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.