XORing

XORing is the operation of applying the exclusive OR (XOR) to two binary inputs. The XOR result is 1 when exactly one of the inputs is 1, and 0 otherwise. In boolean algebra this is represented as a ⊕ b, and the operation is commutative and associative.

Truth table: for bits a and b, a ⊕ b equals 0 if a=b, and 1 if a≠b. The identity element is 0, since a ⊕ 0 = a, and every value is its own inverse since a ⊕ a = 0.

Beyond single bits, XOR is applied bitwise to binary strings and vectors. It is addition modulo 2, and as such XOR forms the additive operation of the finite field GF(2).

Applications: In cryptography, if a random key stream is XORed with plaintext (a one-time pad, if the key is truly random and as long as the plaintext), the ciphertext is theoretically secure. Reusing a key stream breaks this security. XOR is also fundamental in digital logic (XOR gates) and error detection (computing simple parity across data). It is used in simple checksums and in algorithms that rely on parity calculations.

In programming, XOR is often used for bitwise manipulation, including the common two-variable swap technique that avoids a temporary variable, though it can be less readable and risky when the same variable is involved. An example: 1011 ⊕ 1101 = 0110.