Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 

 

440 
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Some Results of Data Encryption and Decryption using Euler’s Totient 

Function 

 

Gobburi Rekha1, V. Srinivas2 

1 Department of Humanities and Sciences, Malla Reddy College of Engineering and Technology 

2 Department of Mathematics, Osmania university 

 

Article History: 

Received: 10-05-2024 

Revised: 20-06-2024 

Accepted: 02-07-2024 

Abstract: 

 In this paper, we discuss about the Hill Cipher cryptosystem, the RSA public-key 

cryptosystem, developed by Rivest, Shamir, Adleman for text encoding and decoding. To 

encrypt and decrypt the message, we use the Euler Phi-function, congruences, and matrix 

multiplication. Further, we employ two keys for coding and decoding. 

Introduction: In present, computer networks, internet and mobile communications are 

important and inevitable part of our society and security of information from hackers is a 

big task. One of the most widely used applications for information security is 

Cryptography. Usually in Hill cipher there are two keys, of which one acts as encryption 

key and the other one acts as decryption key.  In this paper we discuss about Hill Cipher 

using only one key for data encryption and data decryption. Further, we extend our result 

with dual encryption and decryption process which is explained by using the RSA public 

key cryptosystem and Hill Cipher and they together execute a powerful cryptosystem to 

secure the data. 

Objectives: The purpose of this paper is to obtain and compare the result of data encryption 

and data decryption using: 

• A matrix which is invertible. 

• A matrix which is Involutory. 

•  An RSA cryptosystem which is applied for Hill Cipher where the matrix is Involutory. 

Methods: Using Hill Cipher and thereafter Hill Cipher - RSA together we have encrypted 

and decrypted the data under modulo 255. 

Results: The text message is converted to coded message and then original message is 

retrieved by using the three different techniques. 

Conclusions: As the paper involves two stages of encryption and decryption where public 

key and private key which are unthinkable by the third parities, the security of information 

is highly appreciable. 

Keywords: Cryptography, Congruence, Euler-Phi function, Hill Cipher, RSA 

cryptosystem and matrix, ASCII code. 

 

1. Introduction 

In present, computer networks, internet and mobile communications are important and inevitable part 

of our society and security of information from hackers is a big task. One of the most widely used 

applications for information security is Cryptography. Usually in Hill cipher there are two keys, of 

which one acts as encryption key and the other one acts as decryption key.  In this paper we discuss 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 

 

441 
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about Hill Cipher using only one key for data encryption and data decryption. Further, we extend our 

paper with dual encryption and decryption process which is explained using the RSA public key 

cryptosystem and Hill Cipher cryptosystem which together execute a powerful cryptosystem technique 

to secure the data. This paper extends the result of [1] by using involutory matrix for RSA 

cryptosystem. 

2. Standard Definitions 

Plain Text: The message which is to be converted for the purpose of security. 

Cipher Text: The message which is in unreadable language. 

Encryption: Encryption is a form of data security in which information is converted to ciphertext.  

Decryption: Decryption is the process of converting encrypted data into recognizable information.  

Hill Cipher: Invented by Lester S. Hill which is a polygrahic substitution cipher based on linear 

algebra.  

Involutory Matrix: Any square matrix, A which satisfies the condition 𝐴2 = 𝐼. 

Linear Congruence: A linear congruence is a congruence relation of the form ax ≡ b (mod m) where 

a, b, m ∈ 𝑍 +and m > 0. A solution is an integer x which makes the congruence relation true and x is 

a least residue (mod m) (that is, 0 ≤ x ≤ m−1). 

RSA: The Rivest-Shamir-Adleman (RSA) encryption algorithm is an asymmetric 

encryption algorithm. Asymmetric encryption uses a key pair that is linked to encrypt and decrypt data. 

A public key is generated and made available to everybody, while the private key is kept confidential 

and known only to the person who generated the key pair.  

Euler -Phi function: Euler’s phi (or totient) function of a positive integer n is the number of integers 

in {1,2, 3..., n} which are relatively prime to n which is denoted ∅(𝑛). 

Let n be the product of two individual primes p and q. 

Let us define K = (n,p,q,e,d):ed≡1(mod(∅(𝑛)). 

We define ek(x) ≡ xe(mod n) and dk(y) ≡ ye(mod n) where p, q and d are used as public keys. 

3. Methods 

In this we discuss three different methods for encryption and decryption of the data.  

Method I: Using Invertible matrix 

• Let us consider nxn invertible matrix say A in which the matrix A acts as an encryption key 

and its inverse acts as a decryption key. 

• Convert the original plain text into numbers using the ASCII values and arrange the numbers 

in matrix form say P. 

• Let E≡AP (mod 255). 

• Now, on converting the numbers in the matrix E into alphabets and symbols we get the 

encrypted message which is to be sent. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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• To decode the message which is sent by the sender, let us convert the alphabets and symbols 

in the matrix E into numbers. 

• Let D ≡ 𝐴−1𝐸 (mod 255), we get the original plaintext. 

Method II: Using Involutory matrix 

• Now we shall continue the explanation of encrypting and decrypting the data by using an 

involutory matrix where we use only one key for encryption and the same key for decryption. 

• Let us consider a nxn involutory matrix say A which acts as an encryption and decryption key. 

• Convert the plaintext in numbers using the latest ASCII values and arrange the numbers in 

matrix form say P. 

• Let E = AP, adjusting the values under modulo 255 we get the encrypted message. 

• Hence, after converting the numbers in E into symbols under the ASCII values, we get the 

Cipher text. 

• To decode the message let us write E in matrix form and multiply by the key A. 

• Let D = AE. Since, A is an involutory matrix where 𝐴2 = 𝐼 which can also be written as 

 𝐴 = 𝐴−1. 

• Now, adjusting the values of D under modulo 255 we can retrieve the original text.  

Method III: Dual encryption and decryption of data using Euler’s totient function in Hill cipher 

• Let us consider a nxn involutory matrix say A such that 𝐴2 = 𝐼 

• Convert the plaintext in numbers using the latest ASCII values and arrange the numbers in 

matrix form say P. 

• Let E = AP, adjusting the values under modulo 255 which gives the first stage of encryption. 

• For further safety purpose let us encrypt E using RSA cryptosystem we define 

 ek(x) ≡ xe(mod n). 

• To decrypt the message using RSA cryptosystem we define dk(y) ≡ ye(mod n) which gives 

the first stage of decryption. 

• Arrange the above decrypted codes in the form a matrix say M and multiply by 𝐴−1 we get D. 

•  On adjusting the values of D under modulo 255 we can retrieve the original text.  

4. Results 

To see how the methodology works, let us consider the following illustrations. 

Illustration I: Let the plaintext to be sent to the receiver by the sender is “Estimation”. 

Using the ASCII values convert the alphabets into numbers “=34, E = 69, s = 115, t = 116, 

 i = 105, m = 109, a =97, t = 116, i = 105, o = 111, n = 110 and” = 34. 

Encryption: 

Let P = [
34 69 115     

105 109 97  
105 111 110  

116
116
34

]. 

 



Communications on Applied Nonlinear Analysis 

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Let A = [
1 0 −1
1 −1 0

−1 1 −1
] be an invertible matrix. 

    Therefore, 𝐴−1 =  [
1 0 −1
1 −1 0

−1 1 −1
]. 

Let E = AP. 

E = [
1 0 −1
1 −1 0

−1 1 −1
] [

34 69 115     
105 109 97  
105 111 110  

116
116
34

]. 

 E ≡ [
−71 −42 5   
−71 −40 18  
−34 −71 −128  

82
0

−34
] (mod 255). 

E =AP =  [
184 213   5    
184 215 18 
221 184 127  

82
0

221
]. 

On converting the numbers in matrix E into symbols we get    ¸ÕENQR¸×DC2 NULݸDELÝ which 

acts as the cipher text.  

Decryption:  

Let the cipher text be transformed into numbers under the ASCII values. 

Let E = [
184 213   5    
184 215 18 
221 184 127  

82
0

221
]. 

Now let D = 𝐴−1E. 

D = [
1 0 −1
1 −1 0

−1 1 −1
] [

184 213   5    
184 215 18 
221 184 127  

82
0

221
]. 

D ≡ [
−221 −186  −140   
−405 −401 −158
−405 −399 −145 

−139
−139
−221

] ( mod 255). 

D = [
34 69   115  

105 109 97 
105 111 110  

116
116
34

]. 

The matrix D is same as that of matrix P. 

Therefore, on converting the numbers of matrix D into alphabets we get the message as  

“Estimation”. 

Illustration 2: 

Let the message to be sent is “Taste of life” using the ASCII values covert the alphabets into numbers 

as “= 34, T = 84, a = 97, s = 115, t = 116, e = 101, space = 32, o = 111, f = 102,  

space = 32, l = 108, I = 105, f = 102, e = 101 and’’=34.  

 

 

 



Communications on Applied Nonlinear Analysis 

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Encryption: 

Let P = [
34 84 97   

101 32 111  
108 105 102  

115 116
102 32
101 34

] 

Let A = [
−5 −8 0
3 5 0
1 2 −1

] be an involutory matrix such that 𝐴2 = 𝐼. 

𝐴2 =  [
−5 −8 0
3 5 0
1 2 −1

] [
−5 −8 0
3 5 0
1 2 −1

] = [
1 0 0
0 1 0
0 0 1

]. 

Therefore E = AD  

E = [
−5 −8 0
3 5 0
1 2 −1

] [
34 84 97   

101 32 111  
108 105 102  

115 116
102 32
101 34

] 

E = [
−978 −676 −1373   
  607   412   846  
  128   43   217  

 
−1391 −836
  855   508
  218   146

]. 

Adjusting the values under modulo 255. 

E ≡ [
−978 −676 −1373   
  607   412   846  
  128   43   217  

 
−1391 −836
  855   508
  218   146

] (under mod 255). 

We get E =  [
42 89 157  
97 157  81  

  128   43   217  
 

139 184
90   253

  218   146
]. 

 On converting the numbers in E into alphabets we get *Y�‹¸a�QZý€+ÙÚ’ 

Which acts as Cipher text.  

Decryption:  

Let the cipher text be transformed again into numbers under the ASCII values. 

Let E = [
42 89 157  
97 157  81  

  128   43   217  
 
139 184
90   253

  218   146
]. 

Now let D = AE. 

D = [
−5 −8 0
3 5 0
1 2 −1

] [
42 89 157  
97 157  81  

  128   43   217  
 
139 184
90   253

  218   146
]. 

D ≡ [
−986 −1701 −1433  
   611    1052    876 
  108     360   102  

 
−1415 −2944
    867   1817
    101    544

] ( under mod 255). 

D = [
34 84 97   

101 32 111  
108 105 102  

115 116
102 32
101 34

]. 

On converting the numbers of matrix D into alphabets we get the message as “Taste of life”. 

Illustration3: 

We continue the result by applying hill cipher and RSA cryptosystem simultaneously. Now let us 

discuss the illustration which includes dual encryption and decryption of involutory matrix using Euler 

Totient function. 



Communications on Applied Nonlinear Analysis 

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Step 1: Let A be a square matrix of order 3x3 which is an involutory matrix. 

Let A = [
4 3 3

−1 0 −1
−4 −4 −3

] then A2 =  [
4 3 3

−1 0 −1
−4 −4 −3

] [
4 3 3

−1 0 −1
−4 −4 −3

] =[
1 0 0
0 1 0
0 0 1

]. 

Step 2: Transform each plaintext into a column vector B by taking the ASCII values. 

Let us consider the message to be encrypted as Now or Never. 

 P = 78 111 119 32 111 114 32 78 101 118 101 114 which is written in matrix form as  

P = [
78 111 119  

111 114 32
101 118 101  

  
32
78

114
].  

𝟏𝒔𝒕 stage of encryption 

Step 3: To encrypt the message, multiply the plaintext matrix P by matrix A to form the matrix  

       E = [AP] (modulo 255). 

      E = AP = [
4 3 3

−1 0 −1
−4 −4 −3

] [
78 111 119  

111 114 32
101 118 101  

  
32
78

114
]. 

     E ≡  [
948 1140 875 

−179 −229 −220
−1059 −1254 −907  

  
704

−146
−782

] (mod 255). 

E = [
183 120 110  
76 26 35

216 21 113
  
194
109
238

]. 

Step 4: For further safety we will code the cipher text E into another text with the help of Euler -Phi   

function: 

    E = [
183 120 110  
76 26 35

216 21 113
  

194
109
238

]. 

𝟐𝒏𝒅 stage of encryption 

Now we use RSA encoding system in the matrix E 

E= (183 120 110 194 76 26 35 109 216 21 113 238). 

For further safety let us consider three primes where N = P *Q* R 

Let P = 11, Q = 13, R= 17 where N = 2431 

Therefore, ∅(N) = (P − 1) ∗ (Q − 1) ∗ (R − 1). 

                            = 10.12.16 =1920. 

                ≡1921 = 113*17. 

Assume the enciphering proponent to be k then the recapture element the individual integer j    fulfilling 

the congruence kj ≡ 1 mod (∅(N) . 



Communications on Applied Nonlinear Analysis 

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Here k = 113 and j = 17 

To code N we require each part of N to be an integer less than 2431 

Let us find encrypted value for the first element     

183 ≡  183 mod 2431    

1834 ≡  443 mod 2431  

1838 ≡ 1769 mod 2431                 

18316 ≡ 664 mod 243     

18332 ≡ 885 mod 2431    

18364 ≡ 443 mod 2431 

Therefore,183113 ≡ 1509 mod 2431. 

Now, we find for  

120 ≡ 120 mod 2431     

1204 ≡ 562 mod 2431 

1208 ≡ 2245 mod 2431     

12016 ≡ 562  mod 2431     

12032 ≡ 2245 mod 2431                   

12064 ≡ 562 mod 2431 

Therefore, 120113 ≡ 1803 mod 2431.  

Similarly, we get the remaining elements as  

110113 ≡ 1419 mod 2431  

194113 ≡ 1520 mod 2431 

76113 ≡ 1385 mod 2431 

26113 ≡ 1131 mod 2431 

35113 ≡ 2109 mod 2431 

109113 ≡ 109 mod 2431 

216113 ≡ 1542 mod 2431 

21113 ≡ 21 mod 2431 

113113 ≡ 1303 mod 2431 

238113 ≡ 816 mod 2431. 

 



Communications on Applied Nonlinear Analysis 

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Therefore, the total coded message sent by the sender is: 

 (1509 1803 1419 1520 1385 1131 2109 109 1542 21 1303 816). 

To decode the coded message the receiver uses RSA cryptosystem. 

Step 5: Now we will decode the message by using recapture element j = 17. 

𝟏𝒔𝒕 stage of decryption 

Now, let us decode the coded values  

First, we find  

1509 ≡ 1509 mod 2431               

15092 ≡ 1665 mod 2431    

15093 ≡ 1262 mod 2431    

15094 ≡ 885 mod 2431    

15095 ≡ 846 mod 2431    

Therefore,150917 ≡ 183 mod 2431. 

1803 ≡ 1803 mod 2431     

18032 ≡ 562 mod 2431    

18033 ≡ 1990 mod 2431                

18034 ≡ 2245 mod 2431     

18035 ≡ 120 mod 2431       

Therefore, 180317 ≡ 120 mod 2431.  

Similarly, we get the remaining elements as 

141917 ≡ 110 mod 2431  

152017 ≡ 194 mod 2431         

138517 ≡ 76 mod 2431            

113117 ≡ 26 mod 2431 

210917 ≡ 35 mod 2431 

10917 ≡ 109 mod 2431 

154217 ≡ 216 mod 2431 

2117 ≡ 21 mod 2431 

130317 ≡ 113 mod 2431 

81617 ≡ 238 mod 2431 

 



Communications on Applied Nonlinear Analysis 

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Now the decoded message after introducing RSA is  

M = ( 183  120  110  194  76  26  35  109  216  21  113  238). 

Step 6: Now, we shall apply the hill cipher cryptosystem for M which is obtained in step 5. 

𝟐𝒏𝒅 stage of decryption 

Let us write M in the form a matrix. 

M = [
183 120 110  
76 26 35

216 21 113
  
194
109
238

]. 

Now we can recapture the text by using inverse matrix 𝐷 =  𝐴−1𝑀 

D =  [
4 3 3

−1 0 −1
−4 −4 −3

] [
183 120 110  
76 26 35

216 21 113
  
194
109
238

]. 

    ≡ [
1608 621 884 
−399 −141 −223

−1684 −647 −919
  

1817
−432

−1926
] (mod 255). 

   = [
78 111 119  

111 114 32
101 118 101  

  
32
78

114
] which is equal to matrix P. 

  Hence, the total decoded message is P = (28 14 15 23 27 15 18 27 14 5 22 5 18 29 27 27) 

retrieved as Now or Never. 

5. Conclusion 

The present paper involves three different methods to encrypt and decrypt the message where the first 

two methods involved Hill cipher using invertible matrix and involutory matrix. Whereas, the third 

method involves two stages of encryption and decryption where public key and private key which are 

untraceable by the third parities, as it is involves large prime numbers which provides security of 

information. 

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