ICS 201: Cryptography and Communication Security
10/19/2007
Solutions to homework 1
Problem 2.2
If encryption is secure then by Definition 2.1 this condition implies that
Pr
[
M
=
m
] =
Pr
[
M
=
m
′
] for all
m, m
′
in the message space, which is obviously not true if
M
is sampled
from
any
distribution over the message space.
Problem 2.3
After throwing out the key
k
= 0
l
the OTP encryption is no longer perfectly secure because
then the keyspace has one fewer element than the message space. A good example is that
for any
c
∈ {
0
,
1
}
l
the message
m
=
c
is impossible, because
O
l
negationslash∈ K
.
But any other
message is possible, which violates for example the condition in lemma 2.3, which says that
encryption is perfectly secure only if for all
c, m
0
, m
1
the probability that
m
0
encrypts to
c
(over keys) is the same as the probability that
m
1
encrypts to
c
(over keys).
Problem 2.4(b)
Since the substitution cipher has keyspace
K
of size 26!, it can provide perfect secrecy only
if the message space
M
has at most 26! elements too. Let
M
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 Fall '07
 Jarecki
 Cryptography, Encryption, Onetime pad, message space, Cryptography and Communication Security

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