XOR Operation

What is XOR?

XOR stands for exclusive or.

It compares two bits and asks: are they different?

Different → 1. Same → 0.

The Rule

Two 0s? Same → 0

Two 1s? Same → 0

One of each? Different → 1

XOR with Binary Numbers

To XOR two binary numbers, compare each column:

Key Properties

1. Anything XOR itself is 0

$A \oplus A = 0$

Same bits cancel out.

2. Anything XOR 0 is itself

$A \oplus 0 = A$

XOR with 0 changes nothing.

3. XOR twice cancels out

$A \oplus B \oplus B = A$

This is the magic property.

Why Cryptography Loves XOR

That third property makes XOR perfect for encryption.

• Take a message
• XOR it with a key → get ciphertext
• XOR ciphertext with the same key → get message back

Same operation encrypts and decrypts.