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Howard Sueing
Damina Townes
October 28, 2009
Fundamentals of Algorithms
HW 5A
1. (a) Breaking the Code I: A message is intercepted believed to be from “evildoers” allegedly containing
the name of an American city that they wish to target. The message is believed to be encoded in Caesar
type shift code with an unknown shift. Your task as a cryptanalyst is to quickly decode the name of the
city so that appropriate precautions may be taken.
The ciphertext received is “FYQFSYF”
What is the corresponding plaintext?
ABCDEFGHIJKLMNOPQRSTUVWXYZ
The deciphered message reads: ATLANTA, therefore the unknown shift it equal to 21.
(b) Breaking the Code II: The evildoers from part (a) realize that their code can be broken and switch to
public key cryptography. They announce their public key as n=221, e=17 and dare anyone to crack
their messages. Believing the cipher to be secure other evildoers send them the following message:
1 37 37 1 139 215 140 139
Taking into account that the terrorists are most likely using the standard encoding (A=1, B=2, C=3,
etc.) crack this code to decipher the message.
Public Key (n, e) = Public Key (221, 17)
e=17
n=221
A
T
T
A
C
K
D
C
mod 221 = 1
mod 221 = 20
mod 221 = 20
mod 221 = 1
mod 221 = 3
mod 221 = 11
mod 221 = 4
mod 221 = 3
2. (a) Modulus: Find the value of 123456789123456789 mod 1111111111
123456789123456789 mod 1111111111
Applying modular Exponentiation, ab mod n = [(a mod n) (b mod n)] mod n
= [(12345678961728394.5) mod (1111111111)] * [(12345678961728394.5) mod (1111111111)] mod
(1111111111)
=(652822468)*(652822468) mod (1111111111) = 914261588
The blow value returning function was used for the calculation:
public static long modexp2(long x, long y, long z)
{
long result = 1;
while (y > 0)
{
if((y %2) == 1)
{
result = (result * x) % z;
y--;
}
x = (x * x) % z;
y = y / 2;
}
return result;
}
(b) Fermat: Use the Fermat primality test to determine if the following numbers are prime. If you
determine they are not prime please give the prime factorization.
Fermat's little theorem states that if p is prime and 1 ≤ a ≤ n, then
(a) 1,000,000,007
if n = 1,000,000,007
1 ≤ a ≤ 1,000,000,007
a=2
then using the online calculator provided,
Therefore this number is prime.
(b) 1,000,000,011
if n = 1,000,000,011
1 ≤ a ≤ 1,000,000,011
a=2
then using the online calculator provided,
Therefore this number is not prime.
The prime factorizations are: 3, 29, 11494253.
mod n = 1
mod 1000000007 = 1
mod 1000000011 ≠ 1
(c) Suppose we do a Fermat test on a number n and we test 10 bases and n passes the test for each base.
What is the probability that we are wrong and n is actually not prime?
The Little Fermat Theorem applies for half of the numbers from 0 – (n-1). Therefore the probability of
being wrong is 0.5.Finlly,
k =10
(1/2)*k = (1/2)*10 = (1/1024)