Chris Christensen Department of Mathematics and Statistics Northern Kentucky University

Chris Christensen

Department of Mathematics and Statistics

Northern Kentucky University

The Polish Cipher Bureau’s Attack on the German

Enigma Cipher Machine

World War I

Cipher Machines

Polish-Soviet War (1919 – 1921) Polish Mathematician/Codebreakers

Stefan Mazurkiewicz1888 - 1945

Waclaw Sierpinski

1882 - 1969

Stanislaw Lesniewski1886 - 1939

Enigma1926 Reichsmarine1928 Reichswehr1928 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.

Poznan

World War II Polish Mathematician/Codebreakers

Marian Rejewski

1905 - 1980

Jerzy Rozycki

1909 - 1942Henryk Zygalski

1908 - 1978

Marcel Givierge (1871 – 1931)

Zdzislaw Krygowski1872 - 1955

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

Saxon Palace Warsaw

Enigma

Period = 16900

Rotor System 3! = 6 orders

http://en.wikipedia.org/wiki/Enigma_rotor_details

Rotor System 6 26 26 26 105,456

Reflector

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

Reflector Permutation

Plugboard100,391,791,500

Rotor System and Plugboard

Rotors 6 x 26 x 26 x 26 = 105,456Plugboard 100,391,791,500Number of setups

10,586,916,764,424,000

abcdefghijklmnopqrstuvwxyz

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

Enigma Cipher

1Plugboard and Rotors Reflector Plugboard and Rotors

Ground Setting 26 x 26 x 26 = 17,576

Message Setting

1 2 3 4 5 6C V G C V G

N K U G L N R A B

Compose Ciphers AD, BE, CF

1 2 3 4 5 6C V G C V G

N K U G L N R A B

1 42 53 6

G C RL V AN G B

Cipher AD G RCipher BE L ACipher CF N B

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

Message Settings

Composed Permutation AD

AD (a)(bc)(dvpfkxgzyo)(eijumnqlht)(rw)(s)

BE (axt)(blfqveoum)(cgy)(d)(hjpswizrn)(k)

CF (abviktjgfcqny)(duzrekhxwpsmo)

Permutations AD, BE, and CF

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] … .”

Rejewski’s Idea

The Plugboard is Nullified!

1Plugboard Rotor System Plugboard

The Theorem that Won the WarCipher 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.”




Rejewski’s Theorem

Partitions of 13The theoretical possible numberof disjoint cycle structures for eachof AD, BE, and CF is the numberof 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 is105,456.

How Unique?

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

orders and ground settings.

Rejewski’s Catalog

Polish Cyclometer

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.


10+2+1 Notation 6


9+3+1 Notation 9


13 Notation 1

The Catalog 6, 9, 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.

“This job took … over a year.”

21,230 disjoint cycle decompositions appear.

Of these, 11466 (or 54.40%) correspond to a unique setting.

20433 will have 10 or fewer settings to check.

Results

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

Too Bad

And, more and more changes until1 September 1939.

Documents

Chris Christensen Department of Mathematics and Statistics Northern Kentucky University