GF2m

GF(2^m), often written as GF(2^m) or GF2m, denotes the finite field with 2^m elements, a binary finite field. It is the extension field of the two-element field GF(2) of degree m.

Construction and representation: the field is formed by choosing an irreducible polynomial f(x) of degree m over GF(2) and forming the quotient GF(2)[x] / (f(x)). Each element can be represented uniquely by a polynomial of degree less than m with coefficients in {0,1}, equivalently as an m-bit binary vector.

Arithmetic: addition in GF(2^m) is bitwise XOR of the coefficient vectors. Multiplication is polynomial multiplication reduced modulo f(x). The nonzero elements constitute a cyclic multiplicative group of order 2^m − 1; there exists a primitive element, and if f is a primitive polynomial, a root of f(x) generates this group.

Structure and properties: as a vector space over GF(2), GF(2^m) has dimension m and contains 2^m elements. The field has characteristic 2; the additive identity is 0 and the multiplicative identity is 1. The nonzero elements form a cyclic group under multiplication.

Applications: GF(2^m) is widely used in error correction codes (for example Reed–Solomon codes and QR codes), data storage formats, and digital communications. It also underpins certain elliptic-curve cryptosystems and other cryptographic constructions that operate over binary finite fields.

Examples: GF(2^3) uses an irreducible polynomial such as x^3 + x + 1, yielding eight elements. Larger fields, including GF(2^8) with the polynomial x^8 + x^4 + x^3 + x + 1, are common in practical applications such as storage standards and cryptography.