Exercises about Symmetric Encryptions.
Exercises for the class on November 9 in Košice and December 4 in Utrecht:
1. Choose a Slovak word and a Caesar-key. Encrypt the word and exchange it with your
neighbor. Try to find the other’s word. (Utrecht students: if you do not know enough
Slovak words for this exercise, you may use a Dutch word in stead.)
2. (Assume letters are numbered from a=1 to z=26). Encrypt the word CAESAR with
the affine cipher, where the key is (3, 8).
3. Prove that the Affine system is perfectly secure if only a single letter is encrypted.
Is it better that the Caesar system in this case?
4. Compute the entropy of the language of Fibonacci strings: bit strings that do not
contain 2 consecutive 1’s.
Can you compute the entropy of the set of strings without three consecutive 1’s?
5. Alice and Bob exchange emails in their highly redundant natural language: the entropy
is only 20%.
What is the unicity distance when they use a cryptosystem with 256-bit keys (such as
the modern AES algorithm)? Compare this to the size of a small text email.
6. In the situation described in the previous exercise, Alice and Bob decide to use a text
compression program that provides a compression ratio of 1:3. What is the entropy
and redundancy of the resulting messages? How does this influence the unicity
distance?
7. The DES-cracking machine by the EFF is based on search units, each of which can
test 2.5 million keys per second. A single deep crack chip has 24 search units. One
board contains 64 of these chips, and a chassis has 12 boards. For a machine with six
chasses, how long does it take on average to find a DES key with a known plaintext
attack?
8. See previous question. The deep crack chip costs 5 dollars. How much do all chips of
the DES-cracker cost together?
9. Discuss the feasibility of building a DES-cracker in the imaginary situation that DES
were designed to have 112-bit keys.
10. There are DES keys that are self-inverting, meaning that encryption with k is exactly
the same function as decrypting with k. Can you give two of such keys?
11. Discuss the feasibility of a Hellman attack against a cryptosystem with 112-bit keys.
12. One GSM frame represents 4.6ms of conversation and has a 22-bit sequence number.
After how much time of conversation is a sequence number reused?
13. Prove that, using the Diffie-Hellman key exchange, Alice and Bob find the same
number to be used as a key.