observable universe).

- P(306.76) ≈ 1 (password certainly cracked after burning the mass-energy of the
  observable universe)

- P(310) ≈ 0.11 (about one in ten)

- P(326.6) ≈ 1.1×10<sup>-6</sup> (about one in a million)

If your threat model is a bit smaller, simulate putting a smaller object into the
MOAC's furnace. For example, the Earth has a mass of 5.972×10<sup>24</sup> kg; this
gives the MOAC a one-in-ten-trillion chance of cracking a password with 256 entropy
bits and a 100% chance of cracking a 213-bit password.

Conclusion/TLDR
---------------

Question: How much entropy should a password have to ensure it will *never* be
vulnerable to a brute-force attack? Can an impossibly efficient computer--the
MOAC--crack your password?

Answer: limited only by energy, if a computer with the highest level of efficiency
physically possible is made of matter, does work to compute, and obeys the
conservation of energy:

- A password with 256 bits of entropy is practically immune to brute-force attacks
  large enough to quite literally burn the world, but is quite trivial to crack with
  a universe-scale fuel source.
- A password with 327 bits of entropy is nearly impossible to crack even if you burn
  the whole observable universe trying to do so.

At that point, a formidable threat would rather use [other
means](https://xkcd.com/538/) to unlock your secrets.

Further reading: an alternative approach
----------------------------------------

Check out Scott Aaronson's article, [Cosmology and
Complexity](https://www.scottaaronson.com/democritus/lec20.html). He uses an
alternative approach to finding the maximum bits we can work with: he simply inverts
the [cosmological constant](https://en.wikipedia.org/wiki/Cosmological_constant).

This model takes into account more than just the mass of the observable universe.
While we previously found that the MOAC can brute-force a password with 306.76
entropy bits, this model allows the same for up to 405.3 bits.

Another well-known approach to calculating physical limits of computation is
[Bremermann's limit](https://en.wikipedia.org/wiki/Bremermann%27s_limit), which
calculates the speed of computation given a finite amount of mass. This article's
approach disregards time, focusing only on mass-energy equivalence.

Acknowledgements
----------------

Thanks to [Barna Zsombor](http://bzsombor.web.elte.hu/) and [Ryan
Coyler](https://rcolyer.net/) for helping me over IRC with my shaky physics and
pointing out the caveats of my approach.

My notes from Thermal Physics weren't enough to write this; various Wikipedia
articles were also quite helpful, most of which were linked in the body of the
article.

[^1]: James Massey (1994). "Guessing and entropy" (PDF). Proceedings of 1994 IEEE
  International Symposium on Information Theory. IEEE. p. 204.

[^2]: Assis, A. K. T.; Neves, M. C. D. (3 July 1995). "History of the 2.7 K
  Temperature Prior to Penzias and Wilson"

[^3]: The MOAC 2 was supposed to be able to consume other sources of energy such as
  dark matter and dark energy. Unfortunately, Intergalactic Business Machines ran out
  of funds since all their previous funds, being made of matter, were consumed by the
  original MOAC.

[^4]: This is a massive oversimplification; there isn't a single answer to the
  question "What is the volume of the observable universe?" Using this speed-of-light
  approach is one of multiple valid perspectives. The absolute size of the observable
  universe is much greater due to the way expansion works, but stuffing that into the
  MOAC's furnace would require moving mass faster than the speed of light.