CMPT 404 — Cryptograph and Protocols
Outline Solutions to Exercises on Exercises on Probability and Perfect Secu
rity.
1.
The text was encrypted using a substitution cipher. Find the plaintext and explain the steps you took to
find it. Punctuation marks are removed from the text.
This is a quote from “The Hobbit”:
“Dead silence fell in the middle of a word. Out went all lights. The fire leaped up in black smokes. Ashes and
cinders were in the eyes of the dwarves, and the wood was filled again with their clamour and their cries. Bilbo
found himself running round and round (as he thought) and calling and calling: ‘Dori, Nori, Ori, Oin, Gloin,
Fili, Kili, Bombur, Bofur, Dwalin, Balin, Thorin Oakenshield’, while people he could not see or feel were doing
the same all round him (with an occasional ‘Bilbo!’ thrown in). But the cries of the others got steadily further
and fainter, and though after a while it seemed to him they changed to yells and cries for help in the distance, all
noise at last died right away, and he was left alone in complete silence and darkness.”
2.
One method to improve the security of substitution ciphers is to compress the plaintext (say, using gzip)
before applying the encryption algorithm. Explain why this simple procedure improves security. You may
think of two explanations, linguistic and mathematical.
In our attack on a substitution cipher we used two methods: (1) frequencies analysis, and (2) short/frequent
words and frequent combinations of letters.
Compressing a text completely removes all the hints on short
words and changes combinations of letters so that they are unrecognizable. This is the ‘linguistic’ reason why
compressing a text improves the security of a substitution cipher. The mathematical reason is the following.
The reason we compress a text is to make every symbol to carry more information, so that the same amount
of information can be transmitted using fewer symbols. In a text every symbol carries the maximum amount
of information if each symbol occurs with the same probability. Thus compressing a text makes its symbol to
appear more uniformly destroying the basis of frequencies analysis.
Compressing a text usually creates symbols that do not belong to the alphabet. Therefore to use a substitution
cipher we have to extend it onto all possible 256 bytes.
3.
Prove that the statistical distance between random variables
X
and
Y
satisfy the following properties:
(a)
Δ(
X, Y
) =
1
2
X
v

Pr[
X
=
v
]

Pr[
Y
=
v
]

;
(b)
(triangle inequality)
for any random variables
Z
,
Δ(
X, Y
)
≤
Δ(
X, Z
) + Δ(
Z, Y
)
.
(a) By definition
Δ(
X, Y
) = max
T

Pr[
X
∈
T
]

Pr[
Y
∈
T
]

, where maximum is taken over all subsets of the
set
V
of values of
X
and
Y
. Observing that
Pr[
X
∈
T
] =
∑
v
∈
T
Pr[
X
=
v
]
and
Pr[
Y
∈
T
] =
∑
v
∈
T
Pr[
Y
=
v
]
we obtain
Pr[
X
∈
T
]

Pr[
Y
∈
T
] =
X
v
∈
T
(Pr[
X
=
v
]

Pr[
Y
=
v
])
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '12
 AndreiA.Bulatov
 Cryptography, Encryption, Alice, ek, plaintexts

Click to edit the document details