An Execution of a Mathematical Example Using Euler’s Phi-Function in Hill Chiper Cryptosystem

In this article, we have explained the RSA public-key cryptosystem initiate by Rivest and Hill Chiper cryptosystem initiate by Lester. Hill for coding and decoding the text. To explain these we apply the Euler-phi function, congruence, and simple Matrix Application in cryptography to decode and encode the message. In our analyses, we were collaborated two cryptosystems which is more assured than standard cryptographic process such as Ceaser cipher. We apply both secret-key cryptography and public-key cryptography which differ from standard cryptography. In our presentation, we use two keys for coding and two for decoding. Keyword: Cryptography, Congruence, Euler’ phi function, Hill Chiper, RSA cryptosystem, and Matrix. INTRODUCTION: Institutions in both the public and private sectors have become progressively based on electronic data processing. Huge volume of digital data are now assemble and preserved in macro scale, computer data circulated between computers and terminal devices connected jointly in transmission networks. Except suitable prevention, these data are permitting to blocking during transference or they might be materially disconnect or duplicate while in storage. This could effect in unwelcome submission of data and future annexation of privacy (Diffine and Hellman, 1976). In this conference paper, we have explained the RSA public-Key cryptosystem and Hill Chiper crypto system and exécute a math metical exemple of combinaient those crypto system to Secure the data (Cohen, 1994 ; Adhikari and Adhikari, 2007). LINEAR CONGRUENCE This form ax ) (mod n b ax  is said to be a linear congruence and by a outcome of this equation we denote an integer 0 x for which ) (mod 0 n b ax  by explanation ) (mod 0 n b ax  and only if b ax n  0 / what quantity to the identical if and only if 0 0 ny b ax   for some integer 0 y . Thus the matter International Journal of Material and Mathematical Sciences, 2(6), 99-103, 2020 Publisher homepage: www.universepg.com, ISSN: 2707-4625 (Online) & 2707-4617 (Print) https://doi.org/10.34104/ijmms.020.0990103 International Journal of Material and Mathematical Sciences Journal homepage: www.universepg.com/journal/ijmms Ahmed and Iqbal / International Journal of Material and Mathematical Sciences, 2(6), 99-103, 2020 UniversePG l www.universepg.com 100 of determinations of all integers holds the linear congruence ) (mod n b ax  is uniform with that of getting all solutions of the linear Diophantine equation . b ny ax   RSA CRYPTOSYSTEM Let n be a multiply of two individual primes p and q. Let p=c= n z . Let us destine – )) ( (mod 1 : ) , , , , ( n ed d e q p n K    Where, ) (n  called Euler’s function is the positive integers less than n which are relatively prime to n for each – ) , , , , ( d e q p n K  We define ) (mod ) ( n x x e e k  and ) (mod ) ( n y y d d k  where n z y x  , . The values q p, and d are used as public key. HILL CIPHER CRYPTOSYSTEM For Hill Cipher we imposed arithmetical values to every plaintext and cipher text letter so that – 12 , 11 , 10 , 09 , 08 , 07 , 06 , 05 , 05 , 04 , 03 , 02 , 01              L K J J I H G F E D C B A 26 , 25 , 24 , 23 , 22 , 21 , 20 , 19 , 18 , 17 , 16 , 15 , 14 , 13               Z Y X W V U T S R Q P O N M With 27 indicating a space between words. Enciphering step 01: Choose a square matrix A of order 2 2 with integer listings to perform the encoding. The matrix has to be revertible modulo m but we will explain later. Enciphering step 02: Grouping sequential plaintext letters into pairs. If we result in with one single letter at the end directly add an arbitrary “chump” letter to complete the next last pair of letters. Enciphering step 03: Transform each plaintext pair 2 1p p into a column vector P. To encrypt the message we multiply our plaintext matrix P by our converted matrix A to form the product AP. The resultant of our matrix multiplication is the cipher text matrix C. This was the encoding technique. Now we decode our enciphered message. Deciphering step 01: Now we sort the consecutive cipher text letters into pairs and transform each cipher text pair 2 1c c into a column vector C. Then construct the cipher text matrix C of all our cipher text column vectors. Deciphering step 02: Multiply the cipher text matrix C with the inverse of our enciphering matrix A to obtain the decoded message. AN EXAMPLE OF THESE SYSTEM If someone wants to convey a plaintext message to the user such as ARREST NOW To convert the message “ARREST NOW” First converts each letter into its digital identical using the replacement indication in Hill cipher cryptosystem (Rivest et al., 1978). This capitulate the plain text number M=01181805 192027141523 Now we take a 5 2 matrix P for the values of M        23 14 20 05 18 15 27 19 18 01 P Let a 2 2 matrix A as        4 3 3 2 A Then           2 3 3 4 1 A Now the encoding matrix E is         4 3 3 2 AP E             23 14 20 05 18 15 27 19 18 01                  23 . 4 15 . 3 14 . 4 27 . 3 20 . 4 19 . 3 5 . 4 18 . 3 8 . 4 1 . 3 23 . 3 15 . 2 19 . 3 27 . 2 20 . 3 19 . 2 5 . 3 18 . 2 18 . 3 1 . 2 Ahmed and Iqbal / International Journal of Material and Mathematical Sciences, 2(6), 99-103, 2020 UniversePG l www.universepg.com 101        137 137 137 74 75 99 96 98 51 56 Now we use congruence we have: ) 27 (mod 02 137 ) 27 (mod 02 137 ) 27 (mod 02 137 ) 27 (mod 20 74 ) 27 (mod 21 75 ) 27 (mod 18 99 ) 27 (mod 15 96 ) 27 (mod 17 98 ) 27 (mod 24 51 ) 27 (mod 02 56

Institutions in both the public and private sectors have become progressively based on electronic data processing. Huge volume of digital data are now assemble and preserved in macro scale, computer data circulated between computers and terminal devices connected jointly in transmission networks. Except suitable prevention, these data are permitting to blocking during transference or they might be materially disconnect or duplicate while in storage. This could effect in unwelcome submission of data and future annexation of privacy (Diffine and Hellman, 1976).
In this conference paper, we have explained the RSA public-Key cryptosystem and Hill Chiper crypto system and exécute a math metical exemple of combinaient those crypto system to Secure the data (Cohen, 1994 ;Adhikari and Adhikari, 2007).

LINEAR CONGRUENCE
This form ax ) (mod n b ax  is said to be a linear congruence and by a outcome of this equation we denote an integer 0 With 27 indicating a space between words.
Enciphering step 01: Choose a square matrix A of order 2 2  with integer listings to perform the encoding. The matrix has to be revertible modulo m but we will explain later.
Enciphering step 02: Grouping sequential plaintext letters into pairs. If we result in with one single letter at the end directly add an arbitrary "chump" letter to complete the next last pair of letters.
Enciphering step 03: Transform each plaintext pair 2 1p p into a column vector P.
To encrypt the message we multiply our plaintext matrix P by our converted matrix A to form the product AP.
The resultant of our matrix multiplication is the cipher text matrix C.
This was the encoding technique. Now we decode our enciphered message.
Deciphering step 01: Now we sort the consecutive cipher text letters into pairs and transform each cipher text pair 2 1c c into a column vector C. Then construct the cipher text matrix C of all our cipher text column vectors.
Deciphering step 02: Multiply the cipher text matrix C with the inverse of our enciphering matrix A to obtain the decoded message.

AN EXAMPLE OF THESE SYSTEM
If someone wants to convey a plaintext message to the user such as

ARREST NOW
To convert the message "ARREST NOW" First converts each letter into its digital identical using the replacement indication in Hill cipher cryptosystem (Rivest et al., 1978   Now to recapture the message translate every number of M into its digital equivalent using the substitution mentioned earlier this capitulate the original plaintext.

CONCLUSION:
The safety of our collaborate Hill cipher and RSA cryptography system is ground on various factors: First is that we utilize a key matrix A, which is only recognized to first party and second party (Koblitz, 1998). The second is that remembering n and k do not authorize you to know the value of j. Third since you recognize n it will be relatively simple to find j if you just divide n to determine the primes p and never less no one has enough time effectively to factor n when n only two very large prime factors and fourth the collaborate of these two cryptosystem gives a safety where the plaintext is entirely unthinkable to find out for the third parties.

ACKNOWLEDGEMENT:
Thanks to the authority of the Department and coauthor that supported with proper assistance and writing to conduct successful research.