All operations

ADFGX

Split every letter into two of five symbols, then shuffle the whole stream with a keyword.

Two stages

The first stage is a Polybius square with its rows and columns labeled A, D, F, G and X instead of 1 to 5. Every plaintext letter becomes two of those symbols.

The second stage writes that stream into columns under the transposition key and reads them off in the alphabetical order of the key letters. The two halves of a letter start side by side and end in different columns, hundreds of characters apart in a long message.

Either stage alone is weak: substitution falls to a frequency count, columnar transposition to anagramming. Together they leave no letter intact to count and no readable word for the anagram to converge toward.

Why A, D, F, G and X

These messages went out by radio in Morse, between units under artillery fire, copied down by hand. The five are A .-, D -.., F ..-., G --. and X -..-, patterns far enough apart that a clipped or noisy character was unlikely to arrive as a different valid one.

The transposition depends on the exact message length, so one dropped character shifts every column and ruins the rest of the decrypt.

The settings above

The square is btalpdhozkqfvsngjcuxmrewy read five letters to a row, with CARGO as the transposition key. Those are the published worked example, so ATTACK AT ONCE gives the printed ciphertext, FAXDFADDDGDGFFFAFAXAFAFX.

The square key takes a keyword or a complete twenty-five letter grid, as here; the transposition key takes letters only. I and J share a cell by default, as they did in the German square. Decoding throws away anything outside A, D, F, G and X, so a message copied out in five-letter groups pastes straight back in.

Georges Painvin

ADFGX, designed by German signals lieutenant Fritz Nebel, went into service on the Western Front on 1 March 1918, three weeks before the German spring offensive. Painvin, a paleontologist from the School of Mines at Saint-Étienne, by then in the French Army’s cipher bureau, had two daily keys out on 5 April. His way in was pairs of messages sent under one key whose stereotyped openings coincided: lining those up gave the column order, and behind the transposition sat a plain substitution.

The keys changed daily, so each solution bought one day. Painvin came out of that spring fifteen kilograms lighter and needed a long convalescence after the Armistice.

ADFGVX is this cipher three months later, with a sixth symbol and room for the digits. Rail fence is the transposition half on its own. Bifid fractionates on the same grid and mixes the pieces without a keyword.