Cryptographic Hash Functions

What’s a Hash Function?

A hash function takes any input and produces a fixed-size output.

A password, a file, an entire novel. Doesn’t matter. The output is always the same size.

Same Input, Same Output

Hash functions are deterministic.

• "hello" → 2cf24dba5fb0a30e...
• "hello" → 2cf24dba5fb0a30e... (same)
• "hello " → 8eb72a0d4c9b7d7c... (different!)

Change one character, and the entire hash changes.

One-Way Street

You can compute the hash from the input. But you can’t reverse it.

• "hello" → 2cf24dba... (easy)
• 2cf24dba... → ??? (impossible)

Think of it like a fingerprint. You can take a fingerprint from a person, but you can’t reconstruct the person from a fingerprint.

The Three Security Properties

A cryptographic hash function must satisfy three properties.

1. Pre-image Resistance

Given a hash, you can’t find any input that produces it.

This should be computationally impossible.

Why it matters: Password storage.

• Databases store hash(password), not the password
• If stolen, attackers see only hashes
• Without pre-image resistance → they reverse hashes → get passwords

2. Second Pre-image Resistance

Given a message, you can’t find a different message with the same hash.

This should be computationally impossible.

Why it matters: Document integrity.

• You sign a contract and publish its hash
• No one can create a different contract with the same hash
• If they could → they could swap your signed document

3. Collision Resistance

You can’t find any two different messages with the same hash.

This should be computationally impossible.

Why it matters: The swap attack.

1. Attacker creates two contracts: one fair, one malicious
2. Both have the same hash (a collision)
3. You sign the fair one
4. Attacker claims you signed the malicious one

With collision resistance, step 2 is impossible.

What Hashes Are Used For

• Password storage — store hash(password), verify by comparing
• File integrity — publish a hash, others verify it wasn’t tampered
• Digital signatures — sign hash(document) instead of the whole thing
• Blockchain — each block contains the hash of the previous block
• Commitments — commit to a value without revealing it