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Chapter 4: Problem 14

Encrypt the message GRIZZLY BEARS using blocks of five letters and the transposition cipher based on the permutation of \(\\{1,2,3,4,5\\}\) with \(\sigma(1)=3, \sigma(2)=5\) , \(\sigma(3)=1, \sigma(4)=2,\) and \(\sigma(5)=4 .\) For this exercise, use the letter \(X\) as many times as necessary to fill out the final block of fewer then five letters.

Short Answer

Expert verified
IZZGRBEALYXXRXS

Step by step solution

01

Create Blocks of Five Letters

Write the message 'GRIZZLY BEARS' in blocks of five letters. If the final block has fewer than five letters, fill it with the letter 'X'.Original message: GRIZZLY BEARSBlocks: GRIZZ LYBEA RSXXX
02

Write the Permutation

Write down the permutation of \(\{1,2,3,4,5\}\) given by \(\sigma(1) = 3, \sigma(2) = 5, \sigma(3) = 1, \sigma(4) = 2, \sigma(5) = 4\). This indicates the new positions of the letters.
03

Encrypt the First Block

Using the permutation, rearrange the letters in the first block 'GRIZZ' as follows:1 -> 3: G moves to the 3rd position2 -> 5: R moves to the 5th position3 -> 1: I moves to the 1st position4 -> 2: Z moves to the 2nd position5 -> 4: Z moves to the 4th positionSo, 'GRIZZ' becomes 'IZZG R'.
04

Encrypt the Second Block

Using the same permutation, rearrange the letters in the second block 'LYBEA' as follows:1 -> 3: L moves to the 3rd position2 -> 5: Y moves to the 5th position3 -> 1: B moves to the 1st position4 -> 2: E moves to the 2nd position5 -> 4: A moves to the 4th positionSo, 'LYBEA' becomes 'BEALY'.
05

Encrypt the Third Block

Using the same permutation, rearrange the letters in the third block 'RSXXX' as follows:1 -> 3: R moves to the 3rd position2 -> 5: S moves to the 5th position3 -> 1: X moves to the 1st position4 -> 2: X moves to the 2nd position5 -> 4: X moves to the 4th positionSo, 'RSXXX' becomes 'XXRXS'.
06

Combine the Encrypted Blocks

Combine the encrypted blocks to get the final encrypted message:Encrypted message: IZZGRBEALYXXRXS

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

encryption
Encryption is the process of converting plain text into a coded format to prevent unauthorized access. This is done using various algorithms and techniques. In our example, we use a transposition cipher for encryption. The main goal is to rearrange the characters in the message according to a defined system to produce an unreadable format without the correct decryption key. By encrypting messages, we can ensure that sensitive information remains confidential and protected.
permutation
Permutation is a core concept in many cryptographic techniques, including the transposition cipher we use in this exercise. In simple terms, permutation involves rearranging elements in a particular order. Here, we have a permutation of the set {1,2,3,4,5} given as: \( \sigma(1)=3, \sigma(2)=5, \sigma(3)=1, \sigma(4)=2, \sigma(5)=4 \). This means each number maps to a new position. In our example, the first element moves to the third position, the second element moves to the fifth position, and so forth. This rearrangement is applied to each block of five letters in the message, thereby changing their original order.
cryptography
Cryptography is the study and practice of methods for secure communication in the presence of third parties. It encompasses principles like encryption, decryption, and authentication to ensure data security. Cryptographic techniques like the transposition cipher rely on mathematical functions and algorithms to achieve their goals. In our exercise, we use the permutation to encrypt our message. Cryptographic systems are crucial in protecting digital information in various applications, from online banking to secure messaging.

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Most popular questions from this chapter

In Exercises \(31-32\) suppose that Alice and Bob have these public keys and corresponding private keys: \(\left(n_{\text { Alice }}, e_{\text { Alice }}\right)=\) \((2867,7)=(61 \cdot 47,7), \quad d_{\text { Alice }}=1183\) and \(\left(n_{\text { Bob }}, e_{\text { Bob }}\right)=\) \((3127,21)=(59 \cdot 53,21), d_{\text { Bob }}=1149 .\) First express your answers without carrying out the calculations. Then, using a computational aid, if available, perform the calculation to get the numerical answers. Alice wants to send to Bob the message "BUY NOW" so that he knows that she sent it and so that only Bob can read it. What should she send to Bob, assuming she signs the message and then encrypts it using Bob's public key?

How many zeros are there at the end of \(100 ! ?\)

Write an algorithm in pseudocode for generating a sequence of pseudorandom numbers using a linear congruential generator. The middle-square method for generating pseudorandom numbers begins with an \(n\) -digit integer. This number is squared, initial zeros are appended to ensure that the result has 2\(n\) digits, and its middle \(n\) digits are used to form the next number in the sequence. This process is repeated to generate additional terms.

Prove or disprove that \(p_{1} p_{2} \cdots p_{n}+1\) is prime for every positive integer \(n,\) where \(p_{1}, p_{2}, \ldots, p_{n}\) are the \(n\) smallest prime numbers.

What are the greatest common divisors of these pairs of integers? $$ \begin{array}{l}{\text { a) } 2^{2} \cdot 3^{3} \cdot 5^{5}, 2^{5} \cdot 3^{3} \cdot 5^{2}} \\ {\text { b) } 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13,2^{11} \cdot 3^{9} \cdot 11 \cdot 17^{14}} \\ {\text { c) } 17,17^{17} \quad \text { d) } 2^{2} \cdot 7,5^{3} \cdot 13} \\ {\text { e) } 0,5 \quad \text { f) } 2 \cdot 3 \cdot 5 \cdot 7,2 \cdot 3 \cdot 5 \cdot 7}\end{array} $$
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