7.4: Exercises
Encode
IXLOVEXMATH
using the cryptosystem in Example \(7.1\).
Decode
ZLOOA WKLVA EHARQ WKHA ILQDO
, which was encoded using the cryptosystem in Example \(7.1\).
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
What is the significance of this message in the history of cryptography?
What is the total number of possible monoalphabetic cryptosystems? How secure are such cryptosystems?
Prove that a \(2 \times 2\) matrix \(A\) with entries in \({\mathbb Z}_{26}\) is invertible if and only if \(\gcd( \det(A), 26 ) = 1\text{.}\)
Given the matrix
\[ A = \begin{pmatrix} 3 & 4 \\ 2 & 3 \end{pmatrix}\text{,} \nonumber \]
use the encryption function \(f({\mathbf p}) = A {\mathbf p} + {\mathbf b}\) to encode the message
CRYPTOLOGY
, where \({\mathbf b} = ( 2, 5)^\transpose\text{.}\) What is the decoding function?
Encrypt each of the following RSA messages \(x\) so that \(x\) is divided into blocks of integers of length \(2\text{;}\) that is, if \(x = 142528\text{,}\) encode \(14\text{,}\) \(25\text{,}\) and \(28\) separately.
- \(\displaystyle n = 3551, E = 629, x = 31\)
- \(\displaystyle n = 2257, E = 47, x = 23\)
- \(\displaystyle n = 120979, E = 13251, x = 142371\)
- \(\displaystyle n = 45629, E = 781, x = 231561\)
Compute the decoding key \(D\) for each of the encoding keys in Exercise \(7.4.7\).
Decrypt each of the following RSA messages \(y\text{.}\)
- \(\displaystyle n = 3551, D = 1997, y = 2791\)
- \(\displaystyle n = 5893, D = 81, y = 34\)
- \(\displaystyle n = 120979, D = 27331, y = 112135\)
- \(\displaystyle n = 79403, D = 671, y = 129381\)
For each of the following encryption keys \((n, E)\) in the RSA cryptosystem, compute \(D\text{.}\)
- \(\displaystyle (n, E) = (451, 231)\)
- \(\displaystyle (n, E) = (3053, 1921)\)
- \(\displaystyle (n, E) = (37986733, 12371)\)
- \(\displaystyle (n, E) = (16394854313, 34578451)\)
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 \(n\) letters?
Find integers \(n\text{,}\) \(E\text{,}\) and \(X\) such that
\[ X^E \equiv X \pmod{n}\text{.} \nonumber \]
Is this a potential problem in the RSA cryptosystem?
Every person in the class should construct an RSA cryptosystem using primes that are \(10\) to \(15\) digits long. Hand in \((n, E)\) and an encoded message. Keep \(D\) secret. See if you can break one another's codes.