Name:______________________________     Alpha:________________________________ 

1. [10pts] True/False questions.
   • TRUE/FALSE
     If an encryption scheme is not IND-CPA secure, then it's not IND-CCA secure.
   • TRUE/FALSE
     Properly used, a MAC can provide both confidentiality and authenticity.
   • TRUE/FALSE
     Suppose that M1 and M2 are encrypted using counter mode (CTR). Since CTR mode is IND-CPA, even if the same initialization vector (IV) is used for encrypting both messages, every ciphertex block in the resulting ciphertexts will be different.
   • TRUE/FALSE
     Computing a hash for a data item using a cryptographic hash function such as SHA-256 requires possession of the correct secret key.
   • TRUE/FALSE
     When computing a HMAC, it is vital to incorporate a randomized Initialization vector that is unpredictable by an attacker.

2. [10pts] Answer the following:

   • \( -3 \bmod 11 {=} \)

   • \( \mathbb{Z}_7^* {=} \)

   • \( \phi(7) {=} \)

   • \( \mathbb{Z}_{14}^* {=} \)

   • \( \phi(14) {=} \)

3. [4pts] Fill in the blanks: In general, for a prime number \(p \), it holds that \(\phi(p) = \) .

   In addition, for a product of two distinct prime numbers \(p , q \), it holds that \(\phi(pq) = \) .

4. [2pts] True/False: For any positive integer $a, b$, if $a \lt b$, and $a$ is coprime to $b$, it holds that $a \in \ZZ_b^*$.

5. [6pts] Answer the following:
   1. What is $3^{-1} \pmod{ 11 }$?

   2. What is $4^{-1} \pmod{ 8 }$?

   3. What is $5^{-1} \pmod{ 8 }$?
6. [4pts] Answer the following:
   • What is \( 5^{150} \bmod 7 \)? Don't use a computer for this.

   • What is \( 4^{161} \bmod 15 \)? Don't use a compuer for this.