Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions. Division of this type is efficiently realised in hardware by a modified shift register, and in software by a series of equivalent algorithms, starting with simple code close to the mathematics and becoming faster (and arguably more obfuscated) through byte-wise parallelism and space–time tradeoffs.
Various CRC standards extend the polynomial division algorithm by specifying an initial shift register value, a final Exclusive-Or step and, most critically, a bit ordering (endianness). As a result, the code seen in practice deviates confusingly from "pure" division, and the register may shift left or right.
Example
As an example of implementing polynomial division in hardware, suppose that we are trying to compute an 8-bit CRC of an 8-bit message made of the ASCII character "W", which is binary 010101112, decimal 8710, or hexadecimal 5716. For illustration, we will use the CRC-8-ATM (HEC) polynomial . Writing the first bit transmitted (the coefficient of the highest power of ) on the left, this corresponds to the 9-bit string "100000111".
The byte value 5716 can be transmitted in two different orders, depending on the bit ordering convention used. Each one generates a different message polynomial . Msbit-first, this is = 01010111, while lsbit-first, it is = 11101010. These can then be multiplied by to produce two 16-bit message polynomials .
Computing the remainder then consists of subtracting multiples of the generator polynomial . This is just like decimal long division, but even simpler because the only possible multiples at each step are 0 and 1, and the subtractions borrow "from infinity" instead of reducing the upper digits. Because we do not care about the quotient, there is no need to record it.
Most-significant bit first Least-significant bit first 0101011100000000−000000000=0101011100000000 −100000111=0001011011000000 −000000000=0001011011000000 −100000111=0000011010110000 −000000000=0000011010110000 −100000111=0000001010101100 −100000111=0000000010100010 −000000000=0000000010100010 1110101000000000−100000111=0110100110000000 −100000111=0010100001000000 −100000111=0000100010100000 −000000000=0000100010100000 −100000111=0000000010011000 −000000000=0000000010011000 −000000000=0000000010011000 −000000000=0000000010011000 Observe that after each subtraction, the bits are divided into three groups: at the beginning, a group which is all zero; at the end, a group which is unchanged from the original; and a blue shaded group in the middle which is "interesting". The "interesting" group is 8 bits long, matching the degree of the polynomial. Every step, the appropriate multiple of the polynomial is subtracted to make the zero group one bit longer, and the unchanged group becomes one bit shorter, until only the final remainder is left.
In the msbit-first example, the remainder polynomial is . Converting to a hexadecimal number using the convention that the highest power of x is the msbit; this is A216. In the lsbit-first, the remainder is . Converting to hexadecimal using the convention that the highest power of x is the lsbit, this is 1916.
Implementation Writing out the full message at each step, as done in the example above, is very tedious. Efficient implementations use an -bit shift register to hold only the interesting bits. Multiplying the polynomial by is equivalent to shifting the register by one place, as the coefficients do not change in value but only move up to the next term of the polynomial.
Here is a first draft of some pseudocode for computing an n-bit CRC. It uses a contrived composite data type for polynomials, where x is not an integer variable, but a constructor generating a Polynomial object that can be added, multiplied and exponentiated. To xor two polynomials is to add them, modulo two; that is, to exclusive OR the coefficients of each matching term from both polynomials.
function crc(bit array bitString[1..len], int len) { remainderPolynomial := polynomialForm(bitString[1..n]) // First n bits of the message // A popular variant complements remainderPolynomial here; see below for i from 1 to len { remainderPolynomial := remainderPolynomial * x + bitString[i+n] * x0 // Define bitString[k]=0 for k>len if coefficient of xn of remainderPolynomial = 1 { remainderPolynomial := remainderPolynomial xor generatorPolynomial
// A popular variant complements remainderPolynomial here; see below return remainderPolynomial } Code fragment 1: Simple polynomial division
Note that this example code avoids the need to specify a bit-ordering convention by not using bytes; the input bitString is already in the form of a bit array, and the remainderPolynomial is manipulated in terms of polynomial operations; the multiplication by could be a left or right shift, and the addition of bitString[i+n] is done to the coefficient, which could be the right or left end of the register.
This code has two disadvantages. First, it actually requires an n+1-bit register to hold the remainderPolynomial so that the coefficient can be tested. More significantly, it requires the bitString to be padded with n zero bits.
The first problem can be solved by testing the coefficient of the remainderPolynomial before it is multiplied by .
The second problem could be solved by doing the last n iterations differently, but there is a more subtle optimization which is used universally, in both hardware and software implementations.
Because the XOR operation used to subtract the generator polynomial from the message is commutative and associative, it does not matter in what order the various inputs are combined into the remainderPolynomial. And specifically, a given bit of the bitString does not need to be added to the remainderPolynomial until the very last instant when it is tested to determine whether to xor with the generatorPolynomial.
This eliminates the need to preload the remainderPolynomial with the first n bits of the message, as well:
function crc(bit array bitString[1..len], int len) { remainderPolynomial := 0 // A popular variant complements remainderPolynomial here; see below for i from 1 to len { remainderPolynomial := remainderPolynomial xor (bitstring[i] * xn−1) if (coefficient of xn−1 of remainderPolynomial) = 1 { remainderPolynomial := (remainderPolynomial * x) xor generatorPolynomial } else { remainderPolynomial := (remainderPolynomial * x)
// A popular variant complements remainderPolynomial here; see below return remainderPolynomial } Code fragment 2: Polynomial division with deferred message XORing
This is the standard bit-at-a-time hardware CRC implementation, and is well worthy of study; once you understand why this computes exactly the same result as the first version, the remaining optimizations are quite straightforward. If remainderPolynomial is only n bits long, then the coefficients of it and of generatorPolynomial are simply discarded. This is the reason that you will usually see CRC polynomials written in binary with the leading coefficient omitted.
In software, it is convenient to note that while one may delay the xor of each bit until the very last moment, it is also possible to do it earlier. It is usually convenient to perform the xor a byte at a time, even in a bit-at-a-time implementation. Here, we take the input in 8-bit bytes:
function crc(byte array string[1..len], int len) { remainderPolynomial := 0 // A popular variant complements remainderPolynomial here; see below for i from 1 to len { remainderPolynomial := remainderPolynomial xor polynomialForm(string[i]) * xn−8 for j from 1 to 8 { // Assuming 8 bits per byte if coefficient of xn−1 of remainderPolynomial = 1 { remainderPolynomial := (remainderPolynomial * x) xor generatorPolynomial } else { remainderPolynomial := (remainderPolynomial * x)
// A popular variant complements remainderPolynomial here; see below return remainderPolynomial } Code fragment 3: Polynomial division with bytewise message XORing
This is usually the most compact software implementation, used in microcontrollers when space is at a premium over speed.
Bit ordering (endianness) When implemented in bit serial hardware, the generator polynomial uniquely describes the bit assignment; the first bit transmitted is always the coefficient of the highest power of , and the last bits transmitted are the CRC remainder , starting with the coefficient of and ending with the coefficient of , a.k.a. the coefficient of 1.
However, when bits are processed a byte at a time, such as when using parallel transmission, byte framing as in 8B/10B encoding or RS-232-style asynchronous serial communication, or when implementing a CRC in software, it is necessary to specify the bit ordering (endianness) of the data; which bit in each byte is considered "first" and will be the coefficient of the higher power of .
If the data is destined for serial communication, it is best to use the bit ordering the data will ultimately be sent in. This is because a CRC's ability to detect burst errors is based on proximity in the message polynomial ; if adjacent polynomial terms are not transmitted sequentially, a physical error burst of one length may be seen as a longer burst due to the rearrangement of bits.
For example, both IEEE 802 (ethernet) and RS-232 (serial port) standards specify least-significant bit first (little-endian) transmission, so a software CRC implementation to protect data sent across such a link should map the least significant bits in each byte to coefficients of the highest powers of . On the other hand, floppy disks and most hard drives write the most significant bit of each byte first.
The lsbit-first CRC is slightly simpler to implement in software, so is somewhat more commonly seen, but many programmers find the msbit-first bit ordering easier to follow. Thus, for example, the XMODEM-CRC extension, an early use of CRCs in software, uses an msbit-first CRC.
So far, the pseudocode has avoided specifying the ordering of bits within bytes by describing shifts in the pseudocode as multiplications by and writing explicit conversions from binary to polynomial form. In practice, the CRC is held in a standard binary register using a particular bit-ordering convention. In msbit-first form, the most significant binary bits will be sent first and so contain the higher-order polynomial coefficients, while in lsbit-first form, the least-significant binary bits contain the higher-order coefficients. The above pseudocode can be written in both forms. For concreteness, this uses the 16-bit CRC-16-CCITT polynomial :
// Most significant bit first (big-endian) // (x16)+x12+x5+1 = (1) 0001 0000 0010 0001 = 0x1021 function crc(byte array string[1..len], int len) { rem := 0 // A popular variant complements rem here for i from 1 to len { rem := rem xor (string[i] leftShift (n-8)) // n = 16 in this example for j from 1 to 8 { // Assuming 8 bits per byte if rem and 0x8000 { // Test x15 coefficient rem := (rem leftShift 1) xor 0x1021 } else { rem := rem leftShift 1
rem := rem and 0xffff // Trim remainder to 16 bits
// A popular variant complements rem here return rem } Code fragment 4: Shift register based division, MSB first
// Least significant bit first (little-endian) // 1+x5+x12+(x16) = 1000 0100 0000 1000 (1) = 0x8408 function crc(byte array string[1..len], int len) { rem := 0 // A popular variant complements rem here for i from 1 to len { rem := rem xor string[i] for j from 1 to 8 { // Assuming 8 bits per byte if rem and 0x0001 { // Test x15 coefficient rem := (rem rightShift 1) xor 0x8408 } else { rem := rem rightShift 1