A hardware random number generator (HWRNG) using 2×2×2, 3×3×3, 4×4×4, and 5×5×5 Rubik's Cubes.
There are a number of different methods you can execute by hand for generating cryptographically secure randomness:
- Flipping coins
- Tossing dice
- Shuffling playing cards
- Throwing paper segments
- etc.
This project takes advantage of the sheer number of permutations in a Rubik's Cube. The number of unique permutations per cube is already known:
- 2×2×2: 3.674 million
- 3×3×3: 43.252 quintillion
- 4×4×4: 7.401 quattuordecillion
- 5×5×5: 282.870 trevigintillion
This means when sufficiently scrambled, each cube provides approximately the following in symmetric security:
- 2×2×2: ~21 bits
- 3×3×3: ~65 bits
- 4×4×4: ~152 bits
- 5×5×5: ~247 bits
The trick is knowing how many twists are required to sufficiently scramble a cube. Unfortunately, I don't have that answer—at least not rigorously. However, I think we can get within ballpark of at least making it sufficiently difficult for an adversary to precisely reproduce.
The shortest path to solving the worst case 3×3×3 cube is 20 moves, or 26 quarter turn twists. This is known as "God's Number". This number states that no matter how the cube is scrambled, it can be guaranteed to be solved within 20 moves or less.
The World Cube Association as a program called TNoodle that applies 20 twists to a 3×3×3 cube before the competitor can attempt solving it. In fact, the number of turns TNoodle performs on each cube it supports is:
- 2×2×2: ~11 (average)
- 3×3×3: ~19 (average)
- 4×4×4: ~44 (average)
- 5×5×5: 60
- 6×6×6: 80
- 7×7×7: 100
However, scrambling a cube to be solved might not be the same as a sufficiently-scrambled cube for security. Remember that the goal is to prevent an adversary from recreating the scrambled state. Some research has been done here. The goal is to ensure that every cubelet can occupy every possible position from the solved state as well as every possible orientation. For the 2×2×2 cube, it appears that at least 20 moves are necessary.
So instead, scrambling a cube for security might require 10+ additional moves past God's Number. Something like:
- 2×2×2: 20 moves
- 3×3×3: 30 moves
- 4×4×4: 55 moves
- 5×5×5: 70 moves
Once you've sufficiently scrambled your cube, record the facelet colors in the web app and press the button to calculate a random number from that state. The project uses SHA-256 to whiten the raw data before giving you your random number. Hashing with SHA-256 ensures that any biases present in the permutation are removed.
The "test vectors" for the SHA-256 digest of a solved cube are:
- 2×2×2: 0x09cca8
- 3×3×3: 0x1397cb125ac75bd57
- 4×4×4: 0xe5de84cc8a7174f870b62113b98c5b90a3e9
- 5×5×5: 0x5f3a3d2d2199e13440dcb377f29029b51e59c1b2ae5d4c69bd9d692f96f52b
A solved cube is defined as:
- Up: white
- Left: orange
- Front: green
- Right: red
- Back: blue
- Down: yellow
This project doesn't enforce that the cube is solvable from its scrambled state, although it does enforce color count per cube. This means you could disassemble the cube, placing it in an unsolvable scramble, record the colors in the web app, and calculate your random number.
But once you've received your random number, you will want to destroy the scrambled cube state. If you can solve the cube, this would be the best approach. If you cannot solve it, then continue scrambling the cube to erase the state, preventing an adversary from discovering it. Of course, press the red "Clear cube" button or close your browser tab to prevent the digest from staying on the screen for the same reason.
This project does not intentionally transmit anything over the network or store anything to disk.