Homework 2

Due: October 23, 2024
Points: 100

1. (20 points) An affine cipher has the form c = (am + b) mod n. Suppose m is an integer between 0 and 25, each integer representing a letter.
   1. Let n = 26, a = 3, and b = 123. What is the ciphertext corresponding to the phrase THIS IS A CIPHER MESSAGE.
   2. A requirement for a cipher is that every plaintext letter correspond to a different ciphertext letter. If a is not relatively prime to n, does the affine cipher meet this property? If b is not relatively prime to n, does the affine cipher meet this property? In both cases, either prove it does or present a counterexample.

2. (20 points) Alice and Bob are creating RSA public keys. They select different moduli n_Alice and n_Bob. Unknown to both, n_Alice and n_Bob have a common factor.
   1. How could Eve determine that n_Alice and n_Bob have a common factor without factoring those moduli?
   2. Having determined that factor, show how Eve can now obtain the private keys of both Alice and Bob.

3. (20 points) Consider the following authentication protocol, which uses a symmetric cryptosystem. Alice generates a random message r, enciphers it with the key k she shares with Bob, and sends the enciphered message {r}k to Bob. Bob deciphers it and sends {r + 1}k back to Alice. Alice deciphers the message and compares it with r. If the difference is 1, she knows that her correspondent shares the same key k and is therefore Bob. If not, she assumes that her correspondent does not share the key k and so is not Bob. Does this protocol authenticate Bob to Alice? Why or why not?

4. (24 points) The designers of the UNIX password algorithm used a 12-bit salt to perturb the first and third sets of 12 entries in the E-table of the UNIX hashing function (the DES). The maximum length of a UNIX password is 8 characters selected from a set of 96 characters, and the minimum length is 5 characters. Assume that each user is assigned a salt from a uniform random distribution and that anyone can read the password hashes and salts for the users. Also assume a password can be tested in time t.
   1. What is the worst case time to find all passwords using a dictionary attack?
   2. Assume that eight more characters were added to the password and that the DES algorithm was changed so as to use all 16 password characters — that is, the maximum length of a password was 16 characters and the minimum length is 5. What would be the worst case time to find all passwords using a dictionary attack?
   3. Now assume that the passwords were between 5 and 8 characters long, as before, but that the salt length was increased to 24 bits. What would be the worst case time to find all passwords using a dictionary attack?

5. (16 points) A network consists of n hosts. Assuming that symmetric cryptographic keys are distributed on a per-host-pair basis, compute how many different keys are required.

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ECS 235A, Computer and Information Security Version of October 8, 2024 at 6:58PM

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