Can we use LEGO bricks to construct a cipher algorithm?

Question

I read a paper titled "On the entropy of LEGO", which explains how to calculate the number of ways to combine $n$ $b\times w$ LEGO blocks of the same color. For example, six $2\times4$ bricks have $915103765$ ways to combine. I wonder if could we construct a funny cipher algorithm using LEGO bricks.

Some definitions and symbols:

A $2\times4$ brick $i$ can be defined as: $b_i:=\left( \begin{array}{cc} s_0,s_1,s_2,s_3 \\ s_4,s_5,s_6,s_7 \end{array} \right)$, where $s_{i,k}:=(0/1,-/b_j)$ means if a stud $s_{i,k}$ is occupied by another brick $b_j$, then $s_{i,k}=(1,b_j)$; else if no brick is put on this stud $s_{i,k}$, then $s_{i,k}=(0,-)$.

Then, we have a key space $\mathcal{K}=\{b_1\cup b_2\cup ...\cup b_n\}$, where $n$ is the number of bricks, $\cup$ means the combination of bricks (Sorry, I cannot find an appropriate symbol to present the meaning).

The number $num$ of ways to combine $n$ $2\times4$ bricks is: $num:=(2^{n-1} + 46^{n-1})/2$, therefore, the size of $\mathcal{K}$ is the number $num=|\mathcal{K}|$.

Assuming that there is an one way function $f(k,m)\to c,k\in\mathcal{K}, m\in\mathcal{M},c\in\mathcal{C}$, where $\mathcal{M}$ is the plaintext space, and $\mathcal{C}$ is the ciphertext space.

So far, I still cannot find a method to construct the one-way function and confirm whether the LEGO bricks can be used to construct a cipher algorithm or not.

Answer 1

It seems like what you are actually describing is a way to encode data with Lego bricks, rather than encrypting with them. But, maybe the way that you encode data is hard to invert without knowing a secret key, for example.

I don't think the lego adds security to this encoding, though. Suppose you design a method $f$ to encode your message space $\mathcal{M}$ into possible arrangements of bricks, $\mathcal{B}$. I could design my own correspondence $g$ between bitstrings $\{0,1\}^n$ and $\mathcal{B}$ (for some $n$). Then $g^{-1}\circ f$ maps from $\mathcal{M}$ into $\{0,1\}^n$, turning your lego cipher into a regular cipher on bitstrings. This means that if you encrypt a message into bricks and I find the bricks, I can translate their arrangement into bricks and attack the cipher as though it were a traditional computer cipher. Hence, if your map from messages into brick arrangements is supposed to be secure, it should also be secure when viewed as a map from messages into bitstrings.

From this I think the hard work of designing a normal secure cipher still needs to be done, even if lego is the end goal. But I can still see two uses of lego: (1) steganography: to store a small encrypted message in a way that most people wouldn't notice is important; (2) to help compute a cipher by hand (e.g., by making parts of the "computation" easy to store and move around).

Answer 2

It is possible to build something reminiscent of mechanical cipher machines using Lego Technic gears and less complicated parts:

• "Poly-Alphabetic Cipher Machine"
• "Paper Enigma" (encrypts, but is made of cardboard tubes, paper, and sticky tape)
• "LEGO replica of an Enigma Machine" (Entirely Legos, but alas, this implementation doesn't actually encrypt)
• "Lego Cipher Wheel" is a 5-digit counter built from Lego. Simply counts, doesn't actually encrypt, but it appears surprisingly similar to mechanical cipher machines.

Perhaps https://bricks.stackexchange.com/ would be a better place to ask about implementation.

It's also possible to write messages (or store a secret key) using LEGO bricks, although that's generally considered "encoding" rather than "encrypting". Perhaps this could be used as a kind of steganography.

• "Write Coded Messages with LEGO Bricks" and Build a LEGO Cipher uses 26 different kinds of parts, each letter encoded with a single part. This might be a good illustration of one of the simplest, easiest-to-crack kinds of substitution cipher.
• "LEGO Coding" uses white blocks to represent "1" bits and blue blocks to represent "0" bits, each letter encoded with 7 bits (ASCII).
• "LEGO Morse code" and "Morse reference book in LEGO" uses short bricks for dots and bigger bricks for dashes.

(The Legos mentioned in "Fully homomorphic encryption: Introduction and bootstrapping" are more metaphorical).