Suppose we define an encryption scheme as follows. The key will be four elements k1,k2,k3 and k4 in Z26 . The message space will be sequences of elements of of length 6. The ciphertext space will be the same as the message space.

The encryption algorithm is the following. Given a key and and a message , the corresponding ciphertext is , where:

b1 = k1a1 + k2a2 (mod 26)

b2 = k3a1 + k4a2 (mod 26)

b3 = k1a3 + k2a4 (mod 26)

b4 = k3a3 + k4a4 (mod 26)

b5 = k1a5 + k2a6 (mod 26)

b6 = k3a5 + k4a6 (mod 26)

Suppose you know that the message SIXWIG (thought of as a sequence of numbers in ) corresponds to the ciphertext IGPXUY.

1. Write down the system of equations in the unknowns k1,k2,k3 and k4 that you must solve to recover the key from the known plaintext-ciphertext pair.

2. What numbers in Z26 have multiplicative inverses mod 26?

3. Use your answer to Q2 to solve for one of the key variables.

4. Recover the key used to produce the known plaintext pair.