Chris Christensen Department of Mathematics and Statistics Northern Kentucky University

Stefan Mazurkiewicz Waclaw Sierpinski Stanislaw Lesniewski

1926 Reichsmarine 1928 Reichswehr 1928 the Poles were confronted by messages that – because of the randomness of letters in the messages – were thought to be generated by a machine cipher. Early in 1929 the Cipher Bureau began a cryptology course for mathematics students at Poznań University.

Marian Rejewski Jerzy Rozycki Henryk Zygalski

Zdzislaw Krygowski There seems to be a lot of fuss around our breaking of the Enigma. Yet, we did not do anything but applied the knowledge which as first year students, we had learned from [Zdzislaw] Krygowski and [Kazimierz] Abramowicz [1889 – 1936]. Marian Rejewski

(ae)(bj)(cm)(dz)(fl)(gy)(hx)(iv)(kw)(nr)(oq)(pu)(st)

Rotors6 x 26 x 26 x 26 = 105,456 Plugboard 100,391,791,500 Number of setups 10,586,916,764,424,000

abcdefghijklmnopqrstuvwxyz OHELCPYBSURDZTAFXKINJWVQGM (ao)(bh)(ce)(dl)(fp)(gy)(is)(ju)(kr)(mz)(nt)(qx)(vw)

If we have a sufficient number of messages (about eighty) for a given day, then, in general, all the letters of the alphabet will occur in all six places at the openings of the messages. Marian Rejewski

AD(a)(bc)(dvpfkxgzyo)(eijumnqlht)(rw)(s) BE(axt)(blfqveoum)(cgy)(d)(hjpswizrn)(k) CF(abviktjgfcqny)(duzrekhxwpsmo)

The disjoint cycles for AD, BE, and CF assume a characteristic form “generally different for each day [i.e., for each rotor order and ground setting] ….”

The Theorem that Won the War Cipher A. Devours Afterward to How Polish Mathematicians Deciphered the Enigma Marian Rejewski, July 1981

“If we multiply two permutations, consisting solely of [disjoint] transpositions, then the product has an even number of cycles of the same length.” So, AD, BE, and CF “consist of cycles of the same length in even numbers.” AD(a)(bc)(dvpfkxgzyo)(eijumnqlht)(rw)(s) BE(axt)(blfqveoum)(cgy)(d)(hjpswizrn)(k) CF(abviktjgfcqny)(duzrekhxwpsmo)

The theoretical possible number of disjoint cycle structures for each of AD, BE, and CF is the number of partitions of 13, which is 101.

The theoretical possible number of disjoint cycle structures for each of AD, BE, and CF is the “number of partitions of 13,” which is 101. The triples of composed permutations AD, BE, and CF could theoretically have 101 x 101 x 101 = 1,030,301 possible sets of disjoint cycles. The number of rotor system settings is 105,456.

Determine the disjoint cycle structure for AD, BE, and CF for all 105,456 possible rotor orders and ground settings.

One had to note on a card the position of the drums and the number of bulbs that were lit, and to order the cards themselves in a specified way; for example by the lengths of the cycles. AD(a)(bc)(dvpfkxgzyo)(eijumnqlht)(rw)(s) Notation 6 BE(axt)(blfqveoum)(cgy)(d)(hjpswizrn)(k) 9+3+1Notation 9 CF(abviktjgfcqny)(duzrekhxwpsmo) 13Notation 1

Once all six card catalogues [one for each rotor order] were ready, though, obtaining a daily key was usually a matter of twenty minutes. The card told the drum positions, the box from which the card had been taken told the drum sequence.

21,230 disjoint cycle decompositions appear. Of these, (or 54.40%) correspond to a unique setting will have 10 or fewer settings to check.

Unfortunately, on 2 November 1937, when the card catalogue was ready, the Germans exchanged the reversing drum that they had been using, which they designated letter A, for another drum, a B drum, and consequently, we had to do the whole job over again …. Marian Rejewski

And, more and more changes until 1 September 1939.