Dr. A. Betten
Fall 2009
MATH 360 Mathematics of Information Security
Final – Practice
Problem # 1
Bob chooses the elliptic curve y 2 = x3 + 5x + 7 mod 199. He also chooses the base point
P = (1, 49) of order 212 and the secret integer s = 111.
a) Compute Q = s·P ? b) Alice wants to send the short message “OK” to Bob. Can you help
her encode the message as two points M1 and M2 on the curve (using the Koblitz scheme).
The x-coordinate is x = 7m + j where m is the message and j is in Z7 .
c) Alice then chooses her secret integer k = 57. Using the El-Gamal scheme, can you compute
the ciphertext?
Problem # 2
Bob chooses the elliptic curve y 2 = x3 + 2x + 4 mod 31. He also chooses the base point
P = (0, 2, 1) and the secret integer s = 11. Can you help him compute s · P ?
Problem # 3
Suppose Bob chooses the elliptic curve from 1. again. This time he chooses P = (179, 71)
and Q = sP = (112, 82). Can you break the system and find out s?
Problem # 4
In the finite field F16 generated by the irreducible polynomial X 4 + X 3 + 1, compute:
a) The product 1110 times 1011 (your answer must be reduced).
b) The inverse of 1101.
Problem # 5
Create a finite field with 9 elements. In it, find a primitive element (i.e., a generator for the
multiplicative group).
Problem # 6
Consider the elliptic curve E given by y 2 = x3 − 10x + 21 over F557 . Suppose you know that
P = (2, 3, 1) has order 189, so the number of points N557 is a multiple of 189. Use the Hasse
bound
√
|p + 1 − Np | ≤ 2 p
to determine N557 .
Problem # 7
(a) Find integers s and t such that s · 255 + t · 204 = gcd(255, 204).
(b) Find 177−1 mod 107.
Problem # 8
The following ciphertext was encrypted by an affine cipher mod 26:
fzmxnah.
The plaintext starts g and ends in y. Decrypt the message.
Problem # 9
Solve the following system of congruences:
x ≡ 7 mod 8
x ≡ 3 mod 9
x ≡ −2 mod 25
Problem # 10
Compute
a) Φ(108000)
b) Φ(75260)
Problem # 11
Compute the last two digits of 587331
Problem # 12
Use Miller-Rabin with base 2 to find a factor of 91 001.
Problem # 13
Suppose you know that 3 is a primitive root modulo 31. If 14 = 3x mod 31, what is x? Use
the Pohlig Hellman algorithm.
Problem # 14
Find the inverse using the power method (i.e., not the Euclidean algorithm):
a) The inverse of 78 mod 107.
b) The inverse of 701 mod 5500.
Problem # 15
Factor the number n = 28892177 using the p − 1 method with base 2 and bound B = 13.