GF2

GF(2) is the Galois field with two elements, 0 and 1. It is the smallest finite field and is isomorphic to the integers modulo 2, written Z/2Z. The field has characteristic 2, meaning that 1 + 1 = 0.

Operations in GF(2) are defined modulo 2. Addition corresponds to the exclusive-or (XOR) operation, and multiplication corresponds to ordinary multiplication reduced modulo 2. In this system, 0 is the additive identity and 1 is the multiplicative identity; the only nonzero element is 1, whose multiplicative inverse is itself.

GF(2) serves as the base field for constructing larger finite fields, such as GF(2^m), which are used in coding theory, digital communications, and cryptography. Arithmetic in GF(2) is central to many algorithms, including error-detecting and error-correcting codes where polynomial arithmetic over GF(2) is performed. In practical computing, GF(2) operations map naturally to binary logic: addition is XOR, and multiplication is AND in certain bitwise implementations, with carryless arithmetic used in polynomial computations.

Elements of GF(2) are represented by the binary digits 0 and 1. The simplicity of GF(2) underpins a wide range of applications in computer science, including boolean algebra, fault-tolerant coding schemes, and low-level cryptographic primitives that rely on binary field arithmetic.