CIS 5371 Cryptography Home Assignment 1

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