Suppose that on the planet Binary, the written language uses an alphabet that contains only two letters X and Y. Also, suppose that in the Binarian language, the letter X occurs 75% of the time, while Y occurs 25% of the time. Finally, assume that you have two messages in the Binary language, and the messages are of equal length.
a. If you compare the corresponding letters of the two messages, what fraction of the time will the letters match?
b. Suppose that one of the two messages is encrypted with a simple substitution, where X is encrypted as Y and Y is encrypted as X. If you now compare the corresponding letters of the two messages— one encrypted and one not—what fraction of the time will the letters match?
c. Suppose that both of the messages are encrypted with a simple substitution, where X is encrypted as Y and Y is encrypted as X. If you now compare the corresponding letters of the two messages— both of which are encrypted—what fraction of the time will the letters match?
d. Suppose instead that you are given two randomly generated "messages" that use only the two letters X and Y. If you compare the corresponding letters of the two messages, what fraction of the time will the letters match?
e. What is the index of coincidence (IC)? Hint: See, for example, [148].
f. How can the index of coincidence be used to determine the length of the keyword in a Vigenère cipher (see Problem 24 for the definition of a Vigenère cipher)?