CRC Calculator

Compute a simple binary CRC remainder for a message and generator polynomial. This educational calculator shows the formula, result, and step-by-step interpretation.

Computer ScienceCoding Theory

Calculator

What this calculator teaches

CRC uses polynomial division over GF(2) to create a remainder that can detect many transmission errors.

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.

CRC as Polynomial Division over GF(2)

A cyclic redundancy check treats a bit string as coefficients of a polynomial over the two-element field GF(2). Addition and subtraction are both XOR because 1 + 1 = 0 in this arithmetic. If the generator polynomial has degree r, the sender appends r zero bits to the message, divides the resulting polynomial by the generator using XOR long division, and keeps the r-bit remainder.

The transmitted codeword is the original message followed by that remainder. In the ideal algebraic model, dividing an uncorrupted codeword by the same generator gives remainder zero. A changed codeword can produce a nonzero remainder, allowing many transmission errors to be detected. CRC is primarily an error-detection method; the remainder generally does not identify which bit changed.

StepOperationReason
Generator degreer = length(generator) − 1Determines remainder length
Augment messageM(x)xrCreates room for r check bits
Divide in GF(2)XOR instead of subtractionCoefficients are only 0 or 1
Codewordmessage || remainderPolynomial is divisible by generator

The quality of a CRC depends strongly on the chosen generator polynomial. Different generators detect different patterns of errors, so a classroom polynomial-division example should not be treated as a complete communications-standard CRC unless its bit ordering, initialization, reflection, and final-XOR conventions are also specified.

Formula & Symbols

ConceptFormula or rule
CRC remainderbinary polynomial division remainder
Codewordmessage followed by remainder

Worked example

Example: Divide the padded message by the generator bits using XOR subtraction.
Example 2: Message 1101011011 with generator 10011 has degree 4. Appending four zeros and performing XOR division gives remainder 1110, so the codeword is 11010110111110.
Example 3: The classic message 11010011101100 with generator 1011 gives remainder 100 after binary polynomial division.
Example 4: Message 1011 with generator 1101 gives remainder 100. The three-bit remainder length matches the generator degree 3.
Example 5: Using generator 11 corresponds to x + 1. For message 1010, the remainder is 0 because the message has even parity; this generator is closely related to a parity check.

Common mistakes

⚠️
Using the wrong input format

Keep lists comma separated, matrices as rows separated by semicolons, and modular inputs as integers.

⚠️
Ignoring assumptions

Some methods require positive probabilities, valid moduli, independent trials, or small educational input sizes.

⚠️
Using ordinary subtraction in the long division

CRC polynomial arithmetic is over GF(2). Subtraction has no borrowing and is performed with XOR, so 1 XOR 1 = 0 and 1 XOR 0 = 1.

⚠️
Forgetting to append zeros before division

A degree-r generator requires r trailing zeros on the message before computing the remainder. Those positions are then replaced by the r check bits.

FAQ

Can I use decimals?⌄
Most numerical calculators allow decimals, but modular arithmetic and coding-theory tools usually require integers or binary strings.
Is this for homework checking?⌄
Yes. The page is designed to show both the answer and the reasoning pattern.
Why does the result sometimes say approximate?⌄
Some probability, floating-point, and numerical methods naturally produce approximations.
Why is XOR used in CRC division?⌄
CRC coefficients belong to GF(2), where the only coefficients are 0 and 1 and arithmetic is modulo 2. Addition and subtraction are therefore identical, and bitwise XOR implements both operations without carries or borrows.
Does a zero CRC remainder prove a message has no errors?⌄
No. A zero remainder means the received polynomial is divisible by the generator. Well-chosen CRC generators detect broad classes of errors, but some error patterns are also divisible by the generator and can pass undetected.
Can CRC correct an incorrect bit?⌄
Ordinary CRC is designed for error detection, not general error correction. A nonzero syndrome indicates inconsistency with the generator, but it usually does not uniquely identify the location or pattern of changed bits.
Why do real CRC standards need more parameters than a generator polynomial?⌄
Practical CRC specifications can define the initial register value, whether input or output bits are reflected, a final XOR value, and bit-order conventions. Two implementations using the same polynomial can disagree if those conventions differ.

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Formula Explorer connections

Interpretation: This formula uses integer divisibility, modular arithmetic, finite fields or coding relationships. Assumption: Inputs are usually integers with specific modulus or coprimality requirements. Cryptographic examples are educational and not a substitute for vetted security libraries.

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