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 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 error
is detected at the receiver side.
Implementation:
Below implementation for generating code word from given binary data and key.
C++
#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 dividend, string divisor){ // Number of bits to be XORed at a time. int pick = divisor.length(); // Slicing the dividend to appropriate // length for particular step string tmp = dividend.substr(0, pick); int n = dividend.length(); while (pick < n) { if (tmp[0] == '1') // Replace the dividend by the result // of XOR and pull 1 bit down tmp = xor1(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 = xor1(std::string(pick, '0'), 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 = 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";}// checking if the message received by receiver is correct// or not. If the remainder is all 0 then it is correct,// else wrong.void receiver(string data, string key){ string currxor = mod2div(data.substr(0, key.size()), key); int curr = key.size(); while (curr != data.size()) { if (currxor.size() != key.size()) { currxor.push_back(data[curr++]); } else { currxor = mod2div(currxor, key); } } if (currxor.size() == key.size()) { currxor = mod2div(currxor, key); } if (currxor.find('1') != string::npos) { cout << "there is some error in data" << endl; } else { cout << "correct message received" << endl; }}// Driver codeint main(){ string data = "100100"; string key = "1101"; cout << "Sender side..." << endl; encodeData(data, key); cout << "\nReceiver side..." << endl; receiver(data+mod2div(data+std::string(key.size() - 1, '0'),key), key); return 0;}// This code is contributed by MuskanKalra1 , Mayank Sharma |
Java
// Java code to implement the approachimport java.util.Arrays;class Program { // Returns XOR of 'a' and 'b' // (both of same length) static String Xor(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.charAt(i) == b.charAt(i)) result += "0"; else result += "1"; } return result; } // Performs Modulo-2 division static String Mod2Div(String dividend, String divisor) { // Number of bits to be XORed at a time. int pick = divisor.length(); // Slicing the dividend to appropriate // length for particular step String tmp = dividend.substring(0, pick); int n = dividend.length(); while (pick < n) { if (tmp.charAt(0) == '1') // Replace the dividend by the result // of XOR and pull 1 bit down tmp = Xor(divisor, tmp) + dividend.charAt(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(new String(new char[pick]) .replace("\0", "0"), tmp) + dividend.charAt(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.charAt(0) == '1') tmp = Xor(divisor, tmp); else tmp = Xor(new String(new char[pick]) .replace("\0", "0"), tmp); return tmp; } // Function used at the sender side to encode // data by appending remainder of modular division // at the end of data. static void EncodeData(String data, String key) { int l_key = key.length(); // Appends n-1 zeroes at end of data String appended_data = (data + new String(new char[l_key - 1]) .replace("\0", "0")); String remainder = Mod2Div(appended_data, key); // Append remainder in the original data String codeword = data + remainder; System.out.println("Remainder : " + remainder); System.out.println( "Encoded Data (Data + Remainder) :" + codeword + "\n"); } // checking if the message received by receiver is // correct or not. If the remainder is all 0 then it is // correct, else wrong. static void Receiver(String data, String key) { String currxor = Mod2Div(data.substring(0, key.length()), key); int curr = key.length(); while (curr != data.length()) { if (currxor.length() != key.length()) { currxor += data.charAt(curr++); } else { currxor = Mod2Div(currxor, key); } } if (currxor.length() == key.length()) { currxor = Mod2Div(currxor, key); } if (currxor.contains("1")) { System.out.println( "there is some error in data"); } else { System.out.println("correct message received"); } } // Driver code public static void main(String[] args) { String data = "100100"; String key = "1101"; System.out.println("\nSender side..."); EncodeData(data, key); System.out.println("Receiver side..."); Receiver(data+Mod2Div(data+new String(new char[key.length() - 1]) .replace("\0", "0"),key),key); }}// This code is contributed by phasing17 |
Python3
# 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) |
C#
using System;using System.Linq;class Program { // Returns XOR of 'a' and 'b' // (both of same length) static string Xor(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 division static string Mod2Div(string dividend, string divisor) { // Number of bits to be XORed at a time. int pick = divisor.Length; // Slicing the dividend to appropriate // length for particular step string tmp = dividend.Substring(0, pick); int n = dividend.Length; while (pick < n) { 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(new string('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(new string('0', pick), tmp); return tmp; } // Function used at the sender side to encode // data by appending remainder of modular division // at the end of data. static string EncodeData(string data, string key) { int l_key = key.Length; // Appends n-1 zeroes at end of data string appended_data = (data + new string('0', l_key - 1)); string remainder = Mod2Div(appended_data, key); // Append remainder in the original data string codeword = data + remainder; Console.WriteLine("Remainder : " + remainder); Console.WriteLine( "Encoded Data (Data + Remainder) :" + codeword + "\n"); return codeword; } // checking if the message received by receiver is // correct or not. If the remainder is all 0 then it is // correct, else wrong. static void Receiver(string data, string key) { string currxor = Mod2Div(data.Substring(0, key.Length), key); int curr = key.Length; while (curr != data.Length) { if (currxor.Length != key.Length) { currxor += data[curr++]; } else { currxor = Mod2Div(currxor, key); } } if (currxor.Length == key.Length) { currxor = Mod2Div(currxor, key); } if (currxor.Contains('1')) { Console.WriteLine( "there is some error in data"); } else { Console.WriteLine("correct message received"); } } // Driver code static void Main(string[] args) { string data = "100100"; string key = "1101"; Console.WriteLine("Sender side..."); EncodeData(data, key); Console.WriteLine("Receiver side..."); Receiver(data+Mod2Div(data + new string('0', key.Length - 1),key), key); }}// This code is contributed by phasing17. |
Javascript
<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(dividend, divisor) { // Number of bits to be XORed at a time. let pick = divisor.length; // Slicing the dividend to appropriate // length for particular step let tmp = dividend.substr(0, pick); let n = dividend.length; while (pick < n) { if (tmp[0] == '1') { // Replace the dividend by the result // of XOR and pull 1 bit down tmp = xor1(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. let str = ""; for (let i = 0; i < pick; i++) { str = str.concat('0'); } tmp = xor1(str, 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 = 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
Sender side... Remainder : 001 Encoded Data (Data + Remainder) :100100001 Receiver side... correct message received
Output:
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++
// C++ Program to generate CRC codeword#include <iostream>#include <math.h>#include <stdio.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
// 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
# Python3 program to generate CRC codewordfrom math import log, ceildef 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#
// 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
// 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 |
Output
Remainder: 100 Codeword : 10011101100
Time Complexity: O(n)
Auxiliary Space: O(n)
References:
https://en.wikipedia.org/wiki/Cyclic_redundancy_check
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