4 Algorithms for modular arithmetic

A symmetric encryption scheme allows Alice and Bob to privately exchange a sequence of messages in the presence of an eavesdropper Eve. We will assume that Alice and Bob share a random secret key K. How Alice and Bob managed to share a key without the adversary’s knowledge is not going to be our concern here. The encryption scheme consists of an encryption algorithm E that takes as input the key K and the plaintext message M ∈ {0,1}∗, and outputs the ciphertext. The decryption algorithm D takes as input the key and the ciphertext and reconstructs the plaintext message M. In general the encryption algorithm builds upon a block cipher to accomplish two goals: one is to show how to encrypt arbitrarily long messages using a fixed length block cipher. The other is to make sure that if the same message is sent twice, the ciphertext in the two transmissions is not the same. The encryption algorithm to achieve these goals can either be randomized or stateful it either flips coins during its execution, or its operation depends upon some state information. The decryption algorithm is neither randomized nor stateful.