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 intercept 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 a 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 $`=5000($which is the

bonus she got for making a sale on her business trip$),$ and, while keeping

this discrete logarithm $`$ 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 $=30547.$

$6$

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 your Excel spreadsheets or other software, the solutions to

the questions below are not feasible.

$[2]($a$)$ How did Bob obtain his half h of the Diffie$-$Hellman key that he sent

to Alice?

$[2]($b$)$ How did Alice obtain her half f of the Diffie$-$Hellman key that she

sent to Bob?

$[2]($c$)$ What is the full Diffie$-$Hellman key z that they both came to possess?

And how did each of Alice and Bob obtain it$?$

$[4]($d$)$ What is the combination number x that let Bob get into the house?

And how did Bob figure out the combination number?

$[2]($e$)$ How did Alice create the encrypted value y from the actual combination

number x that she knew?