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ENCRYPTION AND DECRYPTION SCHEME INVOLVING

FINITE STATE MACHINE AND LU DECOMPOSITION By

Ayush Mittal, Dr. Ravindra Kumar Gupta

Department of Computer Science & Engineering

S. R. K. University, Bhopal, (M.P.), India

[email protected], [email protected]

Abstract:

Objectives: The aim of this study is to introduce a new encryption and decryption

scheme involving finite state machine and LU decomposition method.

Methods/Statistical analysis: The designed encryption scheme using finite state

machine and LU decomposition method maintained the secrecy of data for secure

communication. In this proposed technique we also used lower and upper

triangular matrices, which is obtain from square matrix by decomposition. Under

modulation of prime number, the key will be lower triangular matrix in

encryption process, while in decryption process the key will be upper triangular

matrix.

Findings: Due to the chosen finite state machine alongwith different encryption

and decryption key which is generated from randomly chosen a non-singular

square matrix, it is very difficult to extract the original information. Brute force

attack is also difficult because of the size of key.

Application/Improvements: The technique is useful in areas where sensitive

information is to be transferred like banking, military services. If the private key

has got into third party hands, the damage can be huge. To overcome this, the

proposed algorithm used finite state machine and LU decomposition for

encrypting the messages which will provide different keys for encrypting different

data blocks.

Keywords: Encryption, Decryption, Finite State Machine and LU decomposition.

1. INTRODUCTION:

There are two important factors in cryptography, encryption and

decryption. To secure the information both the techniques are mainly used. To

convert plain text into cipher text we used encryption method. The reverse

process of encryption is known as decryption i.e. conversion of cipher text into

plain text. Cryptographer is a person who did practice of encryption and

decryption. When user has a certain piece of secret knowledge then information is

determined. Key is known as secret knowledge, which is send to the receiver.

Changing of information from one form to another form in a secure

manner is known as encryption process in cryptography. Thus this change type of

Journal of Xi'an University of Architecture & Technology

Volume XII, Issue II, 2020

Issn No : 1006-7930

Page No: 1270

2

information can't read by unauthorized people, so cipher text is the value of

encryption. A data which is stored in a computer system or transmitted via

internet confidentiality is the purpose of encryption. The major part in an

encryption is the encryption key. There are two types of key, public and private

key. In encryption and decryption process both two keys are mainly used. Public

key is available for everyone, but private key must be kept as secret. The strength

of encryption is directly related to key size. Thus breaking of encrypted data is so

difficult when key size is increased.

When encrypted text is converted into a text that we or our computer are

able to read and understand, then this process is called decryption. Decryption

describes a process in which data is un-encrypted manually or using the proper

codes or keys. Without knowing the secret key it is very difficult to decrypt the

data. We will get the original text after decryption [1]

.

Application of automaton theory having a wide range in the field of

cryptography. In automata theory, a deterministic finite automaton (DFA) is a

branch of theoretical computer science, which is also known as deterministic

finite state machine. A unique computation is produced by finite state machine for

each input string. Uniqueness of the computation an deterministic having same

meaning. A mathematical model of a system is known as finite automaton with

discrete inputs and outputs. When finite automata allows 0, 1 or more transitions

from a state on the same input symbol then it is called non-deterministic finite

automata. If S is the finite non-empty set of states, then a state or element of S is

the outcome for deterministic automata, while in the case of non-deterministic

automata outcome is the subset of S. Thus a behavior model which composed a

finite number of states and transition between them is known as finite state

machine. To encrypt the data as well as maintain secrecy of the data, finite state

machine is used recently in cryptography.

Finite State Machine:

A finite state machine is also known as a mathematical model of

computation which is used to design both sequential logic circuit and computer

program [2]

. Finite state machine is also an abstract machine and one of the finite

states it can be reside. At a time it can be reside in one state only. A state in which

machine can reside at any given time is called current state. Transition is a

process in which machine can change from one state to another state and it is

initiated by triggering event or condition.

A finite machine F having 6-tuple (R, In, Ou, h, x, t0), where

1. R is a finite set of states (t0, t1, t2, …) whose elements are called state of

machine.

2. t0 is a special element of R referred to as the initial state of the machine.

3. In = {i1, i2, …} is a finite set of input letters.

4. Ou = {o1, o2, …} is a finite set of output letters.



Issn No : 1006-7930

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3

5. h is a function from R In to R called the transition function.

6. x is a function from R to Ou called the output function.

A finite state machine is resided in one of its states at any instant.

According to the transition function the machine will go to another state after

receiving an input symbol. According to the output function, machine produces an

output at each state. Machine is resided in the initial state t0 at very beginning.

There are two types of Finite state machines as follows:

1. Moore machine

2. Mealy Machine

When outputs depended on only the present state then machine is called

Moore machine. On other hand a finite state machine is called Mealy machine

when output depends on the present state as well as the present input. We used

Mealy machine in this paper.

Decomposition method:

In this method a matrix can be expressed as the product of a lower

triangular matrix and an upper triangular matrix where all the principal minors of

the matrix are non-singular [3]

.

Consider a matrix A of order n can be expressed as product of two

triangular matrices, one is lower triangular and another is upper triangular, then

[

] [

] [

]

where [

] and [

]

There are three types of decomposition method viz. Doolittle, Crout and

Cholesky. In this paper we will used Doolittle method. In Doolittle method to

simplify the calculations we choose ( ) = (1, 1,..., 1), therefore

[

] [

] [

]

where [

] and [

]

2. LITERATURE REVIEW:

For encrypting and hiding the data using finite state machines various

researchers [4-6]

are developed an innovative technique. To maintain secrecy of the



Issn No : 1006-7930

Page No: 1272

https://en.wikipedia.org/wiki/Doolittle_decomposition



4

message they also used Fourier sine and cosine transformation, Fibonacci series

and other tools.

3. METHODOLOGY:

In this paper we propose to develop a new cryptographic scheme

involving finite state machine and LU decomposition method following on the

lines of [4-6]

and various researchers. On the other hand, under modulation of

prime number, to encrypt the message we use lower triangular matrix at each

stage and for decrypting the cipher text we use upper triangular matrix at each

stage. Proposed algorithm is based on LU decomposition method of a matrix as

secret key and chosen finite state machine.

On the other hand for double security system we consider n consecutive

plaintext letters and n cipher text letters substitute’s for them. n determine the

substitution, where n is a positive integer. The idea is, in one plaintext element for

n alphabetic characters take m linear combinations and produce n × n constant

matrix C. In this technique we generate the key matrix using LU Decomposition

method A = LU and g.c.d ((det A)mod p, p) = 1. Therefore constant matrix C =

APT ⟹ C = (LU)PT. We use LCT = C ⟹ CT = L−1

C for encryption process and

UPT = CT ⟹ PT = U−1

CT for decryption process, where PT is plain text and CT is

cipher text.

Secrecy is maintained by secret key, private key and Finite state machine.

It is very challenging to break the cipher text without chosen finite state machine

and proper key. In the input sequence the number of elements must be maximum

to avoid the cipher attacks.

4. ALGORITHM:

For substitution, we use the following conversion table for

alphabets/symbols.

Table I

alphabet/

symbol

numerical

value

alphabet/

symbol

numerical

value A 1 P 16

B 2 Q 17

C 3 R 18

D 4 S 19

E 5 T 20

F 6 U 21

G 7 V 22

H 8 W 23

I 9 X 24

J 10 Y 25

K 11 Z 26



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5

L 12 Space 27

M 13 # 28

N 14 @ 0

O 15

Encryption:

1. Take the plain text and using Table I (for which sender and receiver are

agreed) to convert each alphabet of plain text into their corresponding

numeric value. After then plain text is divided into n number of texts. A

plain text matrix is created by arranging the text into a square matrix of

order n, n > 0.

2. After adding all the elements of plain text matrix, convert them into binary

form. This binary form is called secrete key and input also.

3. Consider a Finite state machine. For this in this paper we choose Mealy

machine. Under residue mod p (for which sender and receiver are agreed),

obtained the output from Finite state machine.

4. As key matrix choose a non-singular matrix.

5. Using the following formula, calculate the cipher text matrix at each stage

for all plain text matrix:

Pi = P (= Xi) + [output at ith

state] In n, i = 1, 2, 3, …..;

Ci = APi (mod p);

Xi = L−1

Ci (mod p),

where P is the chosen plain text, Pi is the plain text at ith

stage, Ci is the

constant matrix at ith

stage, Xi is the cipher text at ith

stage, I is identity

matrix of order n and L is the lower triangular matrix as encryption key.

6. Using Table I, numeric value of each element of last cipher text matrix is

converted into corresponding alphabet. Finally, this cipher text is send to

the receiver.

Decryption:

1. Finite state machine, Decryption key, Secret key and Cipher text are

received by receiver.

2. Using Table I, each alphabets/symbols of the cipher text are converted into

corresponding numeric value. After then following formula is applied:

Pi = U−1

Xi (mod p)

Xi = Pi − [output at ith

state] In n, i = 1, 2, 3, …..;

where Pi is the plain text at ith


stage, I is

identity matrix of order n and U is the upper triangular matrix as

decryption key.

3. The original message will be obtain finally.

Illustration:

Encryption Steps:

1. Choose the plain text message as ENCRYPT MESSAGE.



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2. Using above given Table 1, chosen plain text is converted into corresponding

numerical values as follows:

[

]

3. Add all elements of plain text P, we get 224. Convert it into binary form, which

gives 224 = (11100000)2. Choose 11100000 as secret key.

4. Now we define a Mealy machine as follows for input key 11100000 with

output under residue mod 4 (say):

Sl.No. Input Previous

State

Present

State

Output Cipher matrix

1 1 q0 q1 1

[

]

2 1 q1 q2 3

[

]

3 1 q2 q3 3

[

]

4 0 q3 q1 2

[

]

5 0 q1 q0 0

[

]

6 0 q0 q2 0

[

]

7 0 q2 q0 0

[

]

8 0 q0 q0 0

[

]



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7

5. Consider a non-singular matrix A of order 4 4 as the key matrix given by

[

]

Let A = LU

where

[

] [

] [

]

For LU decomposition of a matrix in Dolittle's method we choose

( ) = (1, 1, 1, 1), therefore Therefore

[

] [

] [

]

Hence

[

]

[

]

After equating the corresponding elements in above equation, we get Therefore after simplification, we get



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Hence

[

] [

] [

]

6. Now calculate

Pi = P (= Xi) + [output at ith

state] I4 4, i = 1, 2, 3, 4, 5, 6, 7, 8;

Ci = APi (mod 29);

Xi = L−1

Ci (mod 29),

where P is the chosen plain text, Pi is the plain text at ith

stage, Ci is the constant

matrix at ith


stage and I is identity matrix of order

4. Therefore

(i) P1 = P + [output at 1st state] I4 4

= [

] [

]= [

]

and C1 = AP1(mod 29)

= [

] [

] (mod 29)

= [

]

So X1 = L−1

C1(mod 29)

= [

] [

](mod 29)

= [

]

(ii) P2 = P(= X1) + [output at 2nd

state] I4 4



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9

= [

] [

]= [

]


= [

] [

] (mod 29)

= [

]

So X2 = L−1

C2(mod 29)

= [

] [

](mod 29)

= [

]

(iii) P3 = P(= X2) + [output at 3rd

state] I4 4

= [

] [

]= [

]


= [

] [

] (mod 29)

= [

]

So X3 = L−1

C3(mod 29)

= [

] [

](mod 29)

= [

]



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10

(iv) P4 = P(= X3) + [output at 4th

state] I4 4

= [

] [

]= [

]


= [

] [

] (mod 29)

= [

]

So X4 = L−1

C4(mod 29)

= [

] [

](mod 29)

= [

]

(v) P5 = P(= X4) + [output at 5th

state] I4 4

= [

]


= [

] [

] (mod 29)

= [

]

So X5 = L−1

C5(mod 29)

= [

] [

](mod 29)



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= [

]

(vi) P6 = P(= X5) + [output at 6th

state] I4 4

= [

]


= [

] [

] (mod 29)

= [

]

So X6 = L−1

C6(mod 29)

= [

] [

](mod 29)

= [

]

(vii) P7 = P(= X6) + [output at 7th

state] I4 4

= [

]


= [

] [

] (mod 29)

= [

]

So X7 = L−1

C7(mod 29)



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12

= [

] [

](mod 29)

= [

]

(viii) P8 = P(= X7) + [output at 8th

state] I4 4

= [

]


= [

] [

] (mod 29)

= [

]

So X8 = L−1

C8(mod 29)

= [

] [

](mod 29)

= [

]

Therefore cipher text is

C@ZYRCLGKMYRXWDD

Decryption Steps:

1. Take cipher text C@ZYRCLGKMYRXWDD.

2. Using Table 1, find corresponding numerical values of the above cipher text.

After arranging these numerical values into a matrix of order 4, we get as follows:

[

] X8 (say)

3. Now calculate

Pi = U−1

Xi (mod 29)



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13

Xi = Pi − [output at ith

state] I4 4, i = 1, 2, 3, 4, 5, 6, 7, 8;

where Pi is the plain text at ith


stage, I is

identity matrix of order n and U is the upper triangular matrix as

decryption key.

(i) P8 = U−1

X8 (mod 29)

= [

] [

] (mod 29)

= [

]

X7 = P8 − [output at 8th

state] I4 4

X7 = [

]

(ii) P7 = U−1

X7 (mod 29)

= [

] [

] (mod 29)

= [

]


state] I4 4

= [

]

(iii) P6 = U−1

X6 (mod 29)

= [

] [

] (mod 29)

= [

]


state] I4 4



Issn No : 1006-7930

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= [

]

(iv) P5 = U−1

X5 (mod 29)

= [

] [

] (mod 29)

= [

]


state] I4 4

= [

]

(v) P4 = U−1

X4 (mod 29)

= [

] [

] (mod 29)

= [

]


state] I4 4

= [

]

(vi) P3 = U−1

X3 (mod 29)

= [

] [

] (mod 29)

= [

]

X2 = P3 − [output at 3rd

state] I4 4



Issn No : 1006-7930

Page No: 1283

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= [

]

(vii) P2 = U−1

X2 (mod 29)

= [

] [

] (mod 29)

= [

]

X1 = P2 − [output at 2nd

state] I4 4

= [

]

(viii) P1 = U−1

X1 (mod 29)

= [

] [

] (mod 29)

= [

]

P = P1 − [output at 1st state] I4 4

= [

]

4. Using Table I, the numeric values of each element of above matrix are

converted into corresponding alphabet/symbol. Finally we obtain original plain

text as given below:

ENCRYPT MESSAGE

5. RESULT AND DISCUSSION:

To encrypt and decrypt the message there is no single algorithm is

sufficient to fulfill the purpose. Therefore to remove the deficiency and finding

better solution a number of researchers are working in the field of cryptography.

In this paper we develop a new algorithm which involves finite state machine and

LU decomposition of a square matrix. In LU decomposition method we used



Issn No : 1006-7930

Page No: 1284

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lower triangular matrix as encryption key and upper triangular matrix as

decryption key.

By virtue of chosen finite state machine alongwith different encryption

and decryption key which is generated from randomly chosen a non-singular

square matrix, extraction of original information is very difficult. Due to the size

of key, brute force attack is also difficult.

6. CONCLUSION:

There are various advantages and disadvantages of an algorithm. Since our

algorithm is developed by finite state machine and LU decomposition of a square

matrix, therefore our system proposed a good strategy. To encrypt and decrypt the

messages, symmetric key generation is highly secure due to the decomposition of

the matrix into lower and upper triangle matrices. In the proposed algorithm, for

encryption process the lower triangle matrix is used, while for decryption process

upper triangle matrix is used over residue modulo prime number, therefore for

plaintext attack and cipher text attack the proposed cryptosystem get better

protection.

Hence, our system also proposed a new encryption algorithm, which

provides enough security. Due to chosen finite state machine, secret key, different

operations on matrices and LU decomposition, four levels of security are

maintained. However, algorithm is also known, extraction of original information

from cipher text is fairly tough.

REFERENCES

1. Stallings W.: Cryptography and Network Security: Principles and

Practices, Prentice Hall, 2002.

2. John Hopcroft, Jeffrey Ullman: Introduction to Automata Theory,

Languages, and Computation, Addison-Wesley, 2001.

3. Strang Gilbert: Introduction to Linear Algebra, Wellesley-Cambridge

Press, Fifth Edition, 2016.

4. G. Karudaiyar, S.Karthikeyan and B. Sainath3: Encryption and Decryption

Scheme by Using Finite State Machine, Biosciences Biotechnology

Research ASIA, Vol. 11(3), 2014, pp. 1861-1865.

5. Ray Prasanta Kumar, Dila Gopal Krishna, and Patel Bijan Kumar:

Application of Some Recurrence Relations to Cryptography using Finite

State Machine, International Journal of Computer Science and Electronics

Engineering, Vol. 2(4), 2014, pp. 220-223.

6. Sirivaram Srilakshmi: New Encryption Scheme Using Fourier Sine and

Cosine Transforms and Finite State Machine, International Journal of

Advanced Research in Computer Science, Vol. 8, No. 6, July 2017, pp.

153-155.



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