AATA Exercises

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Exercises 7.4 Exercises

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1.

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Encode IXLOVEXMATH using the cryptosystem in Example 7.1.

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2.

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Decode ZLOOA WKLVA EHARQ WKHA ILQDO, which was encoded using the cryptosystem in Example 7.1.

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3.

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Assuming that monoalphabetic code was used to encode the following secret message, what was the original message?

APHUO EGEHP PEXOV FKEUH CKVUE CHKVE APHUO
EGEHU EXOVL EXDKT VGEFT EHFKE UHCKF TZEXO
VEZDT TVKUE XOVKV ENOHK ZFTEH TEHKQ LEROF
PVEHP PEXOV ERYKP GERYT GVKEG XDRTE RGAGA

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What is the significance of this message in the history of cryptography?

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4.

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What is the total number of possible monoalphabetic cryptosystems? How secure are such cryptosystems?

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5.

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Prove that a

2 \times 2

matrix

A

with entries in

Z_{26}

is invertible if and only if

gcd (det (A), 26) = 1 .

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6.

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Given the matrix

A = (\begin{matrix} 3 & 4 \\ 2 & 3 \end{matrix}),

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use the encryption function

f (p) = A p + b

to encode the message CRYPTOLOGY, where

b = (2, 5)^{t} .

What is the decoding function?

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7.

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Encrypt each of the following RSA messages

x

so that

x

is divided into blocks of integers of length

2;

that is, if

x = 142528,

encode

14,

25,

and

28

separately.

$n = 3551, E = 629, x = 31$
$n = 2257, E = 47, x = 23$
$n = 120979, E = 13251, x = 142371$
$n = 45629, E = 781, x = 231561$

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8.

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Compute the decoding key

D

for each of the encoding keys in Exercise 7.4.7.

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9.

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Decrypt each of the following RSA messages

y .

$n = 3551, D = 1997, y = 2791$
$n = 5893, D = 81, y = 34$
$n = 120979, D = 27331, y = 112135$
$n = 79403, D = 671, y = 129381$

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10.

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For each of the following encryption keys

(n, E)

in the RSA cryptosystem, compute

D .

$(n, E) = (451, 231)$
$(n, E) = (3053, 1921)$
$(n, E) = (37986733, 12371)$
$(n, E) = (16394854313, 34578451)$

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11.

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Encrypted messages are often divided into blocks of

n

letters. A message such as THE WORLD WONDERS WHY might be encrypted as JIW OCFRJ LPOEVYQ IOC but sent as JIW OCF RJL POE VYQ IOC. What are the advantages of using blocks of