Homework 13

\( \def\ZZ{\mathbb{Z}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \) Print this page and give the answers.


Name:______________________________     Alpha:________________________________

[10pts] Answer the following:
- \( \mathbb{Z}_{21}^* {=} \)
- \( \phi(35) {=} \)
- What is \(3^{-1} \pmod{ 21 }\)?
- What is \(4^{-1} \pmod{ 21 }\)?
- What is \( 4^{241} \bmod 35\)? Don't use a compuer for this.
[4pts] Consider Plain RSA encryption scheme. where the RSA modulus \(N\) in the public key is \(N = 15\). What numbers are good to be \(e\) in the public key?
```
a. 2           b. 3          c. 4          d. 5
```
[10pts] Consider Plain RSA encryption scheme. where the RSA modulus \(N = 35\) and \(e = 5\), that is \(pk = (35, 5)\).
- What is the encryption of message \(3\)? Don't use the computer (but a calculator) for this.
```
  
```
- What is the decryption key \(d\) in this case? Don't use the computer for this.
```
  
```
[10pts] Give the answer to the RSA inversion problem:
```
N=648153784676707
e=7
C=251930644046701
```
Write a Python script to answer the question. (Hint: Try to factor N)
```
M = 
```
You can also change an integer (say in variable M) to bytes using the following code:
```
M.to_bytes( (M.bit_length()+7)//8, 'big')
```
What is the message?
```
message = 
```
[6pts]
- TRUE/FALSE: The plain RSA is a deterministic encryption.
- TRUE/FALSE. The plain RSA is IND-CPA secure.
- TRUE/FALSE. The plain RSA is IND-CCA secure.