For each of the following pairs of integers (x, y), first determine whether x−1 mod y exists. Then find x−1(mod y) if it exists. Please do this problem by hand, and show all work.
(a) x=5, y=25
(b) x=24, y=35
(c) x=17, y=101
If an encryption function eK is identical to the decryption function dK, then the key K is said to be an involutory key. Find all the involutory keys in the Shift cipher over Z26.
Suppose K = (5, 21) is a key in an Affine cipher over Z29.
(a) Express the decryption function dK (y) in the form dK = a′y + b′, where a′, b′ ∈ Z29.
(b) Prove that dK(eK(x)) = x for all x ∈ Z29.
The following ciphertext was encrypted using an Affine cipher:
The first two letter of the plaintext are if. Please decrypt.
Alice is sending a message to Bob using the Vigenere cryptosystem. At some point, Alice gets bored, and starts sending plaintext that consists of a single letter (known only to her) repeated a few hundred times. Eve knows that the Vigenere cipher is being used, and that the plaintext consists of a single letter, repeated. Show how Eve can deduce the key.
Evan, an attacker, is on a mission. He is given a (plaintext, ciphertext) pair (relation, ORIENTAL), and his task is to determine the complete cryptographic key (table), if the given pair is generated using:
(a) Permutation cipher,
(b) Substitution cipher.
Please put your “black hat” on, and show Evan how to accomplish this mission, or show why it is impossible. In doing so, please assume that the set of possible plaintexts is equal to the set of possible ciphertexts, and that it is equal to Z26.