Question

(Cryptography) A company wants to transmit data over the telephone, but is concerned that its phones could be tapped. All of the data are transmitted as four-digit integers. The company has asked you to write a program that encrypts the data so that it can be transmitted more securely. Your program should read a four-digit integer and encrypt it as follows: Replace each digit by (the sum of that digit plus 7 ) modulus \(10 .\) Then, swap the first digit with the third, swap the second digit with the fourth and print the encrypted integer. Write a separate program that inputs an encrypted fourdigit integer and decrypts it to form the original number.

Short answer

Encrypt by modifying digits and swapping positions. Decrypt by reversing swaps and retrieving original digits.

Step-by-step solution

Read the Four-Digit Integer

Start by reading a four-digit integer input from the user. This number represents the data that needs to be encrypted.

Encrypt Each Digit

For each digit of the four-digit integer, apply the following transformation: - Take the digit, add 7 to it, and then compute the result modulo 10.- For example, if the digit is 3, compute \( (3 + 7) \mod 10 = 0 \) to get the new digit.

Swap Digits

After transforming all four digits, swap the first digit with the third and the second digit with the fourth. This reorders the transformed digits to enhance encryption.

Output the Encrypted Number

Combine the swapped digits into a new four-digit integer. Output this integer as the encrypted data.

Read the Encrypted Integer

To decrypt, begin by reading the encrypted four-digit integer input from the user.

Reverse the Digit Swap

Swap the first digit with the third and the second digit with the fourth to restore their original positions before digit transformation.

Decrypt Each Digit

Reverse the encryption process by decrypting each digit using the formula: - If the digit is less than 7, use \( (digit + 10 - 7) \mod 10 \) to find the original digit.- If the digit is 7 or greater, simply subtract 7.

Output the Original Number

Combine these restored digits back into the original integer and print the decrypted number.

Key concepts

Data encryption

Data encryption is a fundamental principle used to safeguard information by transforming it into a format that cannot be easily understood by unauthorized users. In our context, the encryption process involves modifying each digit of a four-digit number to protect it from being intercepted and misused.

To achieve this, each digit undergoes a simple yet effective transformation:

• First, add 7 to the digit.
• Then, apply the modulo operation with 10 to the result (i.e., find the remainder when divided by 10).

This ensures that the digit is replaced with a new, seemingly random digit, maintaining the security of the data during transmission. For instance, if the original digit is 6:
1. Add 7: 6 + 7 = 13 2. Apply modulo 10: 13 % 10 = 3
Thus, 6 becomes 3 in the encrypted form, securing the data effectively.

Number swapping

Number swapping is a crucial step in enhancing the encrypted output by rearranging the digits. This step adds an additional layer of security.

During the swapping process, the positions of certain digits are exchanged:

• The first digit is swapped with the third digit.
• The second digit is swapped with the fourth digit.

This scrambling of digits disrupts any predictable pattern that may have existed, making it more challenging for an intruder to deduce the original number. For example, let's say our transformed digits are 1, 2, 3, and 4. After swapping, they become:
- Original: 1st, 2nd, 3rd, 4th - Swapped: 3rd, 4th, 1st, 2nd
Hence, the number 1234 would be transmitted as 3412, adding a robust layer to the encryption.

Modulo operation

The modulo operation is an essential mathematical tool used in our encryption and decryption process. It's a simple yet powerful operation that calculates the remainder of a division.

When encrypting each digit:

• You add 7 to the digit.
• Apply the modulo operation with 10, denoted as \( (digit + 7) \mod 10 \).

This operation ensures that each transformed digit stays within the range of 0 through 9, as required for a digit-based encryption system. The modulus operation keeps numbers within a specific range, allowing predictable and reversible results.
Let's explore with a number, say digit = 5:
- Computation: \((5 + 7) \mod 10 = 12 \mod 10 = 2\)
As a result, 5 transforms into 2, providing a reliable method for digit transformation, crucial for both encryption and subsequent decryption.

Decryption process

The decryption process is the reverse of encryption aimed at retrieving the original data from its encrypted form. This step is vital to ensure that the information can be accurately recovered upon reception.

Begin by reading the encrypted digit and reverse the number swapping first:

• Swap back the first digit with the third, and the second with the fourth.

Once the original positions are restored, each digit is then decrypted:

• If a digit is less than 7, the original digit is found using the formula \((digit + 10 - 7) \mod 10\).
• If it is 7 or greater, simply calculate \(digit - 7\).

For instance, if our digit after reordering is 3, since it is less than 7, use:
- Computation: \((3 + 10 - 7) \mod 10 = 6\)
Reversing these steps reliably retrieves the original four-digit number, ensuring data integrity despite the complex transformations during encryption.