Name:______________________________     Alpha:________________________________ 

1. [6pts] What is the smallest safe prime larger than 50? Justify your answer.

   Answer: 
   
   Reason:
   
   
   
   

2. [6pts] In Python, compute the following expressions (copy and paste):

    pow(111111111, 2222222, 33333333333333333333333333333)

    (111111111**2222222) % 33333333333333333333333333333

   • Do they report the same answer?

   • How about the running time? Why? (You may want to check this)

3. [2pts] For $(g, p, y) = (4, 1907, 792)$, find the discrete logarithm $\log_g y \pmod p$. Write a Python script to answer the question.

4. [10pts] The following shows an execution of the Diffie-Hellman key exchange protocol. Fill in the blanks. Don't use a computer for this. You will see a similar problem in the exam.

   prime number p = 47
   generator g = 3
   
   
   
   Alice:                                  A = ____                  Bob:
   exponent a = 4                      --------------->              exponent b = 5             
   
   
                                           B = ____
                                      <---------------
   
   Ka = (___)^(___) mod ___ = ____                                   Kb = (___)^(___) mod ___ = ____
   
   ------
   * Ka must be equal to Kb.