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1. (10 points) What is the number of possible keys of the following cryptosystems:
A. Shift cipher over 26 letter alphabet 26
B. Aﬃne cipher over 26 letter alphabet 26
·
12
C. Substitution cipher over 26 letter alphabet 26!
D. Permutation cipher with block length m
m
!
E. Vigenere cipher with block length m 26
m
2. (10 points) Prove that if
a
≡
b
(mod
m
) then
an
≡
bn
(mod
mn
).
a
≡
b
(mod
m
) means that there is some integer
q
such that
a
=
qm
+
b
. Thus,
an
=
q
(
mn
) +
bn
, which means that
an
≡
bn
(mod
mn
).
3. A. (5 points)
Suppose Alice and Bob uses the shift cipher, to send messages that consists of 100 letters.
Each time Alice wants to send a message, they choose a key uniformly at random from
Z
26
=
{
0
,
1
,...,
25
}
, (they agree on the key over a secure channel) and Alice encrypts
the entire message (100 letters long) using the same key.
What is the set of possible plaintexts, the set of possible ciphertexts and the set of possible
keys? What is the cardinality of each of these sets? Does this achieve perfect secrecy? 
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 Fall '10
 GAL

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