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A cyclic redundancy check (CRC) is a short value calculated from a block of data so a receiver or storage system can detect many accidental changes. The sender adds the CRC to the data; the receiver calculates the expected value and compares it with the received check. A mismatch signals that the data may be corrupted. A CRC detects errors; it does not identify or repair them.
How does a CRC work?
A CRC calculation treats the message bits as coefficients in a polynomial over GF(2), a binary field. In this arithmetic, addition and subtraction are both XOR operations.
- Choose a generator polynomial. The sender and receiver must use the same polynomial and CRC conventions.
- Append zero bits for the CRC width. Mathematically, this multiplies the message polynomial by x raised to the CRC width.
- Divide by the generator polynomial. The remainder has a degree smaller than the generator’s degree and becomes the check value. As RFC 9260 puts it, “The CRC is computed using polynomial division.”
- Transmit and check. The sender sends the data with its CRC. The receiver follows the agreed procedure and checks whether the received check field is consistent with the data.
The exact bit-level procedure depends on the CRC variant and the protocol using it, so this polynomial description explains the principle rather than every implementation detail.
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A CRC can detect many accidental error patterns. It does not reveal which bit changed, restore the original data, or guarantee detection of every possible corruption. A system that needs recovery must use another mechanism, such as retransmission; a system that must correct errors needs an error-correcting code or another repair method.
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Detection guarantees depend on the generator polynomial and the length of the protected message. IEEE Technology Navigator reports that, under its stated CRC-32 assumptions, a CRC-32 detects all single-bit errors, all double-bit errors, all odd numbers of errors, and all bursts shorter than 33 bits for messages up to 231 bits. These are specific technical guarantees, not a promise that every CRC catches every error pattern.
Why isn’t every CRC-32 interchangeable?
The width tells you how many bits the check value contains, but it does not fully define the algorithm. CRC-8, CRC-16, CRC-32, and CRC-64 refer to widths associated with generator polynomials of those degrees; multiple polynomials and conventions can exist at each width.
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For two implementations to agree, they must match on the relevant details, including:
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- the generator polynomial and CRC width;
- bit ordering and whether bits are reflected;
- initialization and finalization conventions;
- the message length and fields covered by the check; and
- how the protocol carries and validates the CRC.
For example, RFC 9260 specifies CRC32c for SCTP using the Castagnoli polynomial and describes SCTP’s protocol-specific calculation. Ethernet uses its own CRC-32 frame check sequence convention. Do not substitute a CRC merely because it has the same width; follow the specification for the protocol or data format in use.
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Where are CRCs used?
CRCs appear in communication and data-integrity systems. Ethernet frames carry a CRC in the frame check sequence, and USB packets use CRC fields to check packet data. CRCs are also used in storage and other systems that need to detect accidental changes. The exact CRC variant and protected fields depend on the applicable specification.
The USB document cited here is a draft, so it supports the general USB CRC context but should not be treated as confirmation of the current normative USB specification.
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What should you remember?
- A CRC is a compact check value calculated from data to detect accidental changes.
- Its core calculation is polynomial division over GF(2), where addition and subtraction are XOR.
- A CRC detects errors but does not correct them.
- Width alone does not identify a compatible CRC algorithm; the polynomial and protocol conventions matter.
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