Hash Functions: The Mathematics That Secures the Internet

Recall the old fairy tale: a royal messenger carries an important letter, but enemies intercept him on the road, get him drunk, and swap the letter for a forgery. The king receives false news, and disaster follows.

But how do you make a message impossible to tamper with — even if the messenger is captured? The answer to that question gave rise to something the entire modern internet, banking transactions, cryptocurrencies, and possibly the future of humanity depends on: hash functions.

What Are Hash Functions in Cryptography

A hash function is like a grain mill for data. You feed in any text, the mill grinds it up, and out comes a unique number — a hash. It is impossible to recover the original message from that number. The slightest change to even a single character in the source text produces a completely different hash. But run the original text through the same mill again, and you get the exact same hash every time. A clever algorithm indeed.

The Mathematics Inside a Hash Function

The data is split into small chunks — imagine cutting a book into individual pages. Each chunk then passes through a series of complex mathematical operations: multiplication, addition, bit shifts, substitutions. Think of it as chaotically shuffling a deck of cards, except instead of cards you have numbers and symbols. The result of all these transformations is a fixed-length string — say, 256 characters — regardless of how large or small the input was.

The Avalanche Effect: Core Magic of Cryptographic Hash Functions

If you change even the tiniest detail in the source data — add a space, swap one letter — the resulting hash changes beyond recognition. This is called the avalanche effect. Thanks to this property, hash functions are extraordinarily resistant to tampering attempts. And even if you can see the final hash, recovering the original data is impossible — it would be like trying to reconstruct a million-piece puzzle with no picture on the box.

Where Hash Functions in Cryptography Are Used

Everywhere. They protect passwords (only their hashes are stored, never the passwords themselves), verify file integrity (download a file, check its hash), and sit at the very foundation of cryptocurrencies: blockchain is built on hash functions.

So the fairy tale about the intercepted messenger would have to end on page three if hash functions had existed back then.

What should we cover next?

• How did we arrive at hash functions in the first place?
• Real-world examples of hash algorithms
• If hashing is so secure, why are passwords still being cracked today?