Here is a Python dictionary of the relative frequency of letters in English text: { "A": 08167, "B": 01492, "C": 02782, "D": 04253, "E": 12702, "F": 02228, "G": 02015, "H": 06094 "I": 06996 "J'': 00 153 "K": "00772 "L": 04025 "M": 02406, "N": 06749, "O": 07507, "P": 01929, "Q": 00095, "R": 05987, "S": 06327, "T": 09056, "U": 02758, "V": 00978, "W": 02360, "X": 00150, "Y": 01974, "Z": 00074 Here is some plaintext: The population variance of a finite population X of size N and mean mu is given by Var(X) = 1/N sigma^N_i = 1 (x_i - mu)^2 (a) What is the population variance of the relative letter frequencies in English text? (b) What is the population variance of the relative letter frequencies in the given plaintext? (c) For each of the following keys - yz, xyz, wxyz, vwxyz, uwxyz - encrypt the plaintext with a Vigenere cipher and the given key, then calculate and report the population variance of the relative letter frequencies in the resulting ciphertext. Describe and briefly explain the trend in this sequence of variances. (d) Viewing a Vigenere key of length k as a collection of k independent Caesar ciphers, calculate the mean of the frequency variances of the ciphertext for each one. (E.g., for key yz, calculate the frequency variance of the even numbered ciphertext characters and the frequency variance of the odd numbered ciphertext characters. Then take their mean.) Report the result for each key in part (c). Is the mean variance like those observed in part (b)? Part (c)? Briefly explain. (e) Consider the ciphertext that was produced with key uvwxyz. In part (d), you calculated the mean of six variances for this key. Revisit that ciphertext, and calculate the mean of the frequency variances that arise if you had assumed that the key had length 2, 3, 4, and 5. Does this suggest a variant to the Kasiski attack? (Don't say no!) Briefly explain