GF2X

GF2x is the finite field with 2^x elements, commonly written GF(2^x) or GF(2^n). It is a field of characteristic 2 constructed as the quotient ring GF(2)[X] / (P(X)), where P(X) is an irreducible polynomial over GF(2) of degree x. Elements are equivalence classes of polynomials of degree less than x with coefficients in {0,1}, typically represented as x-bit binary strings.

Arithmetic in GF(2^x) is defined by the underlying polynomial operations modulo P(X). Addition and subtraction coincide and are performed bitwise as XOR. Multiplication involves polynomial multiplication followed by modular reduction by P(X). Inversion can be computed with the extended Euclidean algorithm or by exponentiation to the power 2^x - 2. Squaring is especially efficient in characteristic 2, and in some representations reduces to simple linear operations.

Basis representations include the polynomial basis and the normal basis. A polynomial basis uses the standard representation of polynomials modulo P(X), while a normal basis makes squaring a cyclic shift, which can simplify certain implementations. The choice of irreducible polynomial P(X) and basis affects performance, memory usage, and hardware complexity.

Implementation considerations often leverage hardware features such as carryless multiplication (CLMUL) to accelerate block operations, along with tailored reduction steps for the chosen P(X). Depending on the application, different irreducible polynomials and basis choices are selected to optimize speed or resource usage.

Applications span error-correcting codes (such as BCH and Reed-Solomon codes over GF(2^m)), digital communications, and various cryptographic protocols that operate over binary finite fields. The exact interpretation of the term GF2x may vary by context, particularly the value of x and the chosen irreducible polynomial.