# Crc calculation example

## Cyclic Redundancy Check Tutorial With Example

Cyclic Redundancy Check CRC An error detection mechanism in which a special number is appended to a block of data in order to detect any changes introduced during storage or transmission. The CRe is recalculated on retrieval or reception and compared to the value originally transmitted, which can reveal certain types of error. For example, a single corrupted bit in the data results in a one-bit change in the calculated CRC, but multiple corrupt bits may cancel each other out.

Rather it is based on binary division. This remainder is called CRC. It should have exactly one less bit than divisor. Appending the CRC to the end of the data unit should result in the bit sequence which is exactly divisible by the divisor. A string of n as is appended to the data unit.

The newly formed data unit i. Now, string of n Os appended to data unit is replaced by the CRC remainder which is also of n bit.

If the remainder of division is zero, receiver assumes that there is no error in data and it accepts it. If remainder is non-zero then there is an error in data and receiver rejects it. The procedure given below is used: 1.

Error Detection and Correction 2: Cyclic Redundancy Check

String of 3 zeroes is appended to as divisor is of 4 bits. Now newly formed data is Data unit is divided by During this process of division, whenever the leftmost bit of dividend or remainder is 0, we use a string of Os of same length as divisor. Thus in this case divisor is replaced by At the receiver side, data received is This data is again divided by a divisor The remainder obtained is ; it means there is no error. Figure shows the polynomial where all the terms with zero coefficient are removed and x J is replaced by x and XO by 1.

For example here a 6-bit pattern is replaced by 3 terms.Whenever digital data is stored or interfaced, data corruption might occur. Since the beginning of computer science, people have been thinking of ways to deal with this type of problem.

For serial data they came up with the solution to attach a parity bit to each sent byte. This simple detection mechanism works if an odd number of bits in a byte changes, but an even number of false bits in one byte will not be detected by the parity check. To overcome this problem people have searched for mathematical sound mechanisms to detect multiple false bits.

The CRC calculation or cyclic redundancy check was the result of this. Nowadays CRC calculations are used in all types of communications. All packets sent over a network connection are checked with a CRC.

Also each data block on your hard-disk has a CRC value attached to it. Modern computer world cannot do without these CRC calculation. The answer is simple, they are powerful, detect many types of errors and are extremely fast to calculate especially when dedicated hardware chips are used.

One might think, that using a checksum can replace proper CRC calculations. It is certainly easier to calculate a checksum, but checksums do not find all errors. Lets take an example string and calculate a one byte checksum. The one byte checksum of this array can be calculated by adding all values, than dividing it by and keeping the remainder.

You can use the calculator above to check this result. In this example we have used a one byte long checksum which gives us different values. Using a two byte checksum will result in 65, possible different checksum values and when a four byte value is used there are more than four billion possible values.

We might conclude that with a four byte checksum the chance that we accidentally do not detect an error is less than 1 to 4 billion. Seems rather good, but this is only theory. In practice, bits do not change purely random during communications.

They often fail in bursts, or due to electrical spikes. The checksum for this new string is stillbut the result is obviously wrong, only after two bits changed. Even if we had used a four byte long checksum we would not have detected this transmission error. The idea behind a check value calculation is simple.

Use a function F bval,cval that inputs one data byte and a check value and outputs a recalculated check value. In fact checksum calculations as described above can be defined in this way. Our one byte checksum example could have been calculated with the following function in C language that we call repeatedly for each byte in the input string.

The initial value for cval is 0. The idea behind CRC calculation is to look at the data as one large binary number. This number is divided by a certain value and the remainder of the calculation is called the CRC.This article is the result of the fact that I found finally time to deal with CRC.

After reading Wikipedia and some other articles, I had the feeling to not really understand completely in depth. Therefore I decided to write this article, trying to cover all topics I had difficulties with. And this in exactly the same order I concerned myself with CRC. Please note that this article is not indented to be a full comprehensive CRC guide explaining all details - it should be used as an additional, practical oriented note to all general explanations on the web.

Here's the outline:. A checksum, calculated by CRC, is attached to the data to help the receiver to detect such errors. Refer also to  for a short or to  for a very detailed CRC introduction.

CRC is based on division. The actual input data is interpreted as one long binary bit stream divident which is divided by another fixed binary number divisor. The remainder of this division is the checksum value.

However, reality is a bit more complicated. The binary numbers divident and divisor are not treated as normal integer values, but as binary polyonimals where the actual bits are used as coefficients. XOR truth table. Division of polynomials differs from integer division. For manual calculation, n zero bits are appended to the input data before actual CRC calculation polynomial division is computed. Let's perform an example CRC computation:. The divisor has 9 bits therefore this is a CRC-8 polynomialso append 8 zero bits to the input pattern.

Align the leading '1' of the divisor with the first '1' of the divident and perform a step-by-step school-like division, using XOR operation for each bit:. The remainder is the CRC value which is transmitted along with the input data to the receiver.

The receiver can either verify the received data by computing the CRC and compare the calculated CRC value with the received one. Or, more commonly used, the CRC value is directly appened to the actual data.

Let's do verification according the latter case:. The generator polynomial is statically defined by the used CRC algorithm and so it's known by the receiver.

So we have seen how to calculate the CRC checksum value manually, but how can it be implemented? The computation has to be performed step-by-step and here the concept of a shift register comes into play. A shift register has a fixed width and can shift it's content by one bit, removing the bit at the right or left border and shift in a new bit at the freed position.

The bit position of the least significant bit is free: here the next bit of the input stream is shifted in. CRC-8 register initialized with 0. Left-Shift register by one position. MSB is 0, so nothing do happen, shift in next byte of input stream. Repeat those steps. Left-Shift register. The shift register contains now the CRC value which is 0x0F. This chapter handles different algorithms and their implementations in C for calculating CRC-8 checksum values.Printable PDF.

Unfortunately, the modulo-2 arithmetic used to compute CRCs doesn't map easily into software. Generally speaking, CRCs are most efficiently calculated in dedicated hardware. I'm going to complete my 3-part discussion of checksums by showing you how to implement a CRC in C.

I'll start with a naive implementation and gradually improve the efficiency of the code as I go along.

For most software engineers, the overwhelmingly confusing thing about CRCs is their implementation. Knowing that all CRC algorithms are simply long division algorithms in disguise doesn't help. Modulo-2 binary division doesn't map particularly well to the instruction sets of off-the-shelf processors. For one thing, generally no registers are available to hold the very long bit sequence that is the numerator.

For another, modulo-2 binary division is not the same as ordinary division. So even if your processor has a division instruction, you won't be able to use it. Before writing even one line of code, let's first examine the mechanics of modulo-2 binary division. We'll use the example in Figure 1 to guide us. The number to be divided is the message augmented with zeros at the end. The number of zero bits added to the message is the same as the width of the checksum what I call c ; in this case four bits were added.

What's most important to notice at this point is that we never use any of the information in the quotient, either during or after computing the CRC. So we won't actually need to track the quotient in our software implementation. Also note here that the result of each XOR with the generator polynomial is a remainder that has zero in its most significant bit. So we never lose any information when the next message bit is shifted into the remainder.

Listing 1 contains a naive software implementation of the CRC computation just described. It simply attempts to implement that algorithm as it was described above for this one particular generator polynomial. Even though the unnecessary steps have been eliminated, it's extremely inefficient.

Multiple C statements at least the decrement and compare, binary AND, test for zero, and left shift operations must be executed for each bit in the message. Given that this particular message is only eight bits long, that might not seem too costly. But what if the message contains several hundred bytes, as is typically the case in a real-world application?

## Computation of cyclic redundancy checks

You don't want to execute dozens of processor opcodes for each byte of input data. Before we start making this more efficient, the first thing to do is to clean this naive routine up a bit. In particular, let's start making some assumptions about the applications in which it will most likely be used.

First, let's assume that our CRCs are always going to be 8-,or bit numbers. In other words, that the remainder can be manipulated easily in software. That means that the generator polynomials will be 9, 17, or 33 bits wide, respectively. At first it seems we may be stuck with unnatural sizes and will need special register combinations, but remember these two facts:. Since we already have the information in the uppermost bit and we don't need it for the XOR, the polynomial can also be stored in an 8-,or bit register.

We can simply discard the most significant bit.CRC uses Generator Polynomial which is available on both sender and receiver side. This generator polynomial represents key Receiver Side Check if there are errors introduced in transmission Perform modulo-2 division again and if remainder is 0, then there are no errors.

In this article we will focus only on finding the remainder i. Modulo 2 Division: The process of modulo-2 binary division is the same as the familiar division process we use for decimal numbers.

Just that instead of subtraction, we use XOR here. Since the remainder is not all zeroes, the error is detected at the receiver side. Note that CRC is mainly designed and used to protect against common of errors on communication channels and NOT suitable protection against intentional alteration of data See reasons here. Implementation using Bit Manipulation: CRC codeword generation can also be done using bit manipulation methods as follows: Python3.

This article is contributed by Jay Patel. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute geeksforgeeks. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Writing code in comment? Please use ide. Returns XOR of 'a' and 'b'. Traverse all bits, if bits are. Performs Modulo-2 division. Number of bits to be XORed at a time. Slicing the divident to appropriate. If the leftmost bit of the dividend or the. For the last n bits, we have to carry it out. Index Out of Bounds.

Function used at the sender side to encode. Appends n-1 zeroes at end of data. Append remainder in the original data. Python program to generate CRC codeword. CRC dataword, generator. Load Comments.Med Bouftira is a 23 year-old university student, graduated with a bachelor degree in English studies in Besides, he is interested in Computer Networks since he was a student in high school.

He is currently in the process of authoring a book regarding CCNA. Note : the previous explanation is adequate for CCNA candidates! After laying the ground for a little harder CRC, it is time to delve more profoundly in how CRC is calculated and how errors are detected by using binary system, which is the actual language of computers, as well as its representation in mathematical forms that are designed for human beings.

CRC, error detection mechanism, is based on binary division, and it is often represented in algebraic polynomial for the reasons that it is shorter than the act of writing zeroes and ones to prove the concept mathematically, and also the capacity of being able to be represented in binary pattern. Let us first explain how to transform from polynomial to binary representation by considering this example :.

Rule : The power of each term shows the position of the bit; the coefficient shows the value of the bit. To make this rule more easier to understand, follow these steps. The transformation above has been done by first looking at the left where we have X8, then we start counting.

This figure summaries all that. Now, after knowing how to transform from polynomial to binary, let us employ all that in a CRC operation. To calculate CRC value we need a generator value along with the message to be transmitted. However, the division here is a litle bit different.

Consider this example. As written in the third statement, n is one less than the number of bits in the CRC generator. So we take CRC generator which is and count how many bits there are, and detract one bit. The number of bits is 6 and by detracting 1 bits, it becomes 5 bits, which is the final number of zeroes to be appended at the end of data unit. The whole data unit becomes Third, we divide the newly formed data unit by the divisor CRC generator using binary division.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 algorithmsstarting 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. 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 2decimal 87 10or hexadecimal 57 The byte value 57 16 can be transmitted in two different orders, depending on the bit ordering convention used.

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. 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.

### 10: Ethernet - How to calculate CRC (Cyclic Redundancy Check) ? part 14

Converting to a hexadecimal number using the convention that the highest power of x is the msbit; this is A2 Converting to hexadecimal using the convention that the highest power of x is the lsbit, this is 19 Writing out the full message at each step, as done in the example above, is very tedious.

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. This code has two disadvantages. More significantly, it requires the bitString to be padded with n zero bits. 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 associativeit 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:. 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. This is the reason that you will usually see CRC polynomials written in binary with the leading coefficient omitted.