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Hill Cipher
  • Difficulty Level : Basic
  • Last Updated : 14 Oct, 2019

Hill cipher is a polygraphic substitution cipher based on linear algebra.Each letter is represented by a number modulo 26. Often the simple scheme A = 0, B = 1, …, Z = 25 is used, but this is not an essential feature of the cipher. To encrypt a message, each block of n letters (considered as an n-component vector) is multiplied by an invertible n × n matrix, against modulus 26. To decrypt the message, each block is multiplied by the inverse of the matrix used for encryption.

The matrix used for encryption is the cipher key, and it should be chosen randomly from the set of invertible n × n matrices (modulo 26).

Examples:

Input  : Plaintext: ACT
         Key: GYBNQKURP
Output : Ciphertext: POH
Input  : Plaintext: GFG
         Key: HILLMAGIC 
Output : Ciphertext: SWK

Encryption

We have to encrypt the message ‘ACT’ (n=3).The key is ‘GYBNQKURP’ which can be written as the nxn matrix:
Cipherkey
The message ‘ACT’ is written as vector:
message vector
The enciphered vector is given as:
enciphered vector
which corresponds to ciphertext of ‘POH’



Decryption

To decrypt the message, we turn the ciphertext back into a vector, then simply multiply by the inverse matrix of the key matrix (IFKVIVVMI in letters).The inverse of the matrix used in the previous example is:
inverse matrix
For the previous Ciphertext ‘POH’:
Decrypt
which gives us back ‘ACT’.
Assume that all the alphabets are in upper case.
Below is the implementation of the above idea for n=3.

C++




// C++ code to implement Hill Cipher
#include <iostream>
using namespace std;
  
// Following function generates the
//  key matrix for the key string
void getKeyMatrix(string key, int keyMatrix[][3])
{
    int k = 0;
    for (int i = 0; i < 3; i++) 
    {
        for (int j = 0; j < 3; j++) 
        {
            keyMatrix[i][j] = (key[k]) % 65;
            k++;
        }
    }
}
  
// Following function encrypts the message
void encrypt(int cipherMatrix[][1],
             int keyMatrix[][3], 
             int messageVector[][1])
{
    int x, i, j;
    for (i = 0; i < 3; i++) 
    {
        for (j = 0; j < 1; j++)
        {
            cipherMatrix[i][j] = 0;
           
             for (x = 0; x < 3; x++)
            {
                cipherMatrix[i][j] += 
                     keyMatrix[i][x] * messageVector[x][j];
            }
          
            cipherMatrix[i][j] = cipherMatrix[i][j] % 26;
        }
    }
}
  
// Function to implement Hill Cipher
void HillCipher(string message, string key)
{
    // Get key matrix from the key string
    int keyMatrix[3][3];
    getKeyMatrix(key, keyMatrix);
  
    int messageVector[3][1];
  
    // Generate vector for the message
    for (int i = 0; i < 3; i++)
        messageVector[i][0] = (message[i]) % 65;
  
    int cipherMatrix[3][1];
  
    // Following function generates
    // the encrypted vector
    encrypt(cipherMatrix, keyMatrix, messageVector);
  
    string CipherText;
  
    // Generate the encrypted text from 
    // the encrypted vector
    for (int i = 0; i < 3; i++)
        CipherText += cipherMatrix[i][0] + 65;
  
    // Finally print the ciphertext
    cout << " Ciphertext:" << CipherText;
}
  
// Driver function for above code
int main()
{
    // Get the message to be encrypted
    string message = "ACT";
  
    // Get the key
    string key = "GYBNQKURP";
  
    HillCipher(message, key);
  
    return 0;
}


Java




// Java code to implement Hill Cipher
class GFG 
{
  
// Following function generates the
// key matrix for the key string
static void getKeyMatrix(String key, int keyMatrix[][])
{
    int k = 0;
    for (int i = 0; i < 3; i++) 
    {
        for (int j = 0; j < 3; j++) 
        {
            keyMatrix[i][j] = (key.charAt(k)) % 65;
            k++;
        }
    }
}
  
// Following function encrypts the message
static void encrypt(int cipherMatrix[][],
            int keyMatrix[][], 
            int messageVector[][])
{
    int x, i, j;
    for (i = 0; i < 3; i++) 
    {
        for (j = 0; j < 1; j++)
        {
            cipherMatrix[i][j] = 0;
          
            for (x = 0; x < 3; x++)
            {
                cipherMatrix[i][j] += 
                    keyMatrix[i][x] * messageVector[x][j];
            }
          
            cipherMatrix[i][j] = cipherMatrix[i][j] % 26;
        }
    }
}
  
// Function to implement Hill Cipher
static void HillCipher(String message, String key)
{
    // Get key matrix from the key string
    int [][]keyMatrix = new int[3][3];
    getKeyMatrix(key, keyMatrix);
  
    int [][]messageVector = new int[3][1];
  
    // Generate vector for the message
    for (int i = 0; i < 3; i++)
        messageVector[i][0] = (message.charAt(i)) % 65;
  
    int [][]cipherMatrix = new int[3][1];
  
    // Following function generates
    // the encrypted vector
    encrypt(cipherMatrix, keyMatrix, messageVector);
  
    String CipherText="";
  
    // Generate the encrypted text from 
    // the encrypted vector
    for (int i = 0; i < 3; i++)
        CipherText += (char)(cipherMatrix[i][0] + 65);
  
    // Finally print the ciphertext
    System.out.print(" Ciphertext:" + CipherText);
}
  
// Driver code
public static void main(String[] args) 
{
    // Get the message to be encrypted
    String message = "ACT";
  
    // Get the key
    String key = "GYBNQKURP";
  
    HillCipher(message, key);
    }
}
  
// This code has been contributed by 29AjayKumar


Python3




# Python3 code to implement Hill Cipher
  
keyMatrix = [[0] * 3 for i in range(3)]
  
# Generate vector for the message
messageVector = [[0] for i in range(3)]
  
# Generate vector for the cipher
cipherMatrix = [[0] for i in range(3)]
  
# Following function generates the
# key matrix for the key string
def getKeyMatrix(key):
    k = 0
    for i in range(3):
        for j in range(3):
            keyMatrix[i][j] = ord(key[k]) % 65
            k += 1
  
# Following function encrypts the message
def encrypt(messageVector):
    for i in range(3):
        for j in range(1):
            cipherMatrix[i][j] = 0
            for x in range(3):
                cipherMatrix[i][j] += (keyMatrix[i][x] * 
                                       messageVector[x][j])
            cipherMatrix[i][j] = cipherMatrix[i][j] % 26
  
def HillCipher(message, key):
  
    # Get key matrix from the key string
    getKeyMatrix(key)
  
    # Generate vector for the message
    for i in range(3):
        messageVector[i][0] = ord(message[i]) % 65
  
    # Following function generates
    # the encrypted vector
    encrypt(messageVector)
  
    # Generate the encrypted text 
    # from the encrypted vector
    CipherText = []
    for i in range(3):
        CipherText.append(chr(cipherMatrix[i][0] + 65))
  
    # Finally print the ciphertext
    print("Ciphertext: ", "".join(CipherText))
  
# Driver Code
def main():
  
    # Get the message to 
    # be encrypted
    message = "ACT"
  
    # Get the key
    key = "GYBNQKURP"
  
    HillCipher(message, key)
  
if __name__ == "__main__":
    main()
  
# This code is contributed 
# by Pratik Somwanshi


C#




// C# code to implement Hill Cipher
using System;
  
class GFG 
  
// Following function generates the 
// key matrix for the key string 
static void getKeyMatrix(String key, 
                         int [,]keyMatrix) 
    int k = 0; 
    for (int i = 0; i < 3; i++) 
    
        for (int j = 0; j < 3; j++) 
        
            keyMatrix[i, j] = (key[k]) % 65; 
            k++; 
        
    
  
// Following function encrypts the message 
static void encrypt(int [,]cipherMatrix, 
                    int [,]keyMatrix, 
                    int [,]messageVector) 
    int x, i, j; 
    for (i = 0; i < 3; i++) 
    
        for (j = 0; j < 1; j++) 
        
            cipherMatrix[i, j] = 0; 
          
            for (x = 0; x < 3; x++) 
            
                cipherMatrix[i, j] += keyMatrix[i, x] * 
                                      messageVector[x, j]; 
            
          
            cipherMatrix[i, j] = cipherMatrix[i, j] % 26; 
        
    
  
// Function to implement Hill Cipher 
static void HillCipher(String message, String key) 
      
    // Get key matrix from the key string 
    int [,]keyMatrix = new int[3, 3]; 
    getKeyMatrix(key, keyMatrix); 
  
    int [,]messageVector = new int[3, 1]; 
  
    // Generate vector for the message 
    for (int i = 0; i < 3; i++) 
        messageVector[i, 0] = (message[i]) % 65; 
  
    int [,]cipherMatrix = new int[3, 1]; 
  
    // Following function generates 
    // the encrypted vector 
    encrypt(cipherMatrix, keyMatrix, messageVector); 
  
    String CipherText = ""
  
    // Generate the encrypted text from 
    // the encrypted vector 
    for (int i = 0; i < 3; i++) 
        CipherText += (char)(cipherMatrix[i, 0] + 65); 
  
    // Finally print the ciphertext 
    Console.Write("Ciphertext: " + CipherText); 
  
// Driver code 
public static void Main(String[] args) 
    // Get the message to be encrypted 
    String message = "ACT"
  
    // Get the key 
    String key = "GYBNQKURP"
  
    HillCipher(message, key); 
  
// This code is contributed by Rajput-Ji 


Output:

Ciphertext: POH

In a similar way you can write the code for decrypting the encrypted message by following the steps explained above.
Reference: https://en.wikipedia.org/wiki/Hill_cipher

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