### Assignment #6

##### Due: Never

1. Suppose Bob has an RSA cryptosystem with a large modulus n which is impractical for any adversary to factor. Alice has a message she wishes to send to Bob which is a string of uppercase alphabetical characters without spaces or punctuation. For example, the message might be "HELLOTHERE". In order to encode her message, she first converts each letter to a number between 0 and 25 using the normal method: A is 0, B is 1, C is 2, ... , and Z is 25. This yields a list of numbers, each number being between 0 and 25. Then she encrypts each of these numbers as usual with Bob's RSA public key. For example, if she has the message "HI", this results in the numbers 7 and 8. If Bob's public key is (n=18721, e=25), then the ciphertext is 725 mod 18721, 825 mod 18721.

Explain why this is a poor use of RSA by giving an attack on this system. You cannot assume that you can factor n. Illustrate your attack by decrypting the following ciphertext which was generated using the public key (n=18721, e=25):

```18718, 13444, 4644, 13444, 1437, 0, 17173, 13444
```

2. Suppose we use two different RSA public keys (n, e1) and (n, e2) where e1 is not equal to e2. In other words, the modulus n is the same, but the public exponents are different. Explain how you can recover the plaintext M if you are given C1 = Me1 mod n and C2 = Me2 mod n.

3. As we well-know, CBCMAC is secure only if the number of blocks in the length of the message is fixed. We define a new MAC called XMAC which is a modification of CBCMAC attempting to allow msgs of any length. XMAC is defined as follows: choose two random keys, K and L, where K is a 56-bit DES key and L is a 64-bit string. Let CBCMACK be the CBCMAC over DES with key K. Now define XMAC(M) = CBCMACK(M) xor L. Please answer the following two questions, giving a convincing argument for your answers.

1. Is XMAC secure over messages with a fixed block-length?
2. Is XMAC secure over messages of any block length?

4. Describe how to find the remaining WEP-key bytes K[4..7] by extending the attack we covered in class.