Binary Polynomial Calculator
Calculate binary polynomials over GF(2). Add, subtract, multiply or divide binary coefficient strings and view polynomial notation, degree, quotient, remainder and calculation steps.
| Step | Current Dividend | Shift | Shifted Divisor | New Remainder |
|---|---|---|---|---|
| Division steps appear here. | ||||
What Is a Binary Polynomial?
A binary polynomial is a polynomial whose coefficients are either 0 or 1. The coefficients are normally interpreted over the finite field GF(2).
Each bit identifies whether a particular power of x is present.
= 1x^3 + 0x^2 + 1x + 1
= x^3 + x + 1
Binary Polynomial Representation
| Binary | Polynomial | Degree |
|---|---|---|
| 1 | 1 | 0 |
| 10 | x | 1 |
| 11 | x + 1 | 1 |
| 101 | x^2 + 1 | 2 |
| 111 | x^2 + x + 1 | 2 |
| 1011 | x^3 + x + 1 | 3 |
| 10011 | x^4 + x + 1 | 4 |
GF(2) Polynomial Arithmetic
Addition
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0
Multiplication
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Binary Polynomial Addition
Corresponding coefficients are added modulo 2. There are no carries, so polynomial addition behaves like XOR.
0110
—-
1101
Result:
x^3 + x^2 + 1
Binary Polynomial Subtraction
Polynomial subtraction over GF(2) produces exactly the same result as addition.
because coefficients are calculated modulo 2.
Binary Polynomial Multiplication
Each nonzero term of one polynomial is multiplied by each nonzero term of the other polynomial. Equal powers are combined modulo 2.
= x^2 + 1
B = 11
= x + 1
Product:
1111
= x^3 + x^2 + x + 1
Binary Polynomial Division
Polynomial division aligns the divisor with the leading term of the current dividend. The shifted divisor is then removed using GF(2) subtraction.
The process continues until the remainder has lower degree than the divisor.
= Divisor x Quotient + Remainder
Polynomial Degree
The polynomial degree is the highest exponent whose coefficient is 1.
= x^5 + x^3 + 1
Degree = 5
Binary Integer vs GF(2) Polynomial
| Feature | Binary Integer | Binary Polynomial |
|---|---|---|
| Bit meaning | Numeric position | Polynomial coefficient |
| Addition | Uses carries | No carries |
| 1 + 1 | 10 | 0 |
| Subtraction | May use borrow | Same as addition |
| Multiplication | Integer product | Polynomial product |
| Division | Numeric remainder | Polynomial remainder |
Where Binary Polynomials Are Used
CRC Mathematics
GF(2) polynomial division is fundamental to understanding cyclic redundancy check calculations.
Coding Theory
Binary polynomials are widely used in cyclic codes and algebraic error-control systems.
Finite Fields
Polynomial arithmetic forms a foundation for extensions of GF(2).
Digital Logic
XOR networks and linear feedback shift registers can be described using polynomial notation.
Important Binary Polynomial Notes
GF(2) addition has no carry.
Addition and subtraction produce the same coefficient result.
Multiplication combines equal powers modulo 2.
Polynomial division returns a quotient and remainder.
The zero polynomial cannot be used as a divisor.