Finite Field GF(2ⁿ) Calculator

Finite fields GF(2ⁿ) are used in AES, CRCs, coding theory, and binary algebra.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

Elements of GF(2ⁿ) can be represented as binary polynomials whose coefficients are in GF(2). A bit string such as 1011 represents x³+x+1. Because coefficients are only 0 or 1, addition and subtraction are the same operation: coefficientwise XOR with no carries.

Multiplication first forms the ordinary polynomial product over GF(2), again using XOR when like powers combine. The raw product is then reduced modulo a chosen degree-n irreducible polynomial. This reduction keeps the result degree below n and is what turns the n-bit representations into a field rather than merely a polynomial ring.

For a true GF(2ⁿ), the modulus polynomial must be irreducible over GF(2). If a reducible polynomial is used, nonzero elements may fail to have multiplicative inverses, so the resulting arithmetic is not a field.

Degree gives an immediate range check. With a degree-n modulus, every reduced field element should have polynomial degree below n, so it fits in at most n bits. If a multiplication result still contains an xⁿ term or higher, polynomial reduction is incomplete.

Do not confuse polynomial bits with the binary representation of ordinary integer multiplication. For instance, in GF(2), x·x is x² and coefficients combine by XOR, not by carrying. The same bit patterns can therefore produce different multiplication results from normal integers. This distinction is central to AES arithmetic, Reed–Solomon-style coding ideas, and other binary-field applications. Always state the modulus polynomial when reporting a multiplication result because the same raw product can reduce differently in different fields.