CRC or Cyclic Redundancy Check is a method of detecting accidental changes/errors in communication channel.

CRC uses **Generator Polynomial **which is available on both sender and receiver side. An example generator polynomial is of the form like x^{3} + x + 1. This generator polynomial represents key 1011. Another example is x^{2} + 1 that represents key 101.

n : Number of bits in data to be sent from sender side. k : Number of bits in the key obtained from generator polynomial.

Sender Side (Generation of Encoded Data from Data and Generator Polynomial (or Key)):

- The binary data is first augmented by adding k-1 zeros in the end of the data
- Use
to divide binary data by the key and store remainder of division.**modulo-2 binary division** - Append the remainder at the end of the data to form the encoded data and send the same
- In each step, a copy of the divisor (or data) is XORed with the k bits of the dividend (or key).
- The result of the XOR operation (remainder) is (n-1) bits, which is used for the next step after 1 extra bit is pulled down to make it n bits long.
- When there are no bits left to pull down, we have a result. The (n-1)-bit remainder which is appended at the sender side.

.

**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.e. check word and the code word.

**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.

**Illustration:**

**Example 1 (No error in transmission): **

Data word to be sent - 100100 Key - 1101 [ Or generator polynomial x^{3}+ x + 1] Sender Side: Therefore, the remainder is 001 and hence the encoded data sent is 100100001. Receiver Side: Code word received at the receiver side 100100001 Therefore, the remainder is all zeros. Hence, the data received has no error.

**Example 2: (Error in transmission)**

Data word to be sent - 100100 Key - 1101 Sender Side: Therefore, the remainder is 001 and hence the code word sent is 100100001. Receiver Side Let there be error in transmission media Code word received at the receiver side - 100000001Since the remainder is not all zeroes, the error is detected at the receiver side.

**Implementation**

Below is Python implementation for generating code word from given binary data and key.

# Returns XOR of 'a' and 'b' # (both of same length) def xor(a, b): # initialize result result = [] # Traverse all bits, if bits are # same, then XOR is 0, else 1 for i in range(1, len(b)): if a[i] == b[i]: result.append('0') else: result.append('1') return ''.join(result) # Performs Modulo-2 division def mod2div(divident, divisor): # Number of bits to be XORed at a time. pick = len(divisor) # Slicing the divident to appropriate # length for particular step tmp = divident[0 : pick] while pick < len(divident): if tmp[0] == '1': # replace the divident by the result # of XOR and pull 1 bit down tmp = xor(divisor, tmp) + divident[pick] else: # If leftmost bit is '0' # If the leftmost bit of the dividend (or the # part used in each step) is 0, the step cannot # use the regular divisor; we need to use an # all-0s divisor. tmp = xor('0'*pick, tmp) + divident[pick] # increment pick to move further pick += 1 # For the last n bits, we have to carry it out # normally as increased value of pick will cause # Index Out of Bounds. if tmp[0] == '1': tmp = xor(divisor, tmp) else: tmp = xor('0'*pick, tmp) checkword = tmp return checkword # Function used at the sender side to encode # data by appending remainder of modular divison # at the end of data. def encodeData(data, key): l_key = len(key) # Appends n-1 zeroes at end of data appended_data = data + '0'*(l_key-1) remainder = mod2div(appended_data, key) # Append remainder in the original data codeword = data + remainder print("Remainder : ", remainder) print("Encoded Data (Data + Remainder) : ", codeword) # Driver code data = "100100" key = "1101" encodeData(data, key)

Output:

Remainder : 001 Encoded Data (Data + Remainder) : 100100001

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)

**References:**

https://en.wikipedia.org/wiki/Cyclic_redundancy_check

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