Enigma machine

Enigma machine

I like cryptography, so I will present you a famous machine, named Enigma which helped the nazis to encrypt messages during the world war two. It was invented in 1918 by Arthur Scherbius just for fun. Then it served the German army to deceive the Allies during the war.

Enigma is an heavy wooden box which is composed of an alphabetic keyboard, a connection table, 3 movable rotors with 26 positions and a table of 26 bulbs corresponding to the 26 letters of the alphabet. In a first part, I will tell you about its functioning, then of the number of possible keys and finally about how it has been decrypted and by who.

1. The functioning

The Enigma machine codes sequences of letters and its functioning is based on three rotors which taking 26 different positions.

To encode a message, users must initialize the machine thanks to the secret key, which changes every day. This operation consists in positioning the rotors and to perform a number of connections at the front of the machine. The operator keystrokes message that he wants to be encoded. At each letter entered, a letter lights up on the display above the keyboard and indicates the result of the coding. What made the Enigma machine so special was the fact that every time a letter was pressed, the movable parts of the machine would change position so that the next time the same letter was pressed, it will give a different letter compared to the first time. This meant that it wasn't possible to use traditional methods to try and crack the notorious cipher.

To make things even more difficult, different parts of the machine could be set up in different ways, with each setting producing a unique stream of crypted letters. Unless you knew the exact settings of the machine, you couldn't decrypt the messages.

1. The number of possible keys

        First of all, there were five rotors to choose from and they could be inserted into three positions on the Enigma machine. In your opinion, how many possible ways are there of positioning 5 rotors in 3 slots in the Enigma?

For the first slot, you can choose any one of 5 rotors. For the second, you can choose any one of 4 rotors. For the last, you can choose any one of 3 rotors. 5x4x3=60 or 5!/(5-3)!=60

There are 60 ways of positioning 5 rotors in 3 slots.

        Once you have chosen the order of the rotors, there are a lot of possible ways to set the starting positions of the rotors.

As there are 26 letters of the alphabet, each of the 3 rotors could be set in any one of 26 different starting positions. 26x26x26=17 576

This gives a total of 17 576 distinct starting positions.

        On the front of the machine was another section called the "plugboard". The Enigma machines had 10 cables with which to link up pairs of letters. So we will calculate the number of ways to link up pairs of letters on the Enigma machine.

There are 26 letters, they have to be divided into 6 unpaired letters and 10 pairs of pairwise connected letters.

Suppose that we had ten differently coloured connecting wires: red, blue, green etc etc.

Then there are C(26,2) ways of choosing a pair for the red wire. For each of these there are C(24,2) ways of choosing a pair for the blue wire, and so on, giving the product

C(26,2) x C(24,2) x C(22,2) x ... x C(8,2)

This can be simplified, with many factors cancelling, to 26! / (6! 210)

But in the actual Enigma the wires are not coloured. This means we must divide by the number of ways of permuting the 10 coloured wires, i.e. divide by a further factor of 10!. This gives the answer: 26! / (6! 10! 210) = 150,738,274,937,250.

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