Arriving home after a stressful day at the office, Bob realizes that he has forgotten the combination numbers to the keypad that lets him get into the house, where a vintage glass of pinot noir awaits him. He decides to get the combination number from his wife Alice, who is away on business, by means of his smart phone. Naturally, he does not want the phone service providers, who interpret everything that he texts, to learn the house combination number. After observing that the phone service providers' computers are so busy gouging people with bills that they cannot possibly calculate any discrete logarithms, he decides that this is a perfect time to get the house combination number via a Diffie$-$Hellman key exchange. So$,$ he looks up a table of primes on the laptop that's in his car, and picks the prime p $=50551,$ which is big enough to encompass the five digit combination number to the house. He also checks with his trusty Excel spreadsheet that a primitive root for this prime is g $=3.$ Finally, he thinks about his anniversary year, and picks the number k $=2000$ as his secret discrete logarithm.$$Then he sends the following text message to Alice... Dear A$,$ I forgot the combination number to the house. Can you send me the combination number encrypted by means of Diffie$-$Hellman key exchange? I suggest we use the prime p$=50551$ with primitive root g$=3.$ Using these parameters, my half of the Diffie$-$Hellman key is h$=44691.$ Can you send me your half f of the Diffie$-$Hellman key, as well as the encrypted value y for the house combination number? Love, B$.$ Upon receiving the message, Alice picks the number l $=5000($which is the bonus she got for making the sale on her business trip$),$ and, while keeping the discrete logarithm l only to herself, she replies as follows... Dear B$,$ My half of the Diffie$-$Hellman key is f$=15081.$ The house combination number, encrypted using the full Diffie$-$Hellman key is y $=30547.$ Love, A$.$ Upon receiving the reply from Alice, Bob quickly uses the software on his laptop to decipher y and get the correct combination number x to his house, and soon after sits down to enjoy his well deserved glass of pinot noir. Note: Without Excel spreadsheets or other software, the solutions to the questions below are not feasible. a$)$ How did Bob obtain his half h of the Diffie$-$Hellman key that he sent to Alice?