Prime Numbers and Technology: Securing Secret Messages in the Age of Encryption

25 December، 2023

Institutions and companies often need to send encrypted secret messages to ensure information security. Here, prime numbers play a crucial and effective role in the processes of encryption and decryption of messages.

Understanding Prime Numbers

Prime numbers are defined as positive integers greater than one, which cannot be divided by any number other than themselves and one. The set of prime numbers is an infinite set including 2, 3, 5, 7, 11, 13, 17, and so on.

History of Prime Numbers

The discovery of prime numbers dates back to ancient Greek times. However, they gained significant importance when the scientists Rivest, Shamir, and Adleman discovered the RSA protocol in the 1970s, used in securing electronic communications and the Internet.

RSA Encryption Algorithm

The RSA encryption system involves the analysis of prime numbers. Messages are encrypted using a public key composed of the product of two large prime numbers (p, q). A private key is then used for decryption, making this system a cornerstone in electronic security and data protection.

Operations in the RSA Algorithm

Choosing two large prime numbers p, q.
Calculating the value of n where n = p × q.
Calculating the value of φ (phi) where φ = (p-1) (q-1).
Assuming a value e (a prime number), with the condition e < φ.
Calculating the value of d from the equation e.d mod φ = 1.
The public key is represented by the pair e, n, and the private key by the pair d, n.
Encryption equation: c = (p^e) mod n.
Decryption equation: p = (c^d) mod n.

Importance of Prime Numbers in Technology

The importance of this algorithm is evident when using large prime numbers, making it difficult for computers to derive these numbers. Researchers highlight that advancements in this field contribute to the progress of digital information.

In conclusion, scientific research to date shows that the study of prime numbers remains vibrant, as research continues in discovering new prime numbers, contributing to the development of the digital information field.