Cryptography is the system in which an algorithm is recycled to convert the information into an arrangement that is not readable to anyone except the sender and receiver who participate in this mechanism. The algorithm must be reliable, efficient and easy to understand by the sender and receiver involved in this communication system or encryption process (Saraswata et al., 2016). Here we have used the asymmetric crypto technique RSA with an extended vigenere method to secure the information before transmit. In RSA methodology, every plaintext is -

encrypted by the receivers public key and the encrypted ciphertext is decrypted by using the receivers private key. As the private key does not participate in the transmission so there is no way to identify this private key by the intruder. However, RSA technique alone can make the system secure but slow in the process. To overcome the existing RSA problem we have proposed an extended vigenere based RSA technique to make the system reliable and faster. Here the Fig 1 shows the overall encryption and decryption process of cryptography.

Various techniques are usually used to convert the original message into cipher-text (Nacira and Abdelaziz, 2004). Among them, the most commonly we are using to encrypt data and secure communication technique is the poly-alphabetic substitution technique which enables to provide more security. In this, a character or letter needs not to be replaced with the same character or letter for its occurrence in the whole message like a mono-alphabetic cipher technique. We propose a new table named modified vigenere table of the poly-alphabetic cipher method. In this proposed table we include the numbers, rare character along with the alphabets. Here, the alphabets (A-Z), the digits (0-9), space, rare characters are appended after the total alphabets with values 69. We take common value from the proposed vigenere table with the key addition by the sender and cipher-text will be generated using a modified vigenere table. The general formula of the above-specified process of encryption is C = (P+K) mod 69 (Nacira and Abdelaziz, 2004; Patni, 2013). 

We have proposed a new vigenere table, the general formula for the above decryption process is as follows: P = ((C) - K) mod 69. In the modified vigenere cipher, most frequent letters are put first and then less frequent letters on both the column and row side. In our previous thesis paper (Hossain and Islam, 2018), we have an updated Vigenere table that consists of 69 rows and 69 columns. Here key space is 69! Which is over 1.7112245242×1098 (Menezes et al., 1996; Senthil et al., 2013). If eavesdropper can able to observe 1,000,000 keys per second, it would still need over 5.426×1072 trillion years to check all possible keys, so it is not a practical approach but it is indeed impossible to break the keys by brute force attack.

Fig 2: Block diagram of modified vigenere diagram. Fig 3: Block diagram of hybrid vigenere asymmetric encrypti.

Table 2: Character weight Mapping.

In this technique, the receiver generates two large prime numbers p and q. Depend on those prime numbers, the modulo factor n, φ, public key e, and private key d will be generated and will share the public key (n, e) to the sender. The sender will form a string of sequential characters from the randomly generated table. Every character will be mapped with the corresponding weight of integer value. All the weights will be encrypted by the public key. Encrypted weight will be converted into the binary form of x digit block. The block size is a receiver choice which depends on the large prime factor p and q [2x = pq]. Sender then sends this ciphertext (binary string) to the receiver. The receiver will decrypt the ciphertext and generate the original plain text. For decryption, the ciphertext is divided into blocks of length x and revert the binary value to an integer which will be decrypted using the receivers private key. As an example, the receiver generates two prime numbers p=7, q=11 and compute e=17 and d=53 by using the RSA key generation algorithm and sends public key (n, e) = (77, 17) to sender. The sender generates random vigenere table which is formed as a character sequence according to the following observation.

cabefdighjklmonpqrtsu052134v96w87xzy,.;_”?:!@#/+-(*){[<}>]~|`&$^=% and is mapped to the corresponding weight of the characters as:

3,1,2,5,6,4,9,8,7,10,11,12,13,15,14,16,17,18,20,19,21,27,32,29,28,30,31,22,36,33,23,35,34,24,26,25,37,38,39,40,42,41,43,44, 45,46,47,48,49,50,52,51,53,54,56,58,55,59,57,61,62,63,64,66,68,67,65,69

Sender will encrypt each integer weight by RSA encryption algorithm c = me mod n. Therefore, the encrypted value is -

75,1,18,3,41,16,4,57,28,54,44,45,62,71,42,25,19,72,48,24,21,69,65,50,63,46,26,22,64,66,67,7,34,40,38,9,60,47,30,17,70,13,43,11,12,51,31,27,14,8,68,39,37,10,56,53,55,5,29,52,6,35,15,33,73,23,32,20

Each encrypted integer is now converted as binary number of length x where x is the length of each binary number by using the formula 2x = n. Sender then sends these encrypted binary strings as cipher text to the receiver.

10010110000001001001000000110101001001000000000

01110010011100011011001011000101101011111010001110101010001100100100111001000011000000110000010101100010110000010110010011111101011100011010 0010110 1000000 1000010 1000011000011101000100101

0000100110 0001001011110001011110011110001000110

001100001101010011 00010110001100 01100110011111

0011011000111000010001000010001001110100101 0001

01001110000110101011011100001010011101011010000001100100011000111101000011001001001011101000000010100

The receiver will divide first the received binary string by x digits of the block, convert to decimal and decrypt each decimal integer by using the private key (n, d) = (77, 53) and the decryption formula m = cd mod n. Then the receiver will get back to the character sequence of the vigenere table by replacing the weight value to the corresponding character. Finally, the receiver will generate same vigenere table which is used by the sender to encrypt the original plaintext and decrypt.

F. Mathematical Equation 

It is proved that the quantity of information depends on the possibility of happening to those events. Let us consider, I is the quantity of information of a message m and P is the possibility of the incident of that event then mathematically, the relation between I and P will be, 

I = log2 (1/P).

In another way, we can say that the amount of information in a message is proportional to the time required to transmit the message. Now let us consider in mind that the possibility of endeavors of letters e and q in an English message is Pe and Pq respectively. We can explain it in the following way,

      Pe ≥Pq

=> 1/Pe ≤ 1/Pq

=> log2 (1/Pe) ≤ log2 (1/Pq)

=> Ie ≤ Iq

If the capacity of a channel is C then the time required to transmit e,

Te = Ie/C

Similarly, the time required to transmit 

Tq = Iq/C

From the equation of (1), and (2) we get 

Ie/C≤ Iq/C

Te ≤ Tq

∴ Te ≤ Tq

Now again we consider a ciphertext consists of length N, the period p, and Ic the index of coincidence,

p≈(N*0.027)/((N-1)*Ic+1-N*0.0145)

Let us consider that the texts start, we can define it by the following equation,

∑_(α=A)^Z▒〖M_α^((i))=M〗

Where M_α^((i)) denotes the number of occurrences of the letters in column i if the ciphertext were written in rows of length p and M is the number of rows [9].

The proof then starts, 2Dc = ∑_(i=1)^p▒∑_(α=A)^Z▒M_α^((i) )  (M_α^((i) )-1)+2∑_(i=1)^p▒∑_(j=i+1)^p▒∑_(α=A)^Z▒〖M_α^((i)) M_α^((j)) 〗

Where Dc is the number of pairs of equal letters in the cipher-text. We can further explain it by the following equation (Brassard, 2005).

2Dc≈ M2∗p∗0.065−pM+M2∗0.0145∗p (p−1)

Here the key will be provided according to the demand of the sender to the vigenere table to decrypt the encrypted message. Multiple substitutions of alphabets are found in poly alphabetic cipher. The Vigenère table that is our research topic is probably a well-known example of a poly alphabetic cipher. 

The Enigma machine is more complex but is still fundamentally using poly alphabetic substitution cipher. An example has been given below to convert plain-text to cipher-text and cipher-text to plaintext by the following table using a modified vigenere table.

Table 3: Comparison of vigenere and proposed method.

Fig 4: Rows and columns of vigenere and modified method.

Table 4:  Example of encryption and decryption.

Fig 5: Cryptanalysis comparison by Brute force.

Fig 6: Cipher-text and key comparisons.

The above analysis and description depict that the vigenere table highlights on the poly alphabetic cipher technique. Our research is to extend the vigenere table by including the alphabets, digits, special and rare symbols in the proposed vigenere table as a result numbers, special and rare symbols will be encrypted by the process of proposed table as well as the number of randomly generated vigenere table will be large and almost unpredictable to intruders. It also reduces the sizes of cipher-text, keys and here symbols are arranged by increasing the numbers of rows and columns.  In the previous proposal of our research paper was the symmetric key encryption process. To overcome this problem, we have made the proposal asymmetric by combing our previous proposed modified vigenere method with the RSA algorithm to ensure the security of communication to keep pace with the current world. 

In the future, the concept of hash value based on block chain cryptography will be added with cipher-text to mark the procedure of cryptanalysis more complex.

Many thanks to the co-author supported with proper assistance and help for analysis and writing to conduct successful research study.

The authors declare they have no competing interests with respect to the research.