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CIS 5371 Cryptography Home Assignment 1 Due: At the beginning of the class on Feb 11, 2016 Exercises taken from the course textbook. Jonathan Katz and Yehuda Lindell, Introduction to Modern Cryptography. • 1.3 Consider an improved version of the Vigenère cipher, where instead of using multiple shift ciphers, multiple mono-alphabetic substitution ciphers are used. That is, the key consists of t random permutations of the alphabet, and the plaintext characters in positions i, t + i, 2t + i, and so on, are encrypted using the ith permutation. Show how to break this version of the cipher. • 1.5 Show that the shift, substitution, and Vigenère ciphers are all trivial to break using a known-plaintext attack. (Assuming normal English text is being encrypted in each case.) How much known plaintext is needed to completely recover the key for each of the ciphers (without resorting to any statistics)? • 1.6 Show that the shift, substitution, and Vigenère ciphers are all trivial to break using a chosen-plaintext attack. How much plaintext must be encrypted in order for the adversary to completely recover the key? Compare to the previous question. • 2.2 Prove or refute: For every encryption scheme that is perfectly secret it holds that for every distribution over the message space M, every m, m0 ∈ M, and every c ∈ C: P r[M = m | C = c] = P r[M = m0 | C = c]. • 2.3 When using the one-time pad (Vernam’s cipher) with the key k = 0` , it follows that Enck (m) = k ⊕m = m and the message is effectively sent in the clear! It has therefore been suggested to improve the one-time pad by only encrypting with a key k 6= 0` (i.e., to have Gen choose k uniformly at random from the set of non-zero keys of length `). Is this an improvement? In particular, is it still perfectly secret? Prove your answer. If your answer is positive, explain why the one-time pad is not described in this way. If your answer is negative, reconcile this fact with the fact that encrypting with 0` doesn’t change the plaintext. Mike Burmester 1