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Homework 3 Questions
1. Describe how you would conduct a chosen plaintext attack against an affine cipher. I
remind you that the goal of the attack is to find the e-key, which is the pair of
numbers (, ).
2. How many keys are possible in the affine crypto system? ( I remind you that in this
system x is encoded by x +  (mod 26).) Justify your answer.
3. Alice and Bob are using the one-time pad cryptosystem. Suppose that the plaintext is
10110101110 and the ciphertext is 00010110111. What is the key?
4. One reasonable idea for enhancing the security of a cryptosystem is to use double
encryption. Thus Alice first encrypts the message m getting c1 and then she encrypts
c1 obtaining c2 which she sends to Bob. Suppose Alice is doing this using the Hill
cipher. Is double encryption (with the Hill cipher) any safer than simple encryption?
Say ‘yes’ or ‘no’ and justify your answer.
5. Suppose that in the “Baby DES” cryptosystem we have been able to determine that
the first 4 bits of K_3 are either 1100 or 0101 and the last 4 bits of K_3 are 1110. List
all the possibilities for K.
6. Alice is sending a DES encrypted message to Bob, and one bit in block C15 is
corrupted during the transfer over the network (all the other ciphertext blocks are
correct). What blocks of the plaintext will be garbled when Bob is doing the
decryption and how many bits will be affected if
a. Alice is using the ECB mode.
b. Alice is using CBC.
c. Alice is using CFB, 8-bit variant.
7. Alice wants to use quad-DES by doing four DES encryptions with two keys K1 and
K2 using the formula C = E K1 (E K1 (E K2 (E K2 (P)))). Describe briefly how Eve can
mount an efficient attack of the type “known plaintext.” Suppose Eve can do 256
simple DES encryptions/decryptions in about 20 minutes and also assume that doing
double DES encryptions/decryptions takes two times the time for single DES
encryption/decryption. How long it will take to Eve to break the new scheme
proposed by Alice (ignoring the time for all other operations except DES
encryptions/decryptions)? Justify briefly.
8. We are analyzing the security of the substitution cryptosystem, denoted as usual EK. (K is the
key, which in the case of the substitution cryptosystem is a table indicating how each letter is
substituted). Suppose the message space M consists of all possible combinations of two
letters from the English alphabet, in other words M = {aa, ab, ac, …, zz}, and each message
from M is equally likely to be sent by Alice.
(a) What is Prob (M= ‘aa’)? (here M is a randomly chosen message from M).
(b) What is Prob (M = ‘aa’| EK(M) = ‘bc’)? (Here M is a randomly chosen message
from M, and K is a randomly chosen key).
(c) What do (a) and (b) say about the security of the substitution cryptosystem? More
precisely, is it perfectly secure (according to the formal definition discussed in class)?
9. The alphabet of a certain language has only the 5 letters (0,1,2,3,4). Suppose the
message `42’ is encrypted with an affine cipher and the ciphertext is `12’. (a) Find the
key. (b) The ciphertext 3124 has been obtained using the key from (a). Find the
corresponding plaintext.
.
10. We encrypt with double Vigenere using for the first encryption the key k_1 =
(1,3,5,7) and for the second encryption the key k_2 = (2,4). Show that this is
equivalent to simple Vigenere encryption, and find the key for the single Vigenere
encryption that is equivalent with double Vigenere encryption with the above keys
k_1 and k_2.
11. The plaintext `friday’ is encrypted using the Hill cipher with m=2 to give the
ciphertext `PQCFKU’. Find the the key K. (It is acceptable to write the key K as a
product of two matrices without performing the multiplication or taking the inverse.)
12. Suppose that you are informed that the plaintext `abccab’ has been encrypted with the
Vigenere cipher and the corresponding ciphertext is `bbccab’. What is the key
length?
13. The alphabet is {0,1,2,3,4}. The Hill cipher is used with m=2, and the key is the 2-by-2
matrix
K = (4 1)
(3 1)
Encrypt the message `1 2 3 4’.
14. Apply one round of the Feistel structure to the string P = 1111 1010 and the key K = 0110, where the
function f is defined by f (R,K) = R + K (+ is bitwise XOR).