Sample Cryptography Questions and Answers
1) (10 pts)
Consider a substitution cipher where 52 symbols were used instead of 26. In
particular, each symbol in the cipher text is for either a lowercase English letter, or an
uppercase English letter. (For example, let E be the encryption function then we could
have E(‘S’) = ‘p’ and E(‘s’)=’m’.) Such a modification augments the key space to 52!
Does this provide added security compared to a standard substitution cipher? Why or
why not?
This does not add much security to the system at all. Capital letters usually appear
only at the beginning of words at the beginning of sentences. Thus, the frequencies
of capital letters are quite small in English text. You could simply consider this
while using frequency analysis. Simply put, disregard all the characters of very
small frequencies and concentrate on solving for the characters with the highest
frequencies which will still be the same lowercase letters. Once these are solved for,
there will be enough recovered plaintext to deduce most if not all of the capital
letters in the message.
2)
(10 pts)
You have found one small piece of matching plaintext and ciphertext for a
Hill cipher using a 2x2 matrix key with mod 17 entries. In particular, the plaintext (12, 5)
maps to the ciphertext (14, 10). (Note that these entries typically appear as column
vectors when the encryption is applied.) Using this given information and nothing else,
how many possible keys could there be? List two of these possible keys.
Let the encryption matrix be
d
c
b
a
.
Then we have the following equations:
12a + 5b = 14 mod 17
12c + 5d = 10 mod 17
Solve these equations for a and c respectively:
12a = (14 – 5b) mod 17
a = (12
1
mod 17)(14 – 5b) mod 17
12c = (10 – 5d) mod 17
c = (12
1
mod 17)(10 – 5d) mod 17
Now, notice that plugging in each of the 17 possible values of b yields a solution for a
and plugging in each of the 17 possible values of d yields a solution for c. Thus, the
total possible number of keys seems to be 17*17 = 289. But, some of these keys give a
determinant equivalent to 0 mod 17, which is not permissible. To see this, using the
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 Summer '09
 Cryptography, Prime number

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