Grid-Based Ciphers: A Blog Series

Jul 26

Grid-Based Ciphers: A Blog Series

A collection of short posts exploring the Polybius square, the checkerboard cipher, and their many variations.

Cracking the Code: An Introduction to the Polybius Square

If you've ever wanted to turn letters into numbers, the Polybius square is one of the oldest and simplest ways to do it. Invented by the ancient Greek historian Polybius, this grid-based cipher arranges the alphabet in a 5×5 table and replaces each letter with a pair of coordinates — a row number and a column number.

The Standard Grid

I/J

Figure 1: The classic Polybius square. A = 11, B = 12, and so on through Z = 55.

Because the alphabet has 26 letters and a 5×5 grid only has 25 cells, I and J typically share a cell. To encode a message, you simply look up each letter and write down its row-then-column pair. To decode, you reverse the process. It's simple enough to do by hand, which is exactly why it was — and still is — a favorite teaching tool for cryptography basics.

Spicing Up the Polybius Square with a Keyword

The plain Polybius square is easy to build, but it's just as easy to break — anyone who knows the layout can decode it instantly. A simple fix is to scramble the grid using a keyword before filling in the rest of the alphabet.

Keyword: CIPHER

Figure 1: A Polybius square seeded with the keyword CIPHER, followed by the remaining letters in order.

To build this grid, write out the keyword first (dropping repeated letters), then fill in the rest of the alphabet in sequence, skipping any letters already used. The coordinate system works exactly the same way as the standard square — the only difference is that without knowing the keyword, an attacker has to try all 5×5 permutations, or fall back on frequency analysis, to recover the message.

Another Keyword Variant: The SECRET Polybius Grid

Keyword-based Polybius squares are endlessly reusable — swap in a new keyword and you get a brand new cipher grid, with no change to the underlying method. Here's another example, this time built around the word SECRET.

Keyword: SECRET

I/J

Figure 1: A Polybius square seeded with SECRET (duplicate letters removed), then filled out alphabetically.

Notice how the repeated letters in SECRET collapse down to S, E, C, R, T before the rest of the alphabet fills in. This small trick — deduplicating a keyword — comes up constantly in classical ciphers, from Playfair to the Nihilist cipher. Once you're comfortable with it, reading and building these grids becomes second nature.

Flipping the Grid: Reversed Coordinates in a Polybius Square

Not every variation of the Polybius square touches the letters themselves — sometimes the twist is in how the coordinates are labeled. Here's a version where the row and column numbers run from 5 down to 1 instead of 1 up to 5.

Reversed Numbering

I/J

Figure 1: The same alphabet layout as the standard square, but with coordinates reversed. A is now 55, Z is now 11.

This kind of variation doesn't add much cryptographic strength on its own, but it illustrates an important point about grid ciphers: the security often comes from the *arrangement* being secret, not from the underlying method being complex. Combine a reversed numbering scheme with a scrambled keyword layout, and you've already made casual decoding much harder.

Beyond Letters: The Straddling Checkerboard Cipher

The straddling checkerboard takes the grid idea a step further. Instead of a uniform 5×5 table, it uses an irregular layout with a 10-column top row for the most frequent letters, and two additional rows underneath for the rest of the alphabet. This design was historically used by spies because it produces compact, low-redundancy number strings.

Straddling Checkerboard (English letter frequency)

Figure 1: The eight most common English letters (E T A O N R I S) sit in the top row and encode as single digits. Rows 2 and 8 are 'shift' rows — any digit that starts with 2 or 8 signals a two-digit code follows.

The efficiency comes from letting the most common letters — E, T, A, O, N, R, I, S — use just one digit each, while rarer letters need two. This variable-length encoding shortens the overall message and makes traffic analysis harder, which is why the straddling checkerboard was a staple of mid-20th-century espionage, often paired with a substitution key for extra security.

Personalizing the Checkerboard: A Keyword-Based Layout

Just like the Polybius square, a straddling checkerboard can be seeded with a keyword to make its layout unpredictable. Here's a version built around the word LEGION.

Keyword: LEGION

Figure 1: The keyword LEGION fills the single-digit top row, and the remaining letters are split across the two-digit rows 3 and 7.

Choosing a keyword with several high-frequency letters (like the E, I, O, and N in LEGION) keeps the single-digit row useful, since those are the letters that benefit most from a short code. Without knowing the keyword, decoding this grid means guessing both the keyword and the two 'shift' row digits — a small grid, but a surprisingly deep rabbit hole for anyone trying to crack it cold.

The Nihilist Cipher's Secret Weapon: A Numbered Grid

The Nihilist cipher, used by 19th-century Russian revolutionaries, builds on the Polybius square by adding a second layer: a numeric keyword added digit-by-digit to the coordinates. Here's the grid it starts from.

Keyword: SWORD

I/J

Figure 1: A 5×5 grid seeded with SWORD. Each plaintext letter's coordinates get added (mod 10, or without reduction in the classic version) to a repeating numeric key derived from a second keyword.

What makes the Nihilist cipher interesting is that the grid itself is only half the story — the real security comes from the addition step that follows. Two identical letters in the plaintext can produce completely different ciphertext numbers, depending on where they fall relative to the repeating key. It's a nice example of how a simple grid can be the foundation for something much stronger.

When Grids Meet Morse: A Fractionated Cipher Table

Not all cipher grids deal with letters directly. Some, like fractionated Morse ciphers, use a small grid to convert Morse code symbols into two-character combinations, which are then encrypted again with a substitution or transposition step.

3×3 Fractionation Grid

Figure 1: A 3×3 table mapping Morse elements (dot, dash, and letter-space 'x') into pair combinations for further processing.

This kind of grid shows just how flexible the underlying concept is: any finite set of symbols — letters, digits, Morse elements — can be arranged in a grid and referenced by coordinates. Once you see the pattern, it's easy to recognize the same basic idea at work across dozens of different classical ciphers, from the Polybius square all the way to modern coordinate-based encoding schemes.

Fractionating Twice: An Introduction to the Bifid Cipher

The Bifid cipher, invented by Félix Delastelle around 1901, takes the Polybius square and adds a clever twist: it splits each letter into its two coordinates, but then recombines those coordinates across the *whole message* before converting back to letters. The result is a cipher where changing one letter of plaintext can ripple through and change several letters of ciphertext.

Keyword: BIFID

I/J

Figure 1: A 5x5 Polybius square seeded with the keyword BIFID, used as the basis for the Bifid cipher's fractionation step.

To encrypt, you first write out the row coordinates of every letter in the message, then the column coordinates, as two separate rows of digits. Next, you read those digits back off in pairs, column by column, and use the grid to convert each new pair back into a letter. Because the two coordinate streams get interleaved this way, the resulting cipher mixes information across letter positions — a simple grid producing surprisingly strong diffusion for a pencil-and-paper method.

Adding a Third Dimension: The Trifid Cipher

If the Bifid cipher fractionates letters into two coordinates, the Trifid cipher — also from Delastelle — goes one step further and uses three. Instead of a single 5x5 grid, the Trifid cipher uses three separate 3x3 layers, so each letter (plus a symbol for padding) gets a row, column, and layer number.

Three 3x3 Layers

Figure 1: Layer 1 of the Trifid grid, holding A through I.

The remaining two layers continue the same pattern:

Trifid Cipher, Continued: Layers 2 and 3

Picking up where the three-layer Trifid grid left off, here are the second and third layers that complete the cipher table.

Layer 2

Figure 1: Layer 2 of the Trifid grid, holding J through R.

And the final layer:

Figure 2: Layer 3 of the Trifid grid, holding S through Z, with a period used as a 27th filler symbol.

Just like the Bifid cipher, encrypting with a Trifid grid means writing out each letter's three coordinates, but this time stacking them as three rows instead of two, then reading the digits back off in groups of three, column by column, to look up the new letters. With three interleaved coordinate streams instead of two, the Trifid cipher spreads each plaintext letter's influence even further across the ciphertext, making it noticeably more resistant to manual cryptanalysis than its two-dimensional cousin.

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