# Cyclic Redundancy Check and Modulo-2 Division

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.
- Modular Division
- Trick for modular division ( (x1 * x2 .... xn) / b ) mod (m)
- Calculate 7n/8 without using division and multiplication operators
- Fast average of two numbers without division
- Compute modulus division by a power-of-2-number
- Divide two integers without using multiplication, division and mod operator
- Check whether K-th bit is set or not
- Check whether a given number is even or odd
- Check if one of the numbers is one's complement of the other
- Check whether the number has only first and last bits set
- Check whether bitwise OR of N numbers is Even or Odd
- Check whether the number has only first and last bits set | Set 2
- Check whether the bit at given position is set or unset
- Check whether product of 'n' numbers is even or odd
- Check if a given number is sparse or not

.

**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) ` |

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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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