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Insert this material after Chapter 13 (Gray Code). There is a chapter on ECC that

should follow this chapter.

14–1 Cyclic Redundancy Check

The cyclic redundancy check, or CRC, is a technique for detecting errors in digital

data, but not for making corrections when errors are detected. It is used primarily

in data transmission. In the CRC method, a certain number of check bits, often

called a checksum, are appended to the message being transmitted. The receiver

can determine whether or not the check bits agree with the data, to ascertain with

a certain degree of probability whether or not an error occurred in transmission. If

an error occurred, the receiver sends a “negative acknowledgement” (NAK) back

to the sender, requesting that the message be retransmitted.

The technique is also sometimes applied to data storage devices, such as a

disk drive. In this situation each block on the disk would have check bits, and the

hardware might automatically initiate a reread of the block when an error is

detected, or it might report the error to software.

The material that follows speaks in terms of a “sender” and a “receiver” of a

“message,” but it should be understood that it applies to storage writing and reading

as well.


There are several techniques for generating check bits that can be added to a message.

Perhaps the simplest is to append a single bit, called the “parity bit,” which

makes the total number of 1-bits in the code vector (message with parity bit

appended) even (or odd). If a single bit gets altered in transmission, this will

change the parity from even to odd (or the reverse). The sender generates the parity

bit by simply summing the message bits modulo 2—that is, by exclusive or’ing

them together. It then appends the parity bit (or its complement) to the message.

The receiver can check the message by summing all the message bits modulo 2

and checking that the sum agrees with the parity bit. Equivalently, the receiver can

sum all the bits (message and parity) and check that the result is 0 (if even parity

is being used).

This simple parity technique is often said to detect 1-bit errors. Actually it

detects errors in any odd number of bits (including the parity bit), but it is a small

comfort to know you are detecting 3-bit errors if you are missing 2-bit errors.


For bit serial sending and receiving, the hardware to generate and check a single

parity bit is very simple. It consists of a single exclusive or gate together with

some control circuitry. For bit parallel transmission, an exclusive or tree may be

used, as illustrated in Figure 141. Efficient ways to compute the parity bit in software

are given in Section 52 on page 74.

Other techniques for computing a checksum are to form the exclusive or of all

the bytes in the message, or to compute a sum with end-around carry of all the

bytes. In the latter method the carry from each 8-bit sum is added into the least

significant bit of the accumulator. It is believed that this is more likely to detect

errors than the simple exclusive or, or the sum of the bytes with carry discarded.

A technique that is believed to be quite good in terms of error detection, and

which is easy to implement in hardware, is the cyclic redundancy check. This is

another way to compute a checksum, usually eight, 16, or 32 bits in length, that is

appended to the message. We will briefly review the theory and then give some

algorithms for computing in software a commonly used 32-bit CRC checksum.


The CRC is based on polynomial arithmetic, in particular, on computing the

remainder of dividing one polynomial in GF(2) (Galois field with two elements)

by another. It is a little like treating the message as a very large binary number, and

computing the remainder on dividing it by a fairly large prime such as

Intuitively, one would expect this to give a reliable checksum.

A polynomial in GF(2) is a polynomial in a single variable x whose coefficients

are 0 or 1. Addition and subtraction are done modulo 2that is, they are

both the same as the exclusive or operator. For example, the sum of the polynomials

FIGURE 141. Exclusive or tree.

b7 b6 b5 b4 b3 b2 b1 b0

+ + + +

+ +


Parity bit (even)

232 5.

x3+x+1 and



is as is their difference. These polynomials are not usually written

with minus signs, but they could be, because a coefficient of 1 is equivalent to a

coefficient of 1.

Multiplication of such polynomials is straightforward. The product of one

coefficient by another is the same as their combination by the logical and operator,

and the partial products are summed using exclusive or. Multiplication is not

needed to compute the CRC checksum.

Division of polynomials over GF(2) can be done in much the same way as

long division of polynomials over the integers. Below is an example.

The reader might like to verify that the quotient of multiplied by the

divisor of plus the remainder of equals the dividend.

The CRC method treats the message as a polynomial in GF(2). For example,

the message 11001001, where the order of transmission is from left to right

(110) is treated as a representation of the polynomial The

sender and receiver agree on a certain fixed polynomial called the generator polynomial.

For example, for a 16-bit CRC the CCITT (Comité Consultatif Internationale

Télégraphique et Téléphonique)1 has chosen the polynomial

which is now widely used for a 16-bit CRC checksum. To

compute an r-bit CRC checksum, the generator polynomial must be of degree r.

The sender appends r 0-bits to the m-bit message and divides the resulting polynomial

of degree by the generator polynomial. This produces a remainder

polynomial of degree (or less). The remainder polynomial has r

coefficients, which are the checksum. The quotient polynomial is discarded. The

data transmitted (the code vector) is the original m-bit message followed by the rbit


There are two ways for the receiver to assess the correctness of the transmission.

It can compute the checksum from the first m bits of the received data, and

verify that it agrees with the last r received bits. Alternatively, and following usual

practice, the receiver can divide all the received bits by the generator polynomial

and check that the r-bit remainder is 0. To see that the remainder must be

0, let M be the polynomial representation of the message, and let R be the polynomial

representation of the remainder that was computed by the sender. Then the

transmitted data corresponds to the polynomial (or, equivalently,

1. Since renamed the ITU-TSS (International Telecommunications Union - Telecommunications

Standards Sector).

x4 +x2+1,

x3+x+1) x7 + x6 + x5 + x2 + x

x4 + x3 + 1

x7 + x5 + x4

x6 + x4

x6 + x4 + x3

x3 + x2 + x

x3 + x + 1

x2 + 1


x3+x+1, x2 + 1,




r 1


Mxr R


). By the way R was computed, we know that where G

is the generator polynomial and Q is the quotient (that was discarded). Therefore

the transmitted data, is equal to QG, which is clearly a multiple of G. If

the receiver is built as nearly as possible just like the sender, the receiver will

append r 0-bits to the received data as it computes the remainder R. But the

received data with 0-bits appended is still a multiple of G, so the computed

remainder is still 0.

Thats the basic idea, but in reality the process is altered slightly to correct for

such deficiencies as the fact that the method as described is insensitive to the number

of leading and trailing 0-bits in the data transmitted. In particular, if a failure

occurred that caused the received data, including the checksum, to be all-0, it

would be accepted.

Choosing a goodgenerator polynomial is something of an art, and beyond

the scope of this text. Two simple observations: For an r-bit checksum, G should

be of degree r, because otherwise the first bit of the checksum would always be 0,

which wastes a bit of the checksum. Similarly, the last coefficient should be 1

(that is, G should not be divisible by x), because otherwise the last bit of the

checksum would always be 0 (because if G is divisible by x,

then R must be also). The following facts about generator polynomials are proved

in [PeBr] and/or [Tanen]:

If G contains two or more terms, all single-bit errors are detected.

If G is not divisible by x (that is, if the last term is 1), and e is the least

positive integer such that G evenly divides then all double errors

that are within a frame of e bits are detected. A particularly good polynomial

in this respect is for which

If is a factor of G, all errors consisting of an odd number of bits are


An r-bit CRC checksum detects all burst errors of length (A burst

error of length r is a string of r bits in which the first and last are in error,

and the intermediate bits may or may not be in error.)

The generator polynomial creates a checksum of length 1, which

applies even parity to the message. (Proof hint: For arbitrary what is the

remainder of dividing by ?)

It is interesting to note that if a code of any type can detect all double-bit and

single-bit errors, then it can in principle correct single-bit errors. To see this, suppose

data containing a single-bit error is received. Imagine complementing all the

bits, one at a time. In all cases but one, this results in a double-bit error, which is

detected. But when the erroneous bit is complemented, the data is error-free,

which is recognized. In spite of this, the CRC method does not seem to be used for

single-bit error correction. Instead, the sender is requested to repeat the whole

transmission if any error is detected.

Mxr + R Mxr = QG + R,

Mxr R,

Mxr = QG + R,

xe + 1,

x15+x14+1, e = 32767.

x + 1


r 2

x + 1

k 0,

xk x + 1



Table 141 shows the generator polynomials used by some common CRC standards.

The Hexcolumn shows the hexadecimal representation of the generator

polynomial; the most significant bit is omitted, as it is always 1.

The CRC standards differ in ways other than the choice of generating polynomial.

Most initialize by assuming that the message has been preceded by certain

nonzero bits, others do no such initialization. Most transmit the bits within a

byte least significant bit first, some most significant bit first. Most append the

checksum least significant byte first, others most significant byte first. Some complement

the checksum.

CRC-12 is used for transmission of 6-bit character streams, and the others are

for 8-bit characters, or 8-bit bytes of arbitrary data. CRC-16 is used in IBMs

BISYNCH communication standard. The CRC-CCITT polynomial, also known

as ITU-TSS, is used in communication protocols such as XMODEM, X.25,

IBMs SDLC, and ISOs HDLC [Tanen]. CRC-32 is also known as AUTODIN-II

and ITU-TSS (ITU-TSS has defined both 16- and a 32-bit polynomials). It is used

in PKZip, Ethernet, AAL5 (ATM Adaptation Layer 5), FDDI (Fiber Distributed

Data Interface), the IEEE-802 LAN/MAN standard, and in some DOD applications.

It is the one for which software algorithms are given here.

The first three polynomials in Table 141 have as a factor. The last

(CRC-32) does not.

To detect the error of erroneous insertion or deletion of leading 0s, some protocols

prepend one or more nonzero bits to the message. These dont actually get

transmitted, they are simply used to initialize the key register (described below)

used in the CRC calculation. A value of r 1-bits seems to be universally used. The

receiver initializes its register in the same way.

The problem of trailing 0s is a little more difficult. There would be no problem

if the receiver operated by comparing the remainder based on just the message

bits to the checksum received. But, it seems to be simpler for the receiver to

calculate the remainder for all bits received (message and checksum) plus r

appended 0-bits. The remainder should be 0. But, with a 0 remainder, if the mes-



Name r


Polynomial Hex

CRC-12 12 80F

CRC-16 16 8005

CRC-CCITT 16 1021

CRC-32 32 04C11DB7

x12 x11 x3 x2 + + + +x+1



x32 x26 x23 x22 x16 x12

x11 x10 x8 x7 x5 x4 x2 x 1

+ + + + + +

+ + + + + + + +

x + 1


sage has trailing 0-bits inserted or deleted, the remainder will still be 0, so this

error goes undetected.

The usual solution to this problem is for the sender to complement the checksum

before appending it. Because this makes the remainder calculated by the

receiver nonzero (usually), the remainder will change if trailing 0s are inserted or

deleted. How then does the receiver recognize an error-free transmission?

Using the modnotation for remainder, we know that

Denoting the complementof the polynomial R by we have

Thus the checksum calculated by the receiver for an error-free transmission

should be

This is a constant (for a given G). For CRC-32 this polynomial, called the residual

or residue, is

or hex C704DD7B [Black].


To develop a hardware circuit for computing the CRC checksum, we reduce the

polynomial division process to its essentials.

The process employs a shift register, which we denote by CRC. This is of

length r (the degree of G) bits, not as you might expect. When the subtractions

(exclusive ors) are done, it is not necessary to represent the high-order bit,

because the high-order bits of G and the quantity it is being subtracted from are

both 1. The division process might be described informally as follows:

Initialize the CRC register to all 0-bits.

Get first/next message bit m.

If the high-order bit of CRC is 1,

Shift CRC and m together left 1 position, and XOR the result with the

low-order r bits of G.

(Mxr + R) mod G = 0.


(Mxr + R) mod G=(Mxr +(xr 1+xr 2++1 R)) mod G

=((Mxr + R)+xr 1+xr 2++1) mod G

=(xr 1+xr 2++1) mod G.

(xr 1+xr 2++1) mod G.

x31 x30 x26 x25 x24 x18 x15 x14 x12

x11 x10 x8 x6 x5 x4 x3 x 1,

+ + + + + + + + +

+ + + + + + + +

r + 1



Just shift CRC and m left 1 position.

If there are more message bits, go back to get the next one.

It might seem that the subtraction should be done first, and then the shift. It

would be done that way if the CRC register held the entire generator polynomial,

which in bit form is bits. Instead, the CRC register holds only the low-order

r bits of G, so the shift is done first, to align things properly.

Below is shown the contents of the CRC register for the generator G =

and the message M = Expressed in binary, G =

1011 and M = 11100110.

000 Initial CRC contents. High-order bit is 0, so just shift in first message bit.

001 High-order bit is 0, so just shift in second message bit, giving:

011 High-order bit is 0 again, so just shift in third message bit, giving:

111 High-order bit is 1, so shift and then XOR with 011, giving:

101 High-order bit is 1, so shift and then XOR with 011, giving:

001 High-order bit is 0, so just shift in fifth message bit, giving:

011 High-order bit is 0, so just shift in sixth message bit, giving:

111 High-order bit is 1, so shift and then XOR with 011, giving:

101 There are no more message bits, so this is the remainder.

These steps can be implemented with the (simplified) circuit shown in

Figure 142, which is known as a feedback shift register.

The three boxes in the figure represent the three bits of the CRC register. When a

message bit comes in, if the high-order bit (x2 box) is 0, simultaneously the message

bit is shifted into the x0 box, the bit in x0 is shifted to x1, the bit in x1 is shifted

to x2, and the bit in x2 is discarded. If the high-order bit of the CRC register is 1,

then a 1 is present at the lower input of each of the two exclusive or gates. When

a message bit comes in, the same shifting takes place but the three bits that wind

up in the CRC register have been exclusive ored with binary 011. When all the

message bits have been processed, the CRC holds M mod G.

If the circuit of Figure 142 were used for the CRC calculation, then after

processing the message, r (in this case 3) 0-bits would have to be fed in. Then the

CRC register would have the desired checksum, But, there is a way

to avoid this step with a simple rearrangement of the circuit.

FIGURE 142. Polynomial division circuit for G =

r + 1

x3+x+1 x7+x6+x5+x2+x.

x2 x1 + x0 + Message



Mxr mod G.


Instead of feeding the message in at the right end, feed it in at the left end, r

steps away, as shown in Figure 143. This has the effect of premultiplying the

input message M by xr. But premultiplying and postmultiplying are the same for

polynomials. Therefore, as each message bit comes in, the CRC register contents

are the remainder for the portion of the message processed, as if that portion had

r 0-bits appended.

Figure 144 shows the circuit for the CRC-32 polynomial.


Figure 145 shows a basic implementation of CRC-32 in software. The CRC-32

protocol initializes the CRC register to all 1s, transmits each byte least significant

bit first, and complements the checksum. We assume the message consists of an

integral number of bytes.

To follow Figure 144 as closely as possible, the program uses left shifts.

This requires reversing each message byte and positioning it at the left end of the

32-bit register denoted bytein the program. The word-level reversing pro-

FIGURE 143. CRC circuit for G =

FIGURE 144. CRC circuit for CRC-32.

x2 x1 + x0





31 30 29 28 27 26 + 25 24 23 + 22 + 21 20 19

18 17 16 + 15 14 13 12 + 11 + 10 + 9 8 +

7 + 6 5 + 4 + 3 2 + 1 + 0





gram shown in Figure 71 on page 102 may be used (although this is not very efficient,

because we need to reverse only eight bits).

The code of Figure 145 is shown for illustration only. It can be improved

substantially while still retaining its one-bit-at-a-time character. First, notice that

the eight bits of the reversed byte are used in the inner loops if-statement and

then discarded. Also, the high-order eight bits of crc are not altered in the inner

loop (other than by shifting). Therefore, we can set crc = crc ^ byte ahead of

the inner loop, simplify the if-statement, and omit the left shift of byte at the

bottom of the loop.

The two reversals can be avoided by shifting right instead of left. This

requires reversing the hex constant that represents the CRC-32 polynomial, and

testing the least significant bit of crc. Finally, the if-test can be replaced with

some simple logic, to save branches. The result is shown in Figure 146.

unsigned int crc32(unsigned char *message) {

int i, j;

unsigned int byte, crc;

i = 0;

crc = 0xFFFFFFFF;

while (message[i] != 0) {

byte = message[i]; // Get next byte.

byte = reverse(byte); // 32-bit reversal.

for (j = 0; j <= 7; j++) { // Do eight times.

if ((int)(crc ^ byte) < 0)

crc = (crc << 1) ^ 0x04C11DB7;

else crc = crc << 1;

byte = byte << 1; // Ready next msg bit.


i = i + 1;


return reverse(~crc);


FIGURE 145. Basic CRC-32 algorithm.


It is not unreasonable to unroll the inner loop by the full factor of eight. If this

is done, the program of Figure 146 executes in about 46 instructions per byte of

input message. This includes a load and a branch. (We rely on the compiler to

common the two loads of message[i], and to transform the while-loop so

there is only one branch, at the bottom of the loop.)

Our next version employs table lookup. This is the usual way that CRC-32 is

calculated. Although the programs above work one bit at a time, the table lookup

method (as usually implemented) works one byte at a time. A table of 256 fullword

constants is used.

The inner loop of Figure 146 shifts register crc right eight times, while

doing an exclusive or operation with a constant when the low-order bit of crc is

1. These steps can be replaced by a single right shift of eight positions, followed

by a single exclusive or with a mask which depends on the pattern of 1-bits in the

rightmost eight bits of the crc register.

It turns out that the calculations for setting up the table are the same as those

for computing the CRC of a single byte. The code is shown in Figure 147. To

keep the program self-contained, it includes steps to set up the table on first use. In

practice, these steps would probably be put in a separate function, to keep the

CRC calculation as simple as possible. Alternatively, the table could be defined

by a long sequence of array initialization data. When compiled with GCC to the

basic RISC, the function executes 13 instructions per byte of input. This includes

two loads and one branch instruction.

unsigned int crc32(unsigned char *message) {

int i, j;

unsigned int byte, crc, mask;

i = 0;

crc = 0xFFFFFFFF;

while (message[i] != 0) {

byte = message[i]; // Get next byte.

crc = crc ^ byte;

for (j = 7; j >= 0; j--) { // Do eight times.

mask = -(crc & 1);

crc = (crc >> 1) ^ (0xEDB88320 & mask);


i = i + 1;


return ~crc;


FIGURE 146. Improved bit-at-a-time CRC-32 algorithm.


Faster versions of these programs can be constructed by standard techniques,

but there is nothing dramatic or particularly interesting known to this writer. One

can unroll loops and do careful scheduling of loads that the compiler may not do

automatically. One can load the message string a halfword or a word at a time, to

reduce the number of loads. The table lookup method can process message bytes

two at a time by using a table of size 65536 words. This might make the program

run faster or slower, depending on the size of caches and the penalty for a miss.


[Black] Black, Richard. Web site www.cl.cam.ac.uk/Research/SRG/bluebook/

21/crc/crc.html. University of Cambridge Computer Laboratory Systems

Research Group, February 1994.

unsigned int crc32(unsigned char *message) {

int i, j;

unsigned int byte, crc, mask;

static unsigned int table[256];

/* Set up the table, if necessary. */

if (table[1] == 0) {

for (byte = 0; byte <= 255; byte++) {

crc = byte;

for (j = 7; j >= 0; j--) { // Do eight times.

mask = -(crc & 1);

crc = (crc >> 1) ^ (0xEDB88320 & mask);


table[byte] = crc;



/* Through with table setup, now calculate the CRC. */

i = 0;

crc = 0xFFFFFFFF;

while (message[i] != 0) {

byte = message[i]; // Get next byte.

crc = (crc >> 8) ^ table[(crc ^ byte) & 0xFF];

i = i + 1;


return ~crc;


FIGURE 147. Table lookup CRC algorithm.


[PeBr] Peterson, W. W. and Brown, D.T. Cyclic Codes for Error Detection.In

Proceedings of the IRE, January 1961, 228235.

[Tanen] Tanenbaum, Andrew S. Computer Networks, Second Edition. Prentice

Hall, 1988.

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