Question

The explosive growth of Internet communications and data storage on Internet- connected computers has greatly increased privacy concerns. The field of cryptography is concerned with coding data to make it difficult (and hopefully-with the most advanced schemes - impossible) for unauthorized users to read. In this exercise you'll investigate a simple scheme for encrypting and decrypting data. A company that wants to send data over the Internet has asked you to write a program that will encrypt it so that it may be transmitted more securely. All the data is transmitted as four-digit integers. Your application should read a four-digit integer entered by the user and encrypt it as follows: Replace each digit with the result of adding 7 to the digit and getting the remainder after dividing the new value by \(10 .\) Then swap the first digit with the third, and swap the second digit with the fourth. Then print the encrypted integer. Write a separate application that inputs an encrypted four-digit integer and decrypts it (by reversing the encryption scheme) to form the original number. [Optional reading project: Research "public key cryptography" in general and the PGP (Pretty Good Privacy) specific public key scheme. You may also want to investigate the RSA scheme, which is widely used in industrial-strength applications.]

Short answer

To encrypt, add 7 to each digit, take modulus 10, and swap the first with the third and second with the fourth digits. To decrypt, reverse the swap and subtract 7 from each digit (adding 10 before modding by 10 if necessary).

Step-by-step solution

Encrypt the First and Second Digits

Read the four-digit integer from the user. Let's denote the digits as A, B, C, and D where A is the thousands place, B is the hundreds place, C is the tens place, and D is the ones place. For encryption, add 7 to each digit and then take the remainder when divided by 10: Encrypted A (EA) = (A + 7) % 10, Encrypted B (EB) = (B + 7) % 10.

Encrypt the Third and Fourth Digits

Similarly, add 7 to the third and fourth digits and then calculate the remainder when divided by 10: Encrypted C (EC) = (C + 7) % 10, Encrypted D (ED) = (D + 7) % 10.

Swap the First and Third, and Second and Fourth Digits

Swap the encrypted digits: the first digit with the third, and the second digit with the fourth to get the final encrypted number: Encrypted Number = ECEAEBED.

Decrypt the First and Second Digits

To decrypt an encrypted four-digit integer, swap the first and third digits and the second and fourth digits to get them back to their original positions.

Decrypt the Third and Fourth Digits

After swapping, subtract 7 from each digit. Since we might get negative numbers, add 10 and then take the remainder when divided by 10 to get the original digit: Original A = (EA + 10 - 7) % 10, Original B = (EB + 10 - 7) % 10.

Produce the Final Decrypted Number

Similarly, compute the original third and fourth digits and reconstruct the original four-digit integer: Original Number = AB CD.

Key concepts

Cryptography

Cryptography is the science of securing communication and information through the use of codes so that only those for whom the information is intended can read and process it. It has been used historically to protect military secrets, financial data and personal information. With the advent of the digital age, its importance has greatly increased, as it now secures data transfers across the expansive networks of the internet.

Cryptographic techniques involve creating written or generated codes that allow information to be kept secret. It can be as simple as the Caesar cipher, which shifts the letters in a message, to complex algorithms that scramble data into seemingly unintelligible forms. This field is vital for maintaining privacy, protecting financial transactions, and safeguarding sensitive information from unauthorized access.

Public Key Cryptography

Public key cryptography, also known as asymmetric cryptography, is a method that uses a pair of keys for encryption and decryption—a public key and a private key. The public key can be shared with anyone, while the private key must be kept secret. A message encrypted with a recipient's public key can only be decrypted with their private key, ensuring secure communication.

Public key cryptography allows for not only confidential communication but also authentication, digital signatures, and non-repudiation. It's a cornerstone of many internet security protocols, such as HTTPS, ensuring that sensitive information—like credit card numbers or passwords—passed over the web remains private.

Integer Encryption Algorithms

Integer encryption algorithms are specific cryptographic techniques designed to secure numeric data. As we saw in the given exercise, these algorithms often work by performing arithmetic operations on the individual digits of numbers. For instance, adding a constant value to each digit, and then applying a modulus operation to ensure the result is still a digit (0-9).

These algorithms can also incorporate more sophisticated mathematical structures like modular arithmetic, prime number theory, and group theory. While simple integer encryption algorithms are useful for educational purposes, they do not offer the stability and security needed for real-world applications where large amounts of sensitive data are handled.

PGP (Pretty Good Privacy)

PGP (Pretty Good Privacy) is a data encryption and decryption program that provides cryptographic privacy and authentication for data communication. Phil Zimmermann developed it in 1991, and it has become synonymous with email security. PGP uses a mix of public-key and symmetric key cryptography, where messages are encrypted with a symmetric key, and this key is then encrypted with the receiver's public key.

PGP allows for secure message transmission over unsecured channels and has been an essential tool for whistleblowers, activists, and anyone else needing to protect the integrity of their communications. Despite its name, PGP is considered highly secure and is an industry standard for encrypted communications.

RSA Encryption Scheme

The RSA encryption scheme is one of the first public-key cryptosystems and is widely used for secure data transmission. Developed by Ron Rivest, Adi Shamir, and Leonard Adleman in 1977, RSA stands for the first letters of their surnames. This cryptographic algorithm involves four stages: key generation, key distribution, encryption, and decryption.

It is based on the practical difficulty of factoring the product of two large prime numbers, an issue known as prime factorization. RSA's security comes through the key generation process, which creates the two keys that are mathematically linked but not easily derivable from one another. The encrypting is done with the recipient's public key, while decrypting requires the private key, ensuring that a message is kept confidential and authentic.