Modern cryptography: exercises chapter 2

This post is a work in progress, so if you find it incomplete and not readable probably it's not finished yet. I prefer to publish a little before than leave a post to rust in my drafts.

These are some solved exercises of chapter 2 of the book “Introduction to modern cryptography” by Jonathan Katz and Yehuda Lindell.

I’m not sure are correct but if anyone find any error let me know. It’s a work in progress and more exercises will follow in future, when I finally find time to solve more. Also, enjoy my hand made diagrams :).

Definitions

An encryption scheme \((\hbox{Gen}, \hbox{Enc}, \hbox{Dec})\) with message space \(M\) is perfectly secret if for every probability distribution over \(M\), every message \(m\in M\), and every ciphertext \(c\in C\) for which \(\hbox{Pr}\left[C = c\right] > 0\):

This is equivalent to a prior indistinguishibility:

i.e. the ciphertext’s probability distribution is independent from the message’s probability distribution.

For notation purpose take in mind this way of writing:

Instead the adversarial indistinguishability experiment \(\hbox{PrivK}^{\hbox{eav}}_{A, \Pi}\) is defined as follow:

  1. The adversary \(A\) outputs a pair of messages \(m_0, m_1\in M\).
  2. A key \(k\) is generated using \(\hbox{Gen}\), and a uniform bit \(b\in{0,1}\) is chosen. Ciphertext \(c\leftarrow\hbox{Enc}_k(m_b)\) is computed and given to \(A\). We refer to \(c\) as the challenge ciphertext.
  3. \(A\) outputs a bit \(b^\prime\).
  4. The output of the experiment is defined to be \(1\) if \(b = b^\prime\), and \(0\) otherwise. We write \(\hbox{PrivK}^{\hbox{eav}}_{A, \Pi} = 1 \) if the output of the experiment is \(1\) and in this case we say that \(A\) succeeds.

    Exercises

2.1: Prove that, by redefining the key space, we may assume that the key generation algorithm \(Gen\) choose a key uniformly at random from the key space, without changing \(\hbox{Pr}\left[C = c | M = m\right]\) for any \(m, c\).

solution

The encryption scheme can be described using the following diagram:

where the \(\hbox{Gen}\) algorithm creates a key using a random tape \(r\) from the space \(R\). Redefining \(\hbox{Enc}\) to include \(\hbox{Gen}\) and using \(r\) as new key we have that

This means that we simply shift the key to be the random tape that provides random values to the Turing machine implementing the probabilistic algorithm \(\hbox{Gen}\).

we can verify


2.2: Prove that, by redefining the key space, we may assume that \(Enc\) is deterministic without changing \(\hbox{Pr}\left[C = c \,|\, M = m\right]\) for any \(m, c\).

solution

\(\hbox{Enc}\) can become deterministic using the random tape as a component of the key;

Expanding the relation \(k^\prime=(k, r)\) we obtain


2.3: Prove or refute: an encryption scheme with message space \(M\) is perfectly secret if and only if (\(\iff\)) for every probability distribution over \(M\) and every \(c_0,\,c_1\in C\) we have \(\hbox{Pr}\left[C = c_0\right] = \hbox{Pr}\left[C = c_1\right]\).

solution

For this one I have not yet a solution but I think has to do with the dimension of the cipher and message space: the iff holds if we take the conditions from the Shannon’s theorem, i.e. if the all the spaces have the same dimensions.


2.4: Prove the second direction of Lemma 2.4

solution

We must proof that perfect secrecy implies perfect indistinguishibility; this can be done using the Bayes’ theorem

together with perfect secrecy

to obtain


2.5: Prove Lemma 2.6.

solution

We need to proof that

\(\Rightarrow\)) If we have one half as probability of success in the experiment, then this means that the ciphertext is indipendent from the message, stated with maths

\(\Leftarrow\))


2.6: For each of the following encryption schemes, state wheter the scheme is perfectly secret. Justify your answer in each case.

solution

Let’s calculate for the part (a)

but this is not independent from from the choose of \(m\) so is not perfectly secret.

Instead for the case (b) we have a strange implementation of the one time pad where the messages and ciphertexts have always the last bit set to zero but since that last bit doesn’t carry information we can say that is perfectly secret.


2.7 When using the one-time pad with key \(k = 0^l\), we have \(\hbox{Enc}_k(m) = k\otimes m = m\) and the message is sent in the clear! It has therefore been suggested to modify the one-time pad by only encrypting with \(k\not= 0^l\) (i.e. to have \(\hbox{Gen}\) choose \(k\) uniformly from the set of nonzero keys of length \(l\)). Is this modified scheme still perfectly secret? Explain.

solution

From theorem 2.10 follows that from an encryption scheme with \(|K| < |M|\) you cannot achieve perfect secrecy and in the case where we omit the zero key we are obtaining such case.


2.8 Let \(\Pi\) denote the Vigenère cipher where the message space consists of all 3-character strings (over the English alphabet), and the key is generated by first choosing the period \(t\) uniformly from \({1, 2, 3}\) and then letting the key be a uniform string of length \(t\).

Do you find this post incomplete? probably because it's a work in progress. Let me know how do you want this to be completed