Bifid Cipher

What is the Bifid Cipher?

The Bifid Cipher is a classical polygraphic substitution cipher invented by the French cryptographer Félix Delastelle around 1901. Delastelle was famous for introducing ciphers that combined **fractionation** and **transposition** to achieve high levels of confusion and diffusion, which were exceptionally strong before the advent of computer-based cryptography.

Bifid achieves this by converting each character into row and column coordinates in a **Polybius square** (a 5x5 grid of letters). These coordinates are written out, mixed (transposed) in blocks of a specific size (the period), and then recombined to look up new letters in the same square.

Mathematical Formulation of Fractionation

Let $\mathcal{A}$ be the alphabet of 25 characters (standard is $26$ letters with 'J' replaced by 'I'). The Polybius square maps each character $m \in \mathcal{A}$ to a unique pair of coordinates: $$\Phi(m) = (r, c) \quad \text{where} \quad r, c \in \{0, 1, 2, 3, 4\}$$ For a block of text of period $P$, we denote the plaintext characters as: $$M = (m_1, m_2, \dots, m_P)$$ Each character is fractionated into its coordinates: $$\Phi(m_i) = (r_i, c_i) \quad \text{for} \quad i = 1, \dots, P$$

Block Transposition & Recombination

The coordinates are listed horizontally in a single sequence of length $2P$: $$S = (r_1, r_2, \dots, r_P, c_1, c_2, \dots, c_P)$$ This coordinate stream is then divided into new pairs to read off the ciphertext characters: $$(r'_i, c'_i) = (S_{2i-1}, S_{2i}) \quad \text{for} \quad i = 1, \dots, P$$ Finally, the ciphertext block $C = (c'_1, c'_2, \dots, c'_P)$ is retrieved using the inverse mapping: $$c'_i = \Phi^{-1}(r'_i, c'_i)$$

Step-by-Step Bifid Example

Let's encrypt the word "HELLOWORLD" with the key "BIFID" and period **5**:

1. Construct the Polybius Square: Using the keyphrase "BIFID" and merging "J" to "I", we populate the 5x5 grid:

     0 1 2 3 4
   0 B I F D A
   1 C E G H K
   2 L M N O P
   3 Q R S T U
   4 V W X Y Z

2. Fractionate the first block "HELLO": Find the coordinates of H, E, L, L, O in the grid:
   • H $\to$ Row 1, Col 3
   • E $\to$ Row 1, Col 1
   • L $\to$ Row 2, Col 0
   • L $\to$ Row 2, Col 0
   • O $\to$ Row 2, Col 3
3. Write coordinates horizontally:

   Rows: 1  1  2  2  2
   Cols: 3  1  0  0  3
   Stream: 1, 1, 2, 2, 2, 3, 1, 0, 0, 3

4. Re-group coordinates into pairs:
   • (1, 1) $\to$ **E**
   • (2, 2) $\to$ **N**
   • (2, 3) $\to$ **O**
   • (1, 0) $\to$ **C**
   • (0, 3) $\to$ **D**
   So "HELLO" enciphers to **"ENOCD"**.