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Answer questions and show all the steps
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MA464
Assignment 10
11/09/23
Show your work. Explain what techniques you applied. Cite all tools and sources used. A correct answer
without supporting work may not receive full credit.
You must attach all PARI/GP calculations to your submission. Submissions without shown calculations will
receive a score of zero.
1. Alice is doing a Diffie-Hellman key exchange with Bob, using the modulus N = 23023433 and the generator g = 3. She
has chosen the secret number a = 21493443.
(a) Verify that g is a primitive root mod N .
(b) Find A, the number Alice sends to Bob.
(c) She receives B = 9640631 from Bob. Find their shared secret.
2. Bob wants to send the message m = 124391381655647142888550432419387618228 to Alice using RSA. She has published
her public key: N = 2037035976334486086268445688409378161051468836112720989483703577423031829106828984834874777,
e = 65537.
(a) Compute the ciphertext c Bob will send to Alice.
(b) Using the fact that N is the product of the primes p = 1427247692705959881058285969449495136382746771 and
q = 1427247692705959881058285969449495136382746787, find Bob’s private key d.
(c) Use Bob’s private key to decrypt the ciphertext c.
3. Let n = 75759578351354273, m = 18446744073709551629.
(a) Determine whether 3 is a Miller-Rabin witness for n. State whether this tells you that n is prime, probably prime,
or composite.
(b) Determine whether 3 is a Miller-Rabin witness for m. State whether this tells you that m is prime, probably
prime, or composite.
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