Cyclic Redundancy Check and Modulo-2 Division

01 Jul, 2022
CRC or Cyclic Redundancy Check is a method of detecting accidental changes/errors in the communication channel. CRC uses Generator Polynomial which is available on both sender and receiver side. An example generator polynomial is of the form like x3 + x + 1. This generator polynomial represents key 1011. Another example is x2 + 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 dataUse modulo-2 binary division to divide binary data by the key and store remainder of division.Append the remainder at the end of the data to form the encoded data and send the same
The binary data is first augmented by adding k-1 zeros in the end of the data
Use modulo-2 binary division to divide binary data by the key and store remainder of division.
Append the remainder at the end of the data to form the encoded data and send the same
 Receiver Side (Check if there are errors introduced in transmission)Perform modulo-2 division again and if the 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.
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.
Illustration:Example 1 (No error in transmission): 
Data word to be sent - 100100
Key - 1101 [ Or generator polynomial x3 + x2 + 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 an error in transmission media
Code word received at the receiver side - 100000001
Since the remainder is not all zeroes, the erroris detected at the receiver side. 
 ImplementationBelow implementation for generating code word from given binary data and key.
C++
Python3
Javascript
#include<bits/stdc++.h>using namespace std;  // Returns XOR of 'a' and 'b'// (both of same length)string xor1(string a, string b){         // Initialize result    string result = "";         int n = b.length();         // Traverse all bits, if bits are    // same, then XOR is 0, else 1    for(int i = 1; i < n; i++)    {        if (a[i] == b[i])            result += "0";        else            result += "1";    }    return result;} // Performs Modulo-2 divisionstring mod2div(string divident, string divisor){         // Number of bits to be XORed at a time.    int pick = divisor.length();         // Slicing the divident to appropriate    // length for particular step    string tmp = divident.substr(0, pick);         int n = divident.length();         while (pick < n)    {        if (tmp[0] == '1')                     // Replace the divident by the result            // of XOR and pull 1 bit down            tmp = xor1(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 = xor1(std::string(pick, '0'), 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 = xor1(divisor, tmp);    else        tmp = xor1(std::string(pick, '0'), tmp);             return tmp;} // Function used at the sender side to encode// data by appending remainder of modular division// at the end of data.void encodeData(string data, string key){    int l_key = key.length();         // Appends n-1 zeroes at end of data    string appended_data = (data +                            std::string(                                l_key - 1, '0'));         string remainder = mod2div(appended_data, key);         // Append remainder in the original data    string codeword = data + remainder;    cout << "Remainder : "         << remainder << "\n";    cout << "Encoded Data (Data + Remainder) :"         << codeword << "\n";}  // Driver codeint main(){    string data = "100100";    string key = "1101";         encodeData(data, key);         return 0;} // This code is contributed by MuskanKalra1
# 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 divisiondef mod2div(dividend, divisor):     # Number of bits to be XORed at a time.    pick = len(divisor)     # Slicing the dividend to appropriate    # length for particular step    tmp = dividend[0 : pick]     while pick < len(dividend):         if tmp[0] == '1':             # replace the dividend by the result            # of XOR and pull 1 bit down            tmp = xor(divisor, tmp) + dividend[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) + dividend[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 division# 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 codedata = "100100"key = "1101"encodeData(data, key)
<script>// A JavaScript program for generating code// word from given binary data and key. // Returns XOR of 'a' and 'b'// (both of same length)function xor1(a, b){         // Initialize result        let result = "";         let n = b.length;         // Traverse all bits, if bits are        // same, then XOR is 0, else 1        for (let i = 1; i < n; i++) {                if (a[i] == b[i]) {                        result += "0";                }                else {                        result += "1";                }        }        return result;} // Performs Modulo-2 divisionfunction mod2div(divident, divisor) {         // Number of bits to be XORed at a time.        let pick = divisor.length;         // Slicing the divident to appropriate        // length for particular step        let tmp = divident.substr(0, pick);         let n = divident.length;         while (pick < n)        {                if (tmp[0] == '1')                {                                         // Replace the divident by the result                        // of XOR and pull 1 bit down                        tmp = xor1(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.                        let str = "";                        for (let i = 0; i < pick; i++) {                                str = str.concat('0');                        }                        tmp = xor1(str, 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 = xor1(divisor, tmp);        }        else {                tmp = xor1(string(pick, '0'), tmp);        }        return tmp;} // Function used at the sender side to encode// data by appending remainder of modular division// at the end of data.function encodeData(data, key) {        let l_key = key.length;         // Appends n-1 zeroes at end of data        let str = "";        for (let i = 0; i < l_key - 1; i++) {                str = str.concat('0');        }        console.log(str);        let appended_data = data.concat(str);         let remainder = mod2div(appended_data, key);         // Append remainder in the original data        let codeword = data + remainder;         // Adding the print statements        document.write("Remainder : ", remainder);        document.write("Encoded Data (Data + Remainder) :", codeword);} // Driver code{        let data = "100100";        let key = "1101";         encodeData(data, key);} // This code is contributed by Gautam goel (gautamgoel962)</script>
Output:
Remainder :  001
Encoded Data (Data + Remainder) :  100100001
Time Complexity: O(n)
Auxiliary Space: O(n)
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:
C++
Java
Python3
C#
Javascript
// C++ Program to generate CRC codeword#include<stdio.h>#include<iostream>#include<math.h> using namespace std; // function to convert integer to binary stringstring toBin(long long int num){    string bin = "";    while (num){        if (num & 1)            bin = "1" + bin;        else            bin = "0" + bin;        num = num>>1;    }    return bin;} // function to convert binary string to decimallong long int toDec(string bin){    long long int num = 0;    for (int i=0; i<bin.length(); i++){        if (bin.at(i)=='1')            num += 1 << (bin.length() - i - 1);    }    return num;} // function to compute CRC and codewordvoid CRC(string dataword, string generator){    int l_gen = generator.length();    long long int gen = toDec(generator);     long long int dword = toDec(dataword);      // append 0s to dividend    long long int dividend = dword << (l_gen-1);           // shft specifies the no. of least    // significant bits not being XORed    int shft = (int) ceill(log2l(dividend+1)) - l_gen;     long long int rem;     while ((dividend >= gen) || (shft >= 0)){         // bitwise XOR the MSBs of dividend with generator        // replace the operated MSBs from the dividend with        // remainder generated        rem = (dividend >> shft) ^ gen;                       dividend = (dividend & ((1 << shft) - 1)) | (rem << shft);         // change shft variable        shft = (int) ceill(log2l(dividend + 1)) - l_gen;    }     // finally, AND the initial dividend with the remainder (=dividend)    long long int codeword = (dword << (l_gen - 1)) | dividend;    cout << "Remainder: " << toBin(dividend) << endl;    cout << "Codeword : " << toBin(codeword) << endl;} int main(){    string dataword, generator;    dataword = "10011101";    generator = "1001";    CRC(dataword, generator);    return 0;}
// Java Program to generate CRC codewordclass GFG {   // function to convert integer to binary string  static String toBin(int num)  {    String bin = "";    while (num > 0) {      if ((num & 1) != 0)        bin = "1" + bin;      else        bin = "0" + bin;      num = num >> 1;    }    return bin;  }   // function to convert binary string to decimal  static int toDec(String bin)  {    int num = 0;    for (int i = 0; i < bin.length(); i++) {      if (bin.charAt(i) == '1')        num += 1 << (bin.length() - i - 1);    }    return num;  }   // function to compute CRC and codeword  static void CRC(String dataword, String generator)  {    int l_gen = generator.length();    int gen = toDec(generator);     int dword = toDec(dataword);     // append 0s to dividend    int dividend = dword << (l_gen - 1);     // shft specifies the no. of least    // significant bits not being XORed    int shft = (int)Math.ceil(Math.log(dividend + 1)                              / Math.log(2))      - l_gen;    int rem;     while ((dividend >= gen) || (shft >= 0)) {       // bitwise XOR the MSBs of dividend with      // generator replace the operated MSBs from the      // dividend with remainder generated      rem = (dividend >> shft) ^ gen;      dividend = (dividend & ((1 << shft) - 1))        | (rem << shft);       // change shft variable      shft = (int)Math.ceil(Math.log(dividend + 1)                            / Math.log(2))        - l_gen;    }     // finally, AND the initial dividend with the    // remainder (=dividend)    int codeword = (dword << (l_gen - 1)) | dividend;    System.out.println("Remainder: " + toBin(dividend));    System.out.println("Codeword : " + toBin(codeword));  }   // Driver Code  public static void main(String[] args)  {    String dataword, generator;    dataword = "10011101";    generator = "1001";    CRC(dataword, generator);  }} // This code is contributed by phasing17
# Python3 program to generate CRC codewordfrom math import log, ceil def CRC(dataword, generator):    dword = int(dataword, 2)    l_gen = len(generator)     # append 0s to dividend    dividend = dword << (l_gen - 1)        # shft specifies the no. of least significant    # bits not being XORed    shft = ceil(log(dividend + 1, 2)) - l_gen         # ceil(log(dividend+1 , 2)) is the no. of binary    # digits in dividend    generator = int(generator, 2)     while dividend >= generator or shft >= 0:         # bitwise XOR the MSBs of dividend with generator        # replace the operated MSBs from the dividend with        # remainder generated        rem = (dividend >> shft) ^ generator           dividend = (dividend & ((1 << shft) - 1)) | (rem << shft)                 # change shft variable        shft = ceil(log(dividend+1, 2)) - l_gen     # finally, AND the initial dividend with the remainder (=dividend)    codeword = dword << (l_gen-1)|dividend    print("Remainder:", bin(dividend).lstrip("-0b"))    print("Codeword :", bin(codeword).lstrip("-0b")) # Driver codedataword = "10011101"generator = "1001"CRC(dataword, generator)
// C# Program to generate CRC codewordusing System; class GFG {     // function to convert integer to binary string    static string toBin(int num)    {        string bin = "";        while (num > 0) {            if ((num & 1) != 0)                bin = "1" + bin;            else                bin = "0" + bin;            num = num >> 1;        }        return bin;    }     // function to convert binary string to decimal    static int toDec(string bin)    {        int num = 0;        for (int i = 0; i < bin.Length; i++) {            if (bin[i] == '1')                num += 1 << (bin.Length - i - 1);        }        return num;    }     // function to compute CRC and codeword    static void CRC(string dataword, string generator)    {        int l_gen = generator.Length;        int gen = toDec(generator);         int dword = toDec(dataword);         // append 0s to dividend        int dividend = dword << (l_gen - 1);         // shft specifies the no. of least        // significant bits not being XORed        int shft = (int)Math.Ceiling(Math.Log(dividend + 1)                                     / Math.Log(2))                   - l_gen;        int rem = (dividend >> shft) ^ gen;         while ((dividend >= gen) || (shft >= 0)) {             // bitwise XOR the MSBs of dividend with            // generator replace the operated MSBs from the            // dividend with remainder generated            rem = (dividend >> shft) ^ gen;            dividend = (dividend & ((1 << shft) - 1))                       | (rem << shft);             // change shft variable            shft = (int)Math.Ceiling(Math.Log(dividend + 1)                                     / Math.Log(2))                   - l_gen;        }         // finally, AND the initial dividend with the        // remainder (=dividend)        int codeword = (dword << (l_gen - 1)) | dividend;        Console.WriteLine("Remainder: " + toBin(dividend));        Console.WriteLine("Codeword : " + toBin(codeword));    }     // Driver Code    public static void Main(string[] args)    {        string dataword, generator;        dataword = "10011101";        generator = "1001";        CRC(dataword, generator);    }} // This code is contributed by phasing17
// JavaScript Program to generate CRC codeword // function to convert integer to binary stringfunction toBin(num){    var bin = "";    while (num){        if (num & 1)            bin = "1" + bin;        else            bin = "0" + bin;        num = num>>1;    }    return bin;} // function to convert binary string to decimalfunction toDec(bin){    var num = 0;    for (var i=0; i<bin.length; i++){        if (bin[i]=='1')            num += 1 << (bin.length - i - 1);    }    return num;} // function to compute CRC and codewordfunction CRC(dataword, generator){    var l_gen = generator.length;    var gen = toDec(generator);     var dword = toDec(dataword);      // append 0s to dividend    var dividend = dword << (l_gen-1);           // shft specifies the no. of least    // significant bits not being XORed    var shft = Math.ceil(Math.log2(dividend+1)) - l_gen;     var rem;     while ((dividend >= gen) || (shft >= 0)){         // bitwise XOR the MSBs of dividend with generator        // replace the operated MSBs from the dividend with        // remainder generated        rem = (dividend >> shft) ^ gen;                       dividend = (dividend & ((1 << shft) - 1)) | (rem << shft);         // change shft variable        shft = Math.ceil(Math.log2(dividend + 1)) - l_gen;    }     // finally, AND the initial dividend with the remainder (=dividend)    var codeword = (dword << (l_gen - 1)) | dividend;    console.log( "Remainder:", toBin(dividend));    console.log("Codeword :", toBin(codeword));} //Driver codevar dataword = "10011101";var generator = "1001";CRC(dataword, generator);  //This code is contributed by phasing17
Remainder: 100
Codeword : 10011101100
 
Time Complexity: O(n)
Auxiliary Space: O(n)
References:https://en.wikipedia.org/wiki/Cyclic_redundancy_check
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 review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
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