F2n

F2n is a common informal notation for the finite field with 2^n elements, usually written as GF(2^n) or F_{2^n}. It is the extension field of the binary field GF(2) of degree n and is fundamental in coding theory, cryptography, and digital communications.

Construction and arithmetic: F2n is formed by choosing an irreducible polynomial p(x) of degree n over GF(2) and forming the quotient ring GF(2)[x]/(p(x)). Elements are represented as polynomials of degree less than n with binary coefficients. Addition is coordinate-wise modulo 2 (equivalently, bitwise XOR). Multiplication is performed modulo p(x). The nonzero elements form a cyclic multiplicative group of order 2^n − 1, generated by a primitive element α.

Representations: Elements can be encoded as n-bit vectors under a polynomial basis or using alternative bases such as a normal basis. Different representations affect the efficiency of arithmetic in software or hardware implementations.

Examples and applications: A common instance is F_{2^3} constructed with a chosen irreducible polynomial of degree 3, yielding eight elements used in compact error-correcting schemes. More widely, F_{2^8} is used in the AES encryption standard, which employs a specific irreducible polynomial x^8 + x^4 + x^3 + x + 1 for efficient byte-wise operations. Finite fields of order 2^n underpin many Reed-Solomon and BCH codes, cryptographic schemes, and digital signal processing techniques.

Notation and ambiguity: While F2n often denotes GF(2^n), the compact form can be ambiguous in plain text. When precision is required, GF(2^n) or GF(2)[x]/(p(x)) is preferred. See also finite field, GF(2), irreducible polynomial.