Binary XOR Division Calculator & Steps

This calculator performs binary XOR division, the core operation behind CRC (Cyclic Redundancy Check) error detection. You enter a binary dividend and a binary divisor (both as strings of 1s and 0s), and the calculator returns the quotient and the remainder using XOR instead of ordinary subtraction.

Formula

Binary XOR division is performed bit-by-bit, just like long division, but the subtraction step is replaced by a bitwise XOR (exclusive OR). The process is:

• Align the divisor with the leftmost bit of the dividend.
• If the current leading bit is 1, XOR the divisor with the current bits; if it is 0, skip (or XOR with zeros).
• Bring down the next bit from the dividend.
• Repeat until all bits of the dividend are processed.

The remainder is the final leftover bits (its length is always one less than the divisor length). The quotient is the sequence of 1s and 0s that recorded whether XOR was applied at each step.

For example, with dividend 1101011011 and divisor 10011 (degree 4, because 10011 has 5 bits), the remainder will have at most 4 bits.

Worked Example

Let’s divide 1101011011 by 10011 using XOR.

1. Take the first 5 bits: 11010. Leading bit is 1, so XOR with 10011: 11010 ⊕ 10011 = 01001.
2. Bring down the next bit (the 6th bit of the dividend, which is 1), giving 010011. The leading bit is 0, so skip (quotient bit 0). Bring down the next bit (0), giving 100110.
3. Leading bit is 1, XOR with 10011: 100110 ⊕ 10011 = 000000 (after aligning). Actually, 100110 (6 bits) XOR 10011 (5 bits) – we pad the divisor to 6 bits: 100110. XOR gives 000000.
4. Bring down the next bit (1), giving 000001. Leading bit 0, skip. Bring down the last bit (1), giving 000011.
5. Leading bit 0, skip. No more bits. The remainder is the last 4 bits: 0011.

The quotient bits (from steps where we performed XOR) are 10100 (the first step gave a 1, the second a 0, the third a 1, the fourth a 0, the fifth a 0). So the result is: quotient 10100, remainder 0011.

Verify: 1101011011 = 10011 * 10100 XOR 0011 (in GF(2) arithmetic).

When to Use XOR Division

XOR division is used almost exclusively in CRC error detection. When a sender transmits data, it appends a CRC remainder so that the entire message (data + remainder) is exactly divisible by a chosen polynomial (the divisor). The receiver performs the same division; if the remainder is zero, the message is considered intact. This is used in network protocols (Ethernet, Wi-Fi), storage devices, and compressed file formats.

You would use this calculator when you need to compute a CRC remainder manually, verify a CRC implementation, or learn how the division works step by step.

How to Read the Result

The remainder is the CRC checksum. In practice, the remainder is often appended to the data, and sometimes inverted or bit-reflected depending on the CRC variant. The quotient is usually discarded – it carries no meaning for error detection.

If you are using a standard CRC (like CRC-32), you must also account for the initial value, final XOR, and bit ordering. This calculator gives the raw polynomial remainder, which is the mathematical core.

Common Mistakes

• Using ordinary subtraction – XOR division is not the same as binary long division with borrows. Forgetting this gives a completely different remainder.
• Ignoring leading zeros – The remainder must be padded with leading zeros to match the divisor’s degree. For example, a remainder of 11 with a 5-bit divisor should be written as 0011.
• Misaligning the divisor – The divisor must be aligned under the current leftmost bit, not the first bit of the whole dividend.
• Forgetting the degree – The remainder length is always (divisor length – 1). If you get a longer remainder, you missed a step.

Limits and Alternatives

This calculator works on binary strings of arbitrary length, but it assumes the standard CRC polynomial representation. It does not handle non-binary fields (like GF(2^8) used in some error-correcting codes). For actual CRC computation, you often need to apply bit reflection and final XOR, which are not included here.

If you need a full CRC-32 or CRC-16 checksum, use a dedicated CRC calculator that implements the specific polynomial and parameters. This tool is for understanding and verifying the raw division step.

Frequently Asked Questions

What is the difference between XOR division and normal binary division?

Normal binary division uses subtraction with borrows, which is not the same as XOR. XOR is a bitwise operation that outputs 1 only when the two bits differ. In polynomial arithmetic over GF(2), addition and subtraction are both XOR, so division uses XOR as the “subtract” step.

Why is the remainder always shorter than the divisor?

Because the division stops when the remaining bits are fewer than the divisor’s length. The remainder is the final leftover, and its length is at most (divisor length – 1). This is a property of polynomial long division over GF(2).

Can I use this calculator for CRC-32?

Not directly. CRC-32 uses a specific 33-bit divisor (0x104C11DB7) and also applies initial value, bit reflection, and final XOR. This calculator gives the raw polynomial remainder, which is the core mathematical step. You would need to apply the additional transformations yourself.

What does the quotient represent in CRC?

In CRC, the quotient is discarded. Only the remainder is used as the checksum. The quotient is a byproduct of the division process and has no error-detection role.

How do I convert a polynomial like x^4 + x + 1 to binary?

Write the coefficients of the polynomial from highest degree to lowest. For x^4 + x + 1, the coefficients are 1 (x^4), 0 (x^3), 0 (x^2), 1 (x^1), and 1 (x^0), giving 10011. The degree is the highest exponent, so the binary length is degree+1.

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in dividend | Dividend (binary) | - | 1101011011 | 0
in divisor | Divisor (binary) | - | 10011 | 0
out remainder | Remainder | - | xor_division(dividend, divisor) | 0
out quotient | Quotient | - | xor_division_q(dividend, divisor) | 0