Binary Polynomial Calculator

Binary polynomials are used in CRCs, finite fields, and coding theory.

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.

A binary polynomial uses coefficients from the two-element field GF(2), so every coefficient is either 0 or 1. A bit string records coefficients from highest power to constant term. Thus 1011 represents x³+x+1 because the x² coefficient is zero.

Addition in GF(2) is XOR: 0+0=0, 0+1=1, 1+0=1, and 1+1=0. There is no carry because 1+1 is 0 modulo 2. Polynomial subtraction is identical to addition since −1=1 in GF(2).

Multiplication is carryless polynomial multiplication. Shift one bit pattern for each nonzero term of the other polynomial, then XOR the shifted copies. Unless an irreducible modulus polynomial is also specified, the raw product is not reduced back to a fixed field width. This distinction matters in CRCs and finite-field arithmetic, where a later reduction step is often required.

Degree provides a quick reasonableness check. If nonzero polynomials A and B have degrees m and n, their unreduced product has degree m+n because the leading coefficients are both 1 and their product remains 1 in GF(2). The sum has degree at most max(m,n), and its leading terms can cancel when both inputs have a 1 in the same highest position.

Bitwise notation makes the algebra compact but should not obscure the polynomial meaning. A left shift does not mean ordinary decimal multiplication for the purpose of the field operation; it represents multiplication by x. XOR then combines coefficients of equal powers modulo 2. Keeping this interpretation visible makes CRC and finite-field exercises easier to follow.