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. The documents included two German manuals and two pages of Enigma daily keys. 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 cryptologic history" by using the theory of permutations and groups to work out the Enigma scrambler wiring. 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. He attempted to solve the equations, but failed with inconsistent results. After some thought, he realized 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. Britain's Dilly Knox was astonished when he learned, in July 1939, that the arrangement was so simple. 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 "Enigma doubles". The grill method The Poles now had the machine's wiring secrets, but they still needed to determine the daily keys for the cipher traffic. The Poles would examine the Enigma traffic and use the method of characteristics to determine the six