Question 1: 1. How many different Vigenere keys of length 4 are there for coding the English alphabet? The last string must be of size 2 to get to 26. 2. Say you have an n block of bits message. You split it to 8 bits each (assume that 8 divides n). For each of the 8 bit we choose a permutation of the bits. How many different codes could come out of this? 3. For any two bits xi, yi, define xi XOR yi = xi + yi mod 2. For two vectors, v, u, define v XOR u to be the vector whose i-th bit is ui XOR vi. Say that the plaintext has x bits. For the cyphertext we randomly select a random number r with x bits, (this means that all of the 2x vectors with x bits, have probability 1/2x to be selected). For a vector u, our code message is x XOR r In a known plaintext attack how would you discover the key r. Question 2: 1. Show that if a number ends with the digit 1, then all power of this number end with 1 2. Show that any number that ends in 9 if squared ends at 1. 3. Show that 100197834292is divisible by 10. Question 3: Apply the simple GCD algorithm for 1. a = 112, b = 32 2. a = 114, b = 68 Question 4: For the two a, b in exercise 3, find the x, y so that x a + y b = gcd(a, b) using the generalized gcd algorithm. Question 5 Consider the modular equation: 9x = 15 mod 30. Find all solutions.