Cyclometer

History

Until September 1938 German Army operators enciphered each message key twice at the day’s basic position, and from 1932 Marian Rejewski of the Polish Cipher Bureau saw what that gave away: a day’s first and fourth letters, second and fifth, third and sixth, form three permutations whose cycle lengths the plugboard cannot change. The Poles built the cyclometer to find those lengths for every rotor position, and after more than a year had a card catalogue of all 105,456 settings. The Germans changed reflector A for B on 2 November 1937 and the catalogue was made again; on 15 September 1938 the procedure changed and the catalogue was done.

How it works

A day’s indicators give the products AD, BE and CF; a product a letter or two short is completed the only ways whose cycles pair. Their cycle lengths are looked up in the catalogue, made here as the cyclometer made it, the right rotor stepping and the others standing. For each setting with that characteristic, the plugboard is the one that carries the setting’s products onto the day’s: one letter’s partner guessed, the rest followed. Only the day’s setting takes one. The plugboard then reads every message key, and the rings are found by trying each on a message’s opening until it reads as German. A day whose middle rotor turned within the indicators is found by the same search with the turn.

How it was checked

Reflector A reads the 1930 test message of the Enigma I’s key instructions, indicator PKPJXI to ABLABL. Tony Sale’s worked day gives his rotor order, position and six plugs and no other setting. Simulated days are broken whole, rings and basic position included. Rejewski’s own example of 1980 has 30 settings with its characteristic and none takes a plugboard, with or without a turn: it is an illustration. The catalogue has 21,225 characteristics, 11,578 naming one setting; Kuhl’s recount of 2007 printed 21,230 and 11,466.

Limits

No list of a real day’s indicators from the 1930s is published, so the days here are simulated and say so. The catalogue counts rotors I–III, the only ones in use until December 1938. Kuhl’s figures could not be reproduced by any model of the machine tried; his paper does not say how his program worked, and its own sums disagree. A basic position whose middle letter is its rotor’s turnover letter is found as the position the double step leaves.