Xor

XOR, short for exclusive OR, is a binary operation that takes two input values and returns true or 1 when exactly one of the inputs is true or 1. It is a fundamental operator in digital logic and computer science. In binary terms, XOR is often denoted by the symbol ⊕ or by the caret ^ in many programming languages.

The truth table of XOR for two binary inputs A and B is:

A ⊕ B yields 0 when A and B are both 0 or both 1, and yields 1 when A and B differ. Concretely: 0 ⊕ 0 = 0, 0 ⊕ 1 = 1, 1 ⊕ 0 = 1, 1 ⊕ 1 = 0.

Algebraically, XOR is commutative and associative, and its identity element is 0. Each element is its own inverse under XOR, since A ⊕ A = 0. The operation is equivalent to addition modulo 2, and it forms the addition operation in the finite field GF(2). When applied to bitstrings, XOR performs a bitwise operation: corresponding bits are XORed independently.

XOR plays a central role in both hardware and software. In digital circuits it is implemented with XOR gates and is essential in arithmetic circuits, parity generation, and error detection. In software, XOR is available as a bitwise operator for integers and as a logical operator for booleans in some languages. It is valued for its simplicity, speed, and the property that XORing a value with a key and then XORing the result with the same key returns the original value.

Applications include parity checking, error detection, data obfuscation, and cryptographic constructions such as the one-time pad, where XOR’s reversibility guarantees that encryption and decryption are identical operations.