GF2n

GF(2^n) denotes a finite field with 2^n elements and characteristic 2. It can be constructed as the quotient GF(2)[x]/(f(x)) where f(x) is an irreducible polynomial over GF(2) of degree n. Each element of GF(2^n) can be represented as a polynomial of degree at most n−1 with coefficients in {0,1}, or equivalently as an n-bit binary vector. Different choices of f(x) yield different representations of the same abstract field; all fields GF(2^n) are isomorphic.

Arithmetic in GF(2^n) is defined with respect to this representation. Addition is performed by bitwise XOR of coefficients. Multiplication is polynomial multiplication followed by reduction modulo f(x). Inversion can be computed by the extended Euclidean algorithm. The nonzero elements form a cyclic group of order 2^n−1, so primitive elements exist and generate the multiplicative group. For efficiency, implementations often use a chosen basis (polynomial basis or normal basis) or precomputed log/anti-log tables.

Representations and optimization: GF(2^n) supports different basis choices, which affect hardware and software efficiency for field operations. The field is widely used in coding theory and cryptography. In error-correcting codes, GF(2^n) fields underpin Reed-Solomon codes used in CDs, DVDs, QR codes, and data transmission systems. In cryptography, elliptic curves over GF(2^n) are used in binary-field ECC schemes. A common concrete instance is GF(2^8), used by the AES encryption standard with the irreducible polynomial x^8 + x^4 + x^3 + x + 1. Although specific polynomials may vary, all GF(2^n) fields share the same fundamental algebraic properties.