Math 470
Exam 2 Sample Problems
October 25, 2011
Please note: These problems by no means make an exhaustive list of the types of problems that we have
seen in class or on the homework or that you can expect to see on the exam.
1. David and Eleanor are using RSA to communicate secretly.
(a) David’s public encryption key is (m, e) = (91, 5). Eleanor wants to encode the plaintext ‘3’ to
send to David. What does she send?
(b) On another occasion, you intercept the ciphertext ‘4’ sent to David. What was the plaintext?
2. Alice and Bob are using RSA to communicate secretly. Bob’s public encryption key is (n, e) =
(332621, 103). Eve discovers that φ(n) = 331452. What is the factorization of n?
3. Use the Miller-Rabin Test to show that 561 is composite.
4. The following congruences hold:
34733274656 ≡ 1
53013
274656
≡1
(mod 274657)
(mod 274657)
3473369265 ≡ 71394
5301372015 ≡ 182831
(mod 277061)
(mod 288061)
34734274656 ≡ 60108
87745
274656
≡1
(mod 274657)
(mod 274657)
34733138530 ≡ 12019
(mod 277061)
53013144030 ≡ 288060
(mod 288061)
(1)
(2)
(3)
(4)
(a) What do lines (1) and (2) imply about the primality of 274657? Explain.
(b) What does line (3) imply about the primality of 277061? Explain.
(c) What does line (4) imply about the primality of 288061? Explain.
5. Suppose that n = 4897 is being used for RSA encryption, and suppose that we know that the encryption
exponent is e = 67 and the decryption exponent is d = 71. Suppose further that the following
congruences hold:
21189 ≡ 4896
4
1189
6
1189
≡1
(mod 4897)
(mod 4897)
≡ 4896
(mod 4897)
31189 ≡ 1
≡ 414
1189
≡1
(mod 4897)
1189
5
7
(mod 4897)
(mod 4897).
Use the Universal Exponent Factorization Method to factor 4897.
6. Let n = 10981. Suppose that in performing the quadratic sieve you have found that
912 − n = −2700,
982 − n = −1377,
1052 − n = 44,
1072 − n = 468,
1152 − n = 2244,
1162 − n = 2475.
Use this information to factor n. Show your work.
7. Calculate the discrete logarithms L5 (10), L5 (100), and L25 (15) modulo 17.
8. The number 2 is a primitive root modulo 29. Use the Pohlig-Hellman algorithm to find L2 (3).