### Assignment #2

##### Due: Feb 25th, 2003 at 11am MST

1. Fix some integer n > 0. Let the set A = [0..2n - 1]. As we know, a function F : A -> A is a permutation if and only if F is one-to-one and onto. A permutation P : A -> A is a bit mixing permutation if and only if for all x in A, P(x) is a reordering of the bits of x. More precisely, if we write out x in binary as b(1) b(2) b(3) ... b(n), then P(x) = b(Q(1)) b(Q(2)) .... b(Q(n)) for some permutation Q : [1..n] -> [1..n].

Prove the following: if P is a bit mixing permutation then for all x and y in A, P(x xor y) = P(x) xor P(y).

2. In class, we discussed the fact that DES implements a tiny fraction of all possible permutations on 64-bit binary strings. Let's model the set of permutations realized by DES as 256 distinct permutations, and call this set of permutations D. Now, what is the probability that a randomly-chosen 64-bit permutation (from the space of all possible permutations) is contained in D?

3. This problem has two parts; the first part is the easier.

• Define c(x) as the one's complement of x. Prove that for all 56-bit keys K and all 64-bit inputs X, we have that DES(K, X) = c(DES(c(K),c(X))).
• Describe an attack on DES that uses this property to cut down the number of keys we have to try when exhaustively searching the keyspace.

4. We saw in class that 256 DES keys was not enough: specialized hardware could be built which finds the key by exhaustive search in about 35 mins on average (for about \$1M US, in 1998). We might attempt to increase the length of the 56-bit DES key by making a new block cipher DES+ with a 120-bit key as follows:

• Take the 120-bit key K and break into two strings: a 56-bit subkey K1, and a 64-bit subkey K2.
• To encipher any 64-bit input block M, compute DES(K1, M xor K2) and output the result
• To decipher any 64-bit ciphertext block C, compute K2 xor DES-1(K1, C) and output the result
Please argue that DES+ is no better at resisting exhaustive key search attacks than DES was. Argue this by showing an attack which uses around 256 DES operations; you may assume you have as many DES+ plaintext-ciphertext pairs as you like.

5. The following problem is very challenging and should not be attempted unless you have successfully completed the 4 preceeding problems and you are prepared to spend some significant time and are fairly comfortable with linear algebra.

We have claimed that the design of the S-boxes in DES was very carefully chosen. This exercise shows that not any design will do.

Let's say you have the ability to change the contents of the S-boxes (not the structure, just the numbers inside). Change them to anything you like and call this new cipher WDES (for Weak DES). Then describe a simple attack on WDES which efficiently extracts the key given one known plaintext-ciphertext pair.