Question

In World War II, Germany used an electromechanical encryption machine called Enigma. Enigma was an excellent machine for the time, and breaking its encryption was an important challenge for the Allied countries. The Enigma machine consisted of a plugboard three (or, near the end of the war, four) rotors, and a reflector (and a keyboard and lights, but these do not affect the security of the system). h. During the course of the war, the Allies captured several Enigma machines and learned several important things: The Germans used five different rotors (later, eight different rotors). Each day, three of the five rotors were placed left to right in the machine (this was part of the daily key). The Allies also learned the wiring of each rotor and were able to copy the rotors. They learned the wiring of the reflector. How many ways could three rotors be selected from five and placed into the machine? (Order mattered; a rotor configuration of 123 operated differently from 132, etc.) i. After learning the wiring of the rotors and the wiring of the reflector, the remaining configuration variables parts of the daily secret key) were (1) the placement of three rotors from five into the machine, (ii) the number of starting positions for three rotors, (m) the position of the two rings (on the leftmost and middle rotors), and (iv) the plugboard configuration. Assuming k= 10 wires were used in the plugboard, how many possible Enigma configurations remained?

Answer 1

To calculate the number of possible Enigma configurations, we need to consider the different variables involved:

1. Selecting three rotors from five: Since order matters (123 is different from 132), we can use the concept of permutations. The number of ways to select three rotors from five can be calculated as 5P3, which is equal to 60.

2. Starting positions for three rotors: Each rotor could be positioned in 26 different starting positions, as there are 26 letters in the English alphabet. Since there are three rotors, the number of possible starting positions is 26^3, which equals 17,576.

3. Position of the two rings: The ring position for each rotor can be set in 26 different positions. As there are two rings, the number of possible ring positions is 26^2, which equals 676.

4. Plugboard configuration: Assuming 10 wires were used in the plugboard, we need to calculate the number of ways to select pairs of wires from these 10. This can be calculated using combinations, specifically 10C2, which is equal to 45.

To determine the total number of possible Enigma configurations, we multiply the number of possibilities for each variable:

60 (rotor selection) x 17,576 (starting positions) x 676 (ring positions) x 45 (plugboard configurations) = 43,679,808,000.

Therefore, there are approximately 43,679,808,000 possible Enigma configurations, given the variables mentioned in the question.