Many mathematical theoretical accounts are taking an active function in the procedure of encoding and opposite of an component or of a map ensuing in decoding procedure. The chief thought behind such buildings is planing a mathematical map in such a manner that following the opposite of the map is non easy. Cryptography is the art of coding & A ; decrypting information. It is used as security mechanism in electronic communicating. The message we want to direct is called ‘plain text ‘ and the cloaked message is called the cypher text. The procedure of change overing field text into cypher text is called ‘encryption ‘ and the contrary procedure is called ‘decryption ‘ .

In this paper, we develop a finite province machine for the procedure of encoding.

The finite zombi is a mathematical theoretical account of a system with distinct inputs and end products. The system can be one of a finite figure of internal constellations or provinces. The finite zombi is a mathematical theoretical account or a system, with distinct inputs and end products. When the finite zombi is modified to let zero, one, or more passages from a province on the same input symbol so it is called a nondeterministic finite zombi. For deterministic zombis the result is a province, i.e. , an component of Q. For nondeterministic zombis the result is a subset of Q, where Q is a finite nonempty set of provinces. Automata theory is the survey of abstract calculating devices or machines. It is a behavior theoretical account composed of a finite figure of provinces, passage between those provinces and actions in which one can inspect the manner logic tallies when certain conditions are met. Recently finite province machines are used in cryptanalysis, non merely to code the message, but besides to keep secretiveness of the message.

In this paper, new secret sharing strategy is proposed utilizing finite province machines.

A finite province machine or finite-state zombi is a mathematical theoretical account of calculation used to plan both computing machine plans and consecutive logic circuits. It is conceived as an abstract machine that can be in one of the finite provinces. The machine is merely in one province at a clip, the province it is in at any given clip is called the current province. It can alter from one province to another when initiated by triping event or status, this called a passage.

Automata theory is a key to package for verifying systems of all types that have a finite figure of distinguishable provinces, such as communicating protocols or protocol for unafraid exchange of information. In Mealy Machine every finite province machine has a fixed end product. Mathematically Mealy machine is a six-tuple machine and is defined as:

M= ( )

: A nonempty finite set of province in Mealy machine

: A nonempty finite set of inputs.

: A nonempty finite set of end products.

: It is a passage map which takes two statements

one is input province and another is input symbol. The out put

of this map is a individual province.

: Is a function map which maps ten to, giving the

end product associated with each passage.

: Is the initial province in Q.

Mealy machine can besides be represented by passage tabular array, every bit good as passage diagram.

Now, we consider a Mealy machine.

Fig.1 Mealy machine

In the above diagram 0/1 represent input/output.

## Recurrence Matrix

Recurrence matrix is a matrix whose elements are taken from a return relation.

The return matrix is a symmetric matrix defined as

Rn=

where n and ‘s are Catalan Numberss taken from Catalan sequence, where,

Cn =2n! / ( n+1 ) ! *n! , n1.

The above expression gives us the Catalan sequence- 1, 1, 2, 5, 14, 42,132

C1, C2, C3, C4, C5, C6, C7aˆ¦aˆ¦aˆ¦ .

## Offset Matrix

We apply the beginning regulation by adding 2 to the input at each phase which decides the value of N of the return matrix obtained from Catalan sequence.

Let us take nine readings to organize a 3×3 matrix P which is considered as field text matrix.

## 2. Proposed Algorithm

## Encoding

Measure 1

Let the field text P be a square matrix of order Ns, n & gt ; 0. P is factorized utilizing one of the factorisation techniques.

Crout ‘s factorisation technique is applied in this instance.

Measure 2

Define Finite province machine through public channel.

Here Mealy machine is publicized.

Measure 3

Define return relation and return matrix.

The return matrix and return relation used in this paper has already been discussed above and Ns of the return matrix depends on input at each phase which is,

n= input+2

Measure 4

Sum of all the rows and columns is calculated and the least two among them are selected as keys. They are converted into the binary signifier, the least among the selected two is sent to receiver one and the other one to receiver two.

Measure 5

Define Cipher Text at Q ( i+1 ) Thursday province.

Cipher text at Q ( i+1 ) Thursday province depends on the input spot. When input spot is ‘0’then cypher text at Q ( i+1 ) Thursday province = Rn + cypher text at Q ( I ) Thursday province.

When end product =0, cipher text at Q ( i+1 ) Thursday state= Rn * cypher text at Q ( I ) Thursday province.

## Decoding

Decrypt the message utilizing the opposite operation and key to acquire the original message.

## 3 Performance analysis

## Mathematical analysis

Algorithm proposed, is a simple application of add-on or generation of two matrices. But the operations are different depending on secret key. It is really hard to interrupt the cypher text without proper key, defined operation and the chosen finite province machine.The key is defined as the amount of specific row or column elements in field text.

## Time complexness

Let the amount of the end product of the finite province machine for thousand spot secret key is r, allow be the clip required for each generation and be the clip required for each add-on. Let secret k spot cardinal consists of figure of ‘0 ‘s and figure of ‘1 ‘s. Then the entire clip required for thousand spot secret key ( ( amount of end product for ‘1 ‘ spot ) ( amount of end product for ‘0 ‘ spot )

## 4. Security analysis

Extracting, the original information from the Cipher text is hard due to the choice of the return matrix, secret key and chosen finite province machines. Brute force onslaught on key is besides hard because of the cardinal size.

Table 1 Security analysis

## S.No

## Name of the onslaught

## Possibility of the onslaught

## Remarks

1

Cipher text onslaught

It is hard to check the cypher text.

Because of the chosen finite province machines and different single keys.

2

Known field text onslaught

Difficult

Because of the chosen finite province machines and different single keys.

3

Chosen plain text onslaught

Difficult

Because of the chosen finite province machines and different

single keys.

4

Adaptive chosen field text onslaught

Difficult

Because of the chosen finite province machines and different single keys.

5

Chosen cypher text onslaught

Difficult to check Cipher text

Because of the chosen finite province machines, different single keys and matrix generation.

6

Adaptive chosen cypher text onslaught

Difficult to check Cipher text

Because of the chosen finite province machines, different single keys and matrix generation.

## 5. Execution

We assign 1 to missive a, 2 to missive B and so on and 26 to the missive omega.

Let us code the word ‘AUTHENTIC ‘ .

As per the algorithm, we construct a square matrix of order N as under:

P==

Where, allow ‘P ‘ be the field text.

Now P is factorized into L and U, where

L= and

U=

Sum of rows and columns of matrix P

R1=42, R2=27, R3=32

C1=29, C2=35, C3=37

R2 & lt ; C1 & lt ; R3 & lt ; C2 & lt ; C3 & lt ; R1

( 27 ) 2=11011, sent to receiver one

( 29 ) 2=11101, sent to receiver two.

The return matrix is

Rn =

The value of N in return matrix depends on input as proposed in the algorithm.

Cipher text at Q ( i+1 ) Thursday province depends on the end product.

Receiver 1

Input signal

End product

( passage )

Recurrence matrix

Cipher text

1

0

1

1

0

0

1

0

1

1

Receiver 2

Input signal

End product

( passage )

Recurrence

matrix

Cipher text

1

0

1

1

1

1

0

0

1

0

Cipher text for receiving system one is

Cipher text for receiving system two is

## Reasoning Remarks

Algorithm proposed, is based on finite province machine and different operations on matrices. Secrecy is maintained at four degrees, the secret key, the chosen finite province machine, the different operations, and the return matrix. The obtained cypher text becomes rather hard to interrupt or to pull out the original information even if the algorithm is known.