The Enigma machine is the most famous cipher machine to date. Nazi Germany used it during World War II

to encrypt messages so that enemies could not understand them. The story of the British cryptanalysts who

successfully deciphered Enigma has become the subject of multiple movies Enigma (2001); The Imitation

Game (2014). In this exercise, we will focus our attention on a simplified version of the Enigma machine,

which we name "Little Enigma." Like the real Enigma machine (picture above), this machine consists of two

key components. First, the Little Enigma machine has 5 different rotors, each of which comes with 10 pins

with numbers ranging from 0 to 9. Second, a component called the plugboard contains 26 holes, corresponding

to the 26 letters of the alphabet. In addition, 13 cables connect all possible pairs of letters. Since a cable has

two ends, one can connect, for example, the letter A with any other of the other 25 letters present in the

plugboard.

To either encode a message or decode an encrypted message, one must provide the Little Enigma machine

with a correct five-digit passcode to align the rotors and a correct configuration of the plugboard. The rotors

are set up just like many combination locks. For example, the passcode 9-4-2-4-9 means that the five rotors

display the numbers 9, 4, 2, 4, and 9 in that order. In addition, the 13 cables connecting the letters in the

plugboard must be appropriately configured. The purpose of the plugboard is thus to scramble the letters. For

example, if B is connected to W, the Little Enigma machine will switch B with W and W with B to encode a

message or decode an encoded message. Thus, a sender types a message on the keyboard, the plugboard

scrambles the letters, and the message is sent in its encrypted form. A receiver decodes the encrypted message

by re-typing it on a paired Little Enigma machine that has the same passcode and plugboard configuration.

Question 1

How many different five-digit passcodes can be set out of the 5 rotors?

Question 2

How many possible configurations does the plugboard provide? In other words, how many ways can 26 letters

be divided into 13 pairs?

Hint: Consider first a case with 4 letters only. When computing the results in R for 26 letters, remember to

compute on the log scale (see page 249 in QSS).

Question 3

Based on the previous two questions, what is the total number of possible settings for the Little Enigma

machine?