User:EVuser/sandbox

= Enigma Permutations Multiplication = Several Articles about the Enigma machine refer to the properties of the product AD if two Enigma permutations A and D are multiplied together. This article supplies further clarification by illustrating the properties of such a multiplication. It makes the properties more plausible by giving a concrete example. It does not supply formal mathematical proofs which are in the domain of a branch of mathematics known as Symmetric Group Theory.

Content

 * 1. Background
 * 1.1 Properties of the product of Enigma permutations multiplication
 * 1.2 Specific example of two Enigma permutations and their product
 * 2. Different notations for the product of Enigma permutations
 * 2.1 The cyclic notation
 * 2.2 A tabular notation
 * 2.3 A graph of the cyclic notation
 * 3. Observations resulting from the graphical representation of cycles
 * 3.1 Observations from the specific example
 * 3.2 Other cyclic structures and the Rejewski characteristic.
 * 3.3 Solving for A and D if one of their eleThe Enigma machine worked by transposing the letters of a plain text message into a cypher text message. The first letter of a plain text message would be changed into another letter.  The first letter of the plain text message could, of course, be any one of 26 letters.  A list of all the 26 letters that could be the first letter in a message, together with the resulting letter to which it would be encoded was called a permutation. After the first letter was encoded, the rightmost rotor of the Enigma machine would rotate one twenty sixth of a revolution to another letter, and this had the effect of completely changing the coding scheme for the second letter.  So for the second letter in the message, the list of all possible 26 letters together with the resulting letter to which it was encoded was a completely different permutation from the first one. After the second letter was encoded, the rightmost rotor would again rotate one twenty sixth of a revolution, and so the permutation for the third letter in the message was different again from the first and second permutations.  The code breakers would often designate these permutations, (lists of 26 letters together with their encoded letter) with the capital letters A, B, C, D, E, F, etc..  The Enigma machine was designed so that it could both encode messages by typing in the plain text and obtaining the cypher text, and also the decode messages by typing in the cypher text and obtaining the plain text.In practice this meant that for specific letters the encoding pair and decoding pair had to be the same.  For example, if the letter "a" encoded into a "p" then the letter "p" had to encode into an "a".  This was true for all letters of all the permutations.  A second (obvious) characteristic of Enigma permutations was that each letter could only encode into one other letter.  (and not into two different letters).  Mathematicians gave a name to permutations which have these two characteristics, they are called "disjoint transpositions"  ments is known.
 * 4. Conclusions
 * 5. See also