How do prime numbers keep your data secure?

1 What is encryption?

Encryption is the process of transforming data into a secret code that only the intended recipient can read. For example, when you send an email or make a payment online, your data is encrypted before it leaves your device, and then decrypted when it reaches the destination. Encryption algorithms use mathematical formulas and keys to perform this transformation. A key is a piece of information that determines how the data is encrypted and decrypted. There are two types of encryption: symmetric and asymmetric.

2 Symmetric encryption

Symmetric encryption uses the same key for both encryption and decryption. This means that both parties need to have the same key and keep it secret from anyone else. For example, if Alice and Bob want to communicate securely, they need to agree on a key and use it to encrypt and decrypt their messages. Symmetric encryption is fast and efficient, but it has a drawback: how can Alice and Bob exchange the key without anyone else intercepting it?

3 Asymmetric encryption

Asymmetric encryption solves this problem by using two different keys: a public key and a private key. A public key is a key that anyone can see and use to encrypt data, but not to decrypt it. A private key is a key that only the owner can see and use to decrypt data, but not to encrypt it. For example, if Alice wants to send a message to Bob, she can use Bob's public key to encrypt it, and then Bob can use his private key to decrypt it. Asymmetric encryption is more secure and flexible than symmetric encryption, but it is also slower and more complex.

4 How prime numbers are used in asymmetric encryption

RSA is an asymmetric encryption algorithm developed by Ron Rivest, Adi Shamir, and Leonard Adleman. It uses prime numbers to generate the public and private keys. To do this, Alice and Bob each choose two large prime numbers, p and q, and multiply them to get n. For example, Alice chooses p = 17 and q = 23, resulting in n = 391. Alice and Bob then calculate phi, which is the product of p-1 and q-1. In this example, Alice calculates phi = (17-1) x (23-1) = 352. Next, they each choose a number called e, smaller than phi and coprime with it. For example, Alice chooses e = 5. Lastly, they calculate d which is the inverse of e modulo phi. This means that d x e divided by phi has a remainder of 1. In this example, Alice calculates d = 141. Finally, they use n and e as their public key and n and d as their private key. For example, Alice's public key is (391, 5) and her private key is (391, 141).

5 How encryption and decryption work with RSA

Alice and Bob can use their keys to encrypt and decrypt messages. To encrypt a message, they use the recipient's public key and raise the message to the power of e modulo n. To decrypt a message, they use their own private key and raise the message to the power of d modulo n. For example, if Alice wants to send the message 100 to Bob, she uses Bob's public key (391, 5) and encrypts the message as 100^5 mod 391 = 234. When Bob receives 234 from Alice, he uses his own private key (391, 141) to decrypt it as 234^141 mod 391 = 100.

6 Why prime numbers are hard to crack

You may wonder why RSA is secure and how hard it is to crack. The answer lies in the difficulty of finding the prime factors of a large number. If an attacker knows n, e, and the encrypted message, they can try to find p and q by dividing n by every possible number until they find the prime factors. However, this is very time-consuming and impractical, especially if n is very large and has hundreds of digits. For example, finding the prime factors of 391 is easy, but finding the prime factors of a 2048-bit number, which is commonly used in RSA, is almost impossible with current technology. This is why prime numbers are so important for keeping your data secure: they make encryption easy and decryption hard, unless you have the right key.