XORed

XORed refers to data or a value that has been processed by the exclusive OR, or XOR, operation. XOR is a binary operation on bits that yields 1 when the two input bits differ and 0 when they are the same. It is denoted by the symbol ⊕ and is implemented in most programming languages as a bitwise operator (often written as ^).

Common properties of XOR include commutativity, associativity, and the existence of an identity element. Specifically, a ⊕ b = b ⊕ a and (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c). The identity element is 0, since a ⊕ 0 = a for any bit pattern a. Each value is its own inverse with respect to XOR: a ⊕ a = 0. These properties enable simple algebraic manipulation and reversible encoding.

Applications: In data processing, XORed data is often used to obfuscate a message by XORing with a key; applying the same operation with the same key again restores the original data. This principle underlies simple stream ciphers and checksums, but it is vulnerable if the key is reused or insufficiently random. In error detection and correction, XOR is used to compute parity bits and to implement RAID 5 parity. In digital logic, XOR gates form the basis of adders and other circuits.

Notes: In practice, the term XORed or XORing usually refers to bitwise operations, but higher-level data can also be XORed in chunks by applying the operation to corresponding bits in each part. Understanding XOR is foundational in fields ranging from computer science to electrical engineering.