Skip to main content

Exercises 12.3 Exercises

1.

Encode IXLOVEXMATH using the cryptosystem in Example 12.1.

2.

Decode ZLOOA WKLVA EHARQ WKHA ILQDO, which was encoded using the cryptosystem in Example 12.1.

3.

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?

4.

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

5.

Prove that a \(2 \times 2\) matrix \(A\) with entries in \({\mathbb Z}_{26}\) is invertible if and only if \(\det(A)\) is a unit in \({\mathbb Z}_{26}\text{.}\)

6.

Given the matrix
\begin{equation*} A = \begin{pmatrix} 3 & 4 \\ 2 & 3 \end{pmatrix}, \end{equation*}
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?

7.

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.
  1. \(\displaystyle n = 3551, E = 629, x = 31\)
  2. \(\displaystyle n = 2257, E = 47, x = 23\)
  3. \(\displaystyle n = 120979, E = 13251, x = 142371\)
  4. \(\displaystyle n = 45629, E = 781, x = 231561\)

8.

Compute the decoding key \(D\) for each of the encoding keys in Exercise 12.3.7.

9.

Decrypt each of the following RSA messages \(y\text{.}\)
  1. \(\displaystyle n = 3551, D = 1997, y = 2791\)
  2. \(\displaystyle n = 5893, D = 81, y = 34\)
  3. \(\displaystyle n = 120979, D = 27331, y = 112135\)
  4. \(\displaystyle n = 79403, D = 671, y = 129381\)

10.

For each of the following encryption keys \((n, E)\) in the RSA cryptosystem, compute \(D\text{.}\)
  1. \(\displaystyle (n, E) = (451, 231)\)
  2. \(\displaystyle (n, E) = (3053, 1921)\)
  3. \(\displaystyle (n, E) = (37986733, 12371)\)
  4. \(\displaystyle (n, E) = (16394854313, 34578451)\)

11.

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?

12.

Find integers \(n\text{,}\) \(E\text{,}\) and \(X\) such that
\begin{equation*} X^E \equiv X \pmod{n}. \end{equation*}
Is this a potential problem in the RSA cryptosystem?

13.

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.

14.

If \(E\) is an elliptic curve over any field \(K\text{,}\) let the 2-torsion of \(E\) be the subgroup \(H = \{P \in E : 2P = O\}\text{.}\) Prove that \(H\) is isomorphic to the trivial group, \(\mathbb Z_2\text{,}\) or \(\mathbb Z_2 \times \mathbb Z_2\text{.}\) Give examples of each of these possibilities for \(K = \mathbb Q\text{.}\)
PrevTopNext