When you send private information online—like your bank details, passwords, or credit card number—it’s not protected by secrecy. It’s protected by math.

That’s right! Cryptography is the science of sending secret messages. And one of the most powerful methods we use today, called RSA encryption, is built on the most fascinating kinds of math: prime numbers.

Behind nearly every secure internet connection is a massive number—so big it can stretch over 600 digits. This number is public. Your browser downloads it. But it guards your secrets because of what lies beneath: two enormous prime numbers that no one else knows.

What's a Prime Number Again?

A prime number is a number that can only be divided by 1 and itself.
Examples: 2, 3, 5, 7, 11, 13...

They're like the atoms of numbers—small, simple, but super important when building something powerful.

RSA: The Two-Way Key System

RSA encryption is what’s known as a public-key encryption system, but it actually uses two keys:

The public key is used to lock (encrypt) messages.
The private key is used to unlock (decrypt) them.

Here’s the catch: both keys are mathematically connected. The public key is based on the product of two large prime numbers—let’s call them p and q. This product, n = p×n becomes part of the public key. The prime numbers are kept secret.

But the private key, used to unlock the message, is derived from knowing the original prime factors, p and q. In other words:

Encryption is built from multiplying primes.
Decryption is only possible if you can factor the result.

That’s the genius of RSA—it’s a one-way function. Easy to compute forward, practically impossible to reverse.

Big Numbers, Tiny Clues

Prime numbers are special because they’re only divisible by 1 and themselves. Think 2, 3, 5, 7... Now imagine primes that are hundreds of digits long. RSA keys often use two such primes, and multiply them to get a number so massive, it's measured in bits, not digits (like 2048-bit keys).

Even though this number is public—used by banks, browsers, and apps—no one can figure out its two prime ingredients. Not without monumental computing power and time.

Why Is This So Hard to Crack?

Here’s what’s easy:

Pick two primes.
Multiply them.

Here’s what’s hard:

Get the product.
Try to figure out which two primes made it.

This process—factorization—is what would allow a hacker to generate the private key and break into an encrypted message. But it’s incredibly time-consuming. A 2048-bit number (about 617 digits) would take thousands of years to factor using today's most powerful computers.

A Simple Example, for Scale

Let’s say:

Your public key uses n=10
That means the primes were 2 and 5 (since 2 × 5 = 10)

If someone wants to crack your message, they must figure out the 2 and 5. Easy in this case—but RSA uses primes that are so large, guessing them is practically impossible.

Fermat’s Little Theorem: The Math That Makes It Work

A 17th-century insight from Pierre de Fermat unexpectedly became a cornerstone of modern encryption. It states:

If you take a number x and a prime p, then:
x^p−x is always divisible by p

Big primes make RSA hard to crack, but it’s Fermat’s Little Theorem that makes it work. It helps the math behind encryption stay predictable, even when the original primes are secret. This is a critical property RSA relies on for securely locking and unlocking messages.

Why RSA Keeps Getting Bigger

RSA’s strength depends on the size of the number and the difficulty of factoring it. In the past:

512-bit keys were breakable
768-bit keys were cracked (after 2 years of effort)
1024-bit keys are now risky

That’s why banks and browsers have moved to 2048-bit encryption. It’s a race between math and machines.

The Quantum Threat

Today’s computers can’t break RSA. But quantum computers might. Algorithms like Shor’s Algorithm could one day factor these enormous products quickly—making the RSA two-key system obsolete. That’s why cryptographers are now racing to build quantum-safe encryption—codes that don’t rely on the difficulty of factorization.

Fun Fact

The biggest prime number ever found has over 41 million digits. That’s longer than the entire Harry Potter series typed out... several times. You’d need over 13,000 pages just to print it..