In this article, we have explained the RSA public-key cryptosystem initiated by Rivest and Hill Chiper cryptosystem initiated 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 have collaborated on two cryptosystems which is more assured than standard cryptographic processes 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.
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 is said to be a linear congruence and by a outcome of this equation we denote an integer for which by explanation and only if what quantity to the identical if and only if for some integer . Thus the matter of determinations of all integers holds the linear congruence is uniform with that of getting all solutions of the linear Diophantine equation
RSA CRYPTOSYSTEM
Let n be a multiply of two individual primes p and q. Let p=c= . Let us destine –
Where, called Eulers function is the positive integers less than n which are relatively prime to n for each –
We define and where . The values 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 –
With 27 indicating a space between words.
Enciphering step 01: Choose a square matrix A of order 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 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 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 matrix P for the values of M
Let a matrix A as
Then
Now the encoding matrix E is
Now we use congruence we have:
Now we use RSA encoding system in the matrix E
Now for the farther safety of the system we will code the cipher text E into another cipher text N with the support of Eulers phi function. For this sample we initially choose primes P=11, Q=17 of an impractical tiny size. In real P and Q would huge enough so that the multiplication of the number n=PQ is impractical (Niven et al., 1980; Burton, 1989). Our enciphering number is –
, and
Modulo.
Assume the enciphering proponent is nominated to be k=23, then the recapture element the individual integer j fulfilling the congruence j=7.
To code N we require each part of N to be an integer less than 187. Now for the first part of the calculation is:
Now for the values
Then,
Now similarly we have
Now the total coded message is:
162 98 63 113 51 162 42 162 35 162
Now we will decoded the message by using recapture element . We can recapture the earliest text:
Now for the number
Now similarly we have
Now the decoded message after introducing R.S.A system is -
M= (02 21 24 20 17 02 15 02 18 02)
Now we can recapture the earlier l text by using the inverse matrix
Now we are applying congruence:
The total decoded message is –
M = (01 18 18 05 19 20 27 14 15 23)
Now to recapture the message translate every number of M into its digital equivalent using the substitution mentioned earlier this capitulate the original plaintext.
ARREST NOW
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.
Thanks to the authority of the Department and co-author that supported with proper assistance and writing to conduct successful research.
The authors declare no conflict of interest.
Academic Editor
Dr. Toansakul Tony Santiboon, Professor, Curtin University of Technology, Bentley, Australia.
Department of Computer Science and Engineering, North Western University (NWU), Khulna, Bangladesh.
Ahmed M, and Iqbal MA. (2020). An execution of a mathematical example using Eulers Phi-function in Hill Chiper cryptosystem, Int. J. Mat. Math. Sci., 2(6), 99-103. https://doi.org/10.34104/ijmms.020.0990103