MATH 470.200/501
Examination 2
November 1, 2011
NAME
SIGNATURE
This exam consists of 6 problems, numbered 1–6. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is permitted on this exam.
The point value for each question is shown next to each question.
You must turn your Exam Note Sheet in with this exam.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 6 PROBLEMS ON
7 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
Points
Possible Credit
1
24
2
16
3
16
4
16
5
16
6
16
Notes
5
Total
109
NAME
1.
MATH 470
Examination 2
Page 2
[24 points] Alice and Bob are using RSA to communicate.
(a) Alice’s encryption key is (n1 , e1 ) = (187, 7). Alice wants to encode the plaintext
‘10’ to send to Bob. What is the ciphertext that she sends?
(b) On another occasion, you intercept the ciphertext ‘5’ sent to Alice. Find Alice’s
decryption key and write down the expression that yields the plaintext. (You do not
need to calculate the plaintext completely.)
(c) Suppose that Bob’s encryption key is (n2 , e2 ) = (989, 12), and suppose that we
have discovered that φ(989) = 924. Use this information to factor 989.
November 1, 2011
NAME
2.
MATH 470
Examination 2
Page 3
[16 points] The following congruences hold:
789495 ≡ 8154 (mod 15841)
7891980 ≡ 218 (mod 15841)
7897920 ≡ 1 (mod 15841)
789990 ≡ 3039 (mod 15841)
7893960 ≡ 1 (mod 15841)
78915840 ≡ 1 (mod 15841)
(1)
(2)
(3)
1231625 ≡ 4852 (mod 13001)
1236500 ≡ 13000 (mod 13001)
1233250 ≡ 10094 (mod 13001)
12313000 ≡ 1 (mod 13001)
(4)
(5)
(a) Using the data in lines (1)–(3) above, apply the Miller-Rabin Primality Test to
n = 15841. What can you conclude about 15841? Explain.
(b) Using the data in lines (4)–(5) above, apply the Miller-Rabin Primality Test to
n = 13001. What can you conclude about 13001? Explain.
November 1, 2011
NAME
3.
MATH 470
Examination 2
Page 4
[16 points] Suppose that n = 1643 is being used for RSA encryption, and suppose that
we have discovered that the encryption exponent is e = 29 and that the decryption
exponent is d = 269. Suppose further that the following congruences hold:
2975 ≡ 1613 (mod 1643)
5975 ≡ 931 (mod 1643)
3975 ≡ 712 (mod 1643)
6975 ≡ 1642 (mod 1643).
Use the Universal Exponent Factorization Method to factor 1643. Show your work.
November 1, 2011
NAME
4.
MATH 470
Examination 2
Page 5
[16 points] Let n = 7991. We know that 7991 is the product of two distinct primes.
Suppose in carrying out the quadratic sieve that you have found that
752 − n = −2366,
842 − n = −935,
1012 − n = 2210,
792 − n = −1750,
942 − n = 845,
1032 − n = 2618.
Use this information to factor 7991. Show your work.
November 1, 2011
NAME
5.
MATH 470
Examination 2
Page 6
[16 points; (a) 6 points, (b) 10 points] Throughout this problem we work modulo
p = 19, with chosen primitive root α = 3.
(a) Show that L3 (5) = 4.
(b) Use the Pohlig-Hellman algorithm to find L3 (2).
November 1, 2011
NAME
6.
MATH 470
Examination 2
Page 7
[16 points; (a) 6 points; (b) 10 points] Alice and Bob are again using RSA to communicate. They are using the same modulus n, so that Alice’s public encryption key
is (n, eA ) and Bob’s is (n, eB ). Alice and Bob happen to have chosen eA and eB that
are relatively prime.
Now Charles wants to send the message m to both Alice and Bob. You may assume
that m < n.
(a) What does Charles send to Alice? What does he send to Bob?
(b) Suppose that Eve intercepts both of these transmissions. Show how Eve can
recover m without factoring n.
November 1, 2011