Editing Cryptanalysis of the Enigma (section)

===The spy and the rotor wiring===
Before Rejewski started work on the Enigma, the French had a spy, [[Hans-Thilo Schmidt]], who worked at Germany's Cipher Office in Berlin and had access to some Enigma documents. Even with the help of those documents, the French did not make progress on breaking the Enigma. The French decided to share the material with their British and Polish allies. In a December 1931 meeting, the French provided [[Gwido Langer]], head of the Polish Cipher Bureau, with copies of some Enigma material. Langer asked the French for more material, and [[Gustave Bertrand]] of French Military Intelligence quickly obliged; Bertrand provided additional material in May and September 1932.<ref>{{harvnb|Sebag-Montefiore|2004|pp=22–23}}</ref> The documents included two German manuals and two pages of Enigma daily keys.<ref>{{Harvnb|Rejewski|Woytak|1984b|p=256}}</ref><ref>The documents were ''Instructions for Using the Enigma Cipher Machine'' and ''Keying Instructions for the Enigma Cipher Machine'', and the pages of Enigma keys were for September and October 1932 which fortunately had different rotor orders.</ref>

In December 1932, the Bureau provided Rejewski with some German manuals and monthly keys. The material enabled Rejewski to achieve "one of the most important breakthroughs in [[cryptology|cryptologic]] history"<ref>{{Harvnb|Kahn|1991|p=974}}</ref> by using the [[permutation group|theory of permutations and groups]] to work out the Enigma scrambler wiring.<ref>{{Harvnb|Wilcox|2001|p=5}}</ref><ref>{{Harvnb|Hodges|1983|p=170}}</ref>

Rejewski could look at a day's cipher traffic and solve for the permutations at the six sequential positions used to encipher the indicator. Since Rejewski had the cipher key for the day, he knew and could factor out the plugboard permutation. He assumed the keyboard permutation was the same as the commercial Enigma, so he factored that out. He knew the rotor order, the ring settings, and the starting position. He developed a set of equations that would allow him to solve for the rightmost rotor wiring assuming the two rotors to the left did not move.<ref>Solve save for an arbitrary rotation.</ref>

He attempted to solve the equations, but failed with inconsistent results. After some thought, he realised one of his assumptions must be wrong.

Rejewski found that the connections between the military Enigma's keyboard and the entry ring were not, as in the commercial Enigma, in the order of the keys on a German typewriter. He made an inspired correct guess that it was in alphabetical order.<ref>{{Harvnb|Gaj|Orlowski|2003}}</ref> Britain's [[Dilly Knox]] was astonished when he learned, in July 1939, that the arrangement was so simple.<ref>{{Harvnb|Copeland|2004|p=234}}</ref><ref>{{Harvnb|Rejewski|Woytak|1984b|p=257}} citing {{citation |last=Fitzgerald |first=Penelope |author-link=Penelope Fitzgerald |year=1977 |title=The Knox Brothers |location=London |publisher=Macmillan |isbn=1-58243-095-0 |url=https://archive.org/details/knoxbrothers00pene}}</ref>

With the new assumption, Rejewski succeeded in solving the wiring of the rightmost rotor. The next month's cipher traffic used a different rotor in the rightmost position, so Rejewski used the same equations to solve for its wiring. With those rotors known, the remaining third rotor and the reflector wiring were determined. Without capturing a single rotor to reverse engineer, Rejewski had determined the logical structure of the machine.

The Polish Cipher Bureau then had some Enigma machine replicas made; the replicas were called [[Polish Enigma double|"Enigma doubles"]].