Trifid Cipher

What is the Trifid Cipher?

The Trifid Cipher is a classical symmetric cipher invented by the French amateur cryptographer Félix Delastelle and published in 1901. It is a direct and ingenious extension of his own **Bifid cipher**. While the Bifid cipher fractionates characters into two components (row and column) using a 2D $5 \times 5$ grid, the Trifid cipher fractionates characters into **three components** using a 3D $3 \times 3 \times 3$ grid (a cube).

By using three-dimensional fractionation, the Trifid cipher achieves even higher diffusion and confusion, making it significantly more difficult to crack using simple frequency analysis compared to other classical pen-and-paper ciphers.

Mathematical Coordinates in 3D Space

Let $\mathcal{A}$ be an alphabet consisting of $27$ characters (typically $26$ letters of the English alphabet plus a symbol such as a period ., space, or plus sign +). The 3D grid maps each character $m \in \mathcal{A}$ to a unique coordinate triplet: $$\Phi(m) = (l, r, c) \quad \text{where} \quad l, r, c \in \{0, 1, 2\}$$ Here, $l$ represents the layer (Z-axis), $r$ represents the row (Y-axis), and $c$ represents the column (X-axis).

For a block of text of period $P$, the characters are denoted as: $$M = (m_1, m_2, \dots, m_P)$$ And their coordinate representations are: $$\Phi(m_i) = (l_i, r_i, c_i) \quad \text{for} \quad i = 1, \dots, P$$

3-Way Transposition and Recombination

The coordinates are written in three separate rows and then read horizontally to form a single sequence of length $3P$: $$S = (l_1, l_2, \dots, l_P, r_1, r_2, \dots, r_P, c_1, c_2, \dots, c_P)$$ We regroup this flat sequence into new coordinate triplets: $$(l'_i, r'_i, c'_i) = (S_{3i-2}, S_{3i-1}, S_{3i}) \quad \text{for} \quad i = 1, \dots, P$$ The corresponding ciphertext characters are retrieved via the inverse mapping: $$c'_i = \Phi^{-1}(l'_i, r'_i, c'_i)$$

Step-by-Step Trifid Example

Let's encrypt the word "HELLOWORLD" using the key "TRIFID", period **5**, and extra character **"."**:

1. Generate the 3x3x3 Cube: Using the keyphrase "TRIFID" and appending the remaining letters + ".", we get the 27-character grid:

   Layer 0:
   T R I F D A B C E
   Layer 1:
   G H K L M N O P Q
   Layer 2:
   S U V W X Y Z . [Space]

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

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

4. Re-group coordinates into triplets:
   • (1, 0, 1) $\to$ **H**
   • (1, 1, 0) $\to$ **L**
   • (2, 1, 1) $\to$ **Y**
   • (2, 1, 2) $\to$ **Z**
   • (0, 0, 0) $\to$ **T**
   So "HELLO" enciphers to **"HLYZT"**.