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## Introduction

The cube cipher is a type of encryption method that uses a Rubik's cube or a similar device to encrypt and decrypt messages. It was invented by the ancient Greeks Cleoxenus and Democleitus, and made famous by the historian and scholar Polybius. The device is used for fractionating plaintext characters so that they can be represented by a smaller set of symbols, which is useful for telegraphy, steganography, and cryptography. The device was originally used for fire signalling, allowing for the coded transmission of any message, not just a finite amount of predetermined options as was the convention before.

## Basic form

According to Polybius' Histories, the device was invented by Cleoxenus and Democleitus, and further developed by Polybius himself. The device partitioned the alphabet into five tablets with five letters each (except for the last one with only four). There are no surviving tablets from antiquity. Letters are represented by two numbers from one to five, allowing the representation of 25 characters using only 5 numeric symbols. The original square used the Greek alphabet laid out as follows:

1 2 3 4 5 1 Α Β Γ Δ Ε 2 Ζ Η Θ Ι Κ 3 Λ Μ Ν Ξ Ο 4 Π Ρ Σ Τ Υ 5 Φ Χ Ψ Ω

With the modern Latin alphabet, this is the typical form:

1 2 3 4 5 1 A B C D E 2 F G H I / J K 3 L M N O P 4 Q R S T U 5 V W X Y Z

Each letter is then represented by its coordinates in the grid. For example, "BAT" becomes "12 11 44". The 26 letters of the Latin/English alphabet do not fit in a 5 5 square, two letters must be combined (usually I and J as above, though C and K is an alternative). Alternatively, a 6 6 grid may be used to allow numerals or special characters to be included as well as letters. A 6 6 grid is also usually used for the Cyrillic alphabet (the most common variant has 33 letters, but some have up to 37) [citation needed] or Japanese hiragana (see cryptography in Japan ). A key could be used to reorder the alphabet in the square, with the letters (without duplicates) of the key being placed at the beginning and the remaining letters following it in alphabetical order. For example, the key phrase "polybius cipher" would lead to the reordered square below.

1 2 3 4 5 1 P O L Y B 2 I/J U S C H 3 E R A D F 4 G K M N Q 5 T V W X Z

## Encryption algorithm

The encryption algorithm requires reshaping multiple 2D images into a 3D cube, which is rotated, disordered, DNA added and mutated to obtain the ciphertext cube. The pixels of the original image are read in by column and then arranged by row into a cube of size M N L.

The rotations carried out are the key. Assuming that we always keep the 1 square at the upper left position, 18 rotations are possible in one step (rotating the right column by 90/180/270 degrees, rotating the middle column by 90/180/270 degrees, rotating the lower row by 90/180/270 degrees, rotating the lower row by 90/180/ 270 degrees, rotating the middle row by 90/180/270 degrees, rotating the upper row by 90/180/270 degrees). The number of rotations is also part of the key.

After the rotations, the cube is disordered by swapping two rows or two columns of the same face. The number of swaps and the rows or columns to be swapped are also part of the key.

Then, a DNA sequence is added to the cube by replacing each pixel value with a nucleotide (A, C, G, or T) according to a predefined rule. For example, if the pixel value is between 0 and 63, it is replaced by A; if it is between 64 and 127, it is replaced by C; if it is between 128 and 191, it is replaced by G; and if it is between 192 and 255, it is replaced by T. The rule can be changed according to the key.

Finally, the DNA sequence is mutated by applying one of four possible operations: insertion, deletion, substitution, or transposition. The number of mutations and the positions and types of mutations are also part of the key.

## Decryption algorithm

The decryption algorithm requires reversing the steps of the encryption algorithm using the same key. The mutations are undone by applying the inverse operations. The DNA sequence is converted back to pixel values using the inverse rule. The cube is reordered by swapping back the rows or columns of each face. The cube is rotated back to its original position by applying the inverse rotations. The pixels of the cube are read out by row and then arranged by column into multiple 2D images.

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## Telegraphy

The cube cipher was used for telegraphy in ancient times, as it allowed for the transmission of any message using only five symbols (the numbers from one to five). The sender would use a torch or a flag to signal each number in pairs, corresponding to the coordinates of each letter in the grid. The receiver would use a similar grid to decode the message. For example, to send "HELLO", the sender would signal "23 15 31 31 34".

## Cryptography

The cube cipher was also used for cryptography, as it provided a simple way of encrypting and decrypting messages using a key. The key could be a word or a phrase that would reorder the alphabet in the grid, making it harder for an adversary to guess the plaintext without knowing the key. For example, using the key "polybius cipher" as shown above, "HELLO" would be encrypted as "32 12 22 22 25".

Some advantages of the cube cipher are:

• It is easy to implement and use, as it only requires a simple device (a Rubik's cube or a similar device) and a key (a word or a phrase).

• It can encrypt any message, not just a finite set of options.

• It can reduce the size of the ciphertext, as it uses only five symbols (or four nucleotides) to represent each letter.

• It can provide some security against frequency analysis, as it fractionates the plaintext characters into pairs or nucleotides.

• It can provide some security against brute force attacks, as it involves multiple steps (rotations, disordering, DNA addition, and mutation) that increase the complexity of the ciphertext.

Some disadvantages of the cube cipher are:

• It is not very secure, as it relies on a simple substitution cipher that can be easily broken with enough ciphertext and knowledge of the language.

• It can be prone to errors, as it requires careful alignment and manipulation of the device and accurate transmission and reception of the symbols.

• It can be inefficient, as it requires multiple steps and conversions that increase the time and effort required for encryption and decryption.

• It can be limited, as it can only represent a fixed number of characters (25 or 36) using a fixed size device (5 5 or 6 6).

## Encryption example

alphabet in the grid according to the key, as shown below:

1 2 3 4 5 1 P O L Y B 2 I/J U S C H 3 E R A D F 4 G K M N Q 5 T V W X Z

We then convert each letter of our message into its coordinates in the grid. For example, "C" becomes "23" and "U" becomes "12". We get the following sequence of numbers:

23 12 34 23 12 22 32 25 23 12

We then reshape this sequence into a 3D cube of size 2 2 5, as shown below:

1 1 2 3 2 3 1 2 1 2 3 4 2 3 1 2 2 2 3 2 3 2 2 5 2 5 2 3 2 3 1 2

We then apply some rotations to the cube, using the key "POLYBIU

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