Secure and Fast Chaos based Encryption System using Digital Logic Circuit

Rapid development of computer and network technology arise the importance of the network management and security. Network security defined over network containing data integrity, authentication, secrecy, data repudiation and data controllability [1], [2]. The data is secured using some techniques like as cryptology. Cryptology is used for studying the cryptosystem and cryptanalysis [3], [4]. The encryption and decryption algorithms are used for converting the data information into unreadable and readable form in cryptographic system for secure communication [5].

A new Chaotic based Cryptographic Technique is proposed in this paper using digital logic circuits. So the speed of encryption is enhanced. Chaotic maps are used to provide a high degree of randomization and hence increase the confusion and diffusion of the system.

The rest of the paper is organized as follows. In section II, chaos theory, cryptography, chaos based cryptography and literature survey are described. In section III, the proposed cryptosystem is described. In section IV, Proposed cryptosystem has been explained by example. Then, in section V, analysis of security of keys of the proposed approach is provided. In section VI, methodology is checked against all four cryptanalysis attacks. Finally, the paper is concluded in section VII.

• II.    Technical description

• A.    Cryptography

A cryptosystem has four important components [5]. Plain text is the original message for transferring. Cryptographic system is the combination of encryption and decryption system. The ciphertext is the yield of applying an encryption technique to the original message. The key is a combination of bits using in encrypting and decrypting processes [6].

A cryptographic system is represented by:

E k ( P ) = C

D k ( C ) = P

Fig. 1. Cryptographic System

• B.    Cryptanalysis

Cryptanalysis is a procedure which is used to split the code and deduce a particular plain text or the key being used. All future and past information about message encrypted with that key are compromised.

Table 1. Ciphertext Attack

Type of Attack	Known to Cryptanalyst
Ciphertext only	•     Encryption algorithm •     Ciphertext to be decoded
Known   plain text	•     Encryption algorithm •     Ciphertext to be decoded •     One or more plain text-cipher text pairs formed with the secret key
Chosen   plain text	•     Encryption algorithm •     Ciphertext to be decoded •     Plain text message  chosen  by cryptanalyst,   together  with  its corresponding cipher text generated with the secret key
Chosen  cipher text	•     Encryption algorithm •     Ciphertext to be decoded •     The purported cipher text chosen by cryptanalyst,   together  with  its corresponding decrypted plain text generated with the secret key
Chosen text	•     Encryption algorithm •     Ciphertext to be decoded •     Plain text  message  chosen  by cryptanalyst,   together  with  its corresponding cipher text generated with the secret key The purported cipher text chosen by cryptanalyst,   together with  its corresponding decrypted plain text generated with the secret key

The table contains the several types of cryptanalysis attacks depend upon the amount of information identified by the cryptanalyst.

• C.    Chaos Theory

Chaotic system or chaos for short is meddling between rigid regularity and randomness based on possibility [7]. Chaos can be defined by some typical characteristics.

Nonlinearity [8] in this property, the smallest change in data at any instant can result in a change in the same or a different data at a later time, that is not depended to the change at the initial time [9], [10].

Determinism it has provided determinism which is controlled by exact rules with no element of chance.

Sensitivity to initial condition Negligible changes in its initial variable can give completely different final variable [11], [12].

Irregularity the behavior of chaos is not predictable. It is provided irregularity.

Long term prediction chaos gives long term uncontrolled dynamic prediction. Chaos can be controlled conditionally [13].

A useful feature of chaotic systems is their capability of producing complex patterns of behavior [14]. This is performed by simple real systems with a small set of evolution equations [14]. These features have made a chaotic system useful for several applications in many disciplines, such as biology, economics, engineering, neural network and others [15].

Fig. 2. Chaos Iterative Function

• D.    Chaos And Cryptology

Chaos and cryptology are related with each other by some features: Both (chaotic and encryption techniques) are deterministic in nature [16]. Both are provided with highly complex and unpredictable variables [17]. It means random nature for any external observer, not having any prior knowledge of the algorithm and initial condition – key [18], [19].

Chaos is sensitive to initial conditions ie. Negligible difference in any variable totally changes the outputs [20]. Cryptography is depending on confusion and diffusion with key dependency ie. Updating of 1 bit of the plaintext or key should change all bits of the cipher text with 60% probability [21].

Chaos is topological transitive and repetitive process and cryptography is multiple round transformations and repetitive transformations with a single logistic map function [22].

Chaos is also different from cryptology: Chaos based systems are defined on bounded continuous space and cryptography is defined over finite discrete space.

Chaos theory has found a way for understanding the asymptotic performance of repetitive process, whereas cryptography based on the properties of first few repetitions [23].

Table 2. Chaos Theory Vs. Cryptography

Chaos Theory	Cryptography
Chaos based system	Pseudo-chaos based system
Random transformation	Random transformation
Infinite number of stages	Finite number of stages
Infinite number of repetitions	Finite number of repetitions
Initial stage	Plain text
Final stage	Ciphertext
Initial circumstances and/or parameters	Key
Asymptotic liberty of initial and final stages	Confusion
Compassion to initial circumstances and parameter mixing	Diffusion

Fig. 3. Relation between Chaos and Cryptography

• E.    Motivation

Chaos theory for cryptosystem has been broadly discussed for secure communication in the last few years. Chaos system is generally defined in a symmetric encryption system with some complex mathematical function [23]. It will illustrate that the keys with negligible change produce different cipher texts. Cryptanalysis shows the resistivity against several attacks and stronger than existing encryption technique [24]. Cryptanalysis shows that there are negligible changes in key generate diffusion. A block encryption technique used dynamic sequences by single and multiple chaotic systems [25], [26]. Several one-dimension logistic maps are used to provide pseudorandom sequences, which are independent, nonlinear and approximately uniform [27]. The chaotic masking technique is used for encrypting the transmitting messages with a binary sequence extracted from a logistic map [28]. A symbolic sequence generated by another skew tent chaotic map is provided the masked message sequence which is tracked [28], [29]. The theoretical and simulation results explain many characteristics such as high speed, easy implementing, accuracy and high security. Therefore, it is suitable for practical use in the secure communication between two private parties [30], [31].

The chaotic properties such as ergodicity, sensitive dependency on initial conditions and system parameters have been properly utilized in encryption [31]. The recent advance researches on information security technique are getting more imperative to the development of network management and security technique. Today, Most of data information is transmitted in the form of a stream of bits over the internet. Jinhong Luo proposes a way of adjusting the corresponding sequence of key encryption and plain message to get the best anti decryption signed and shows its advantages and disadvantages by analyzing this arithmetic [32].

The parameter of the chaotic Lorenz system is also used in a two-channel cryptosystem first the geometrical properties of the Lorenz system and second the parameters which are precisely determined - directly from the cipher text - through the minimization of the average jamming noise power generated by the encryption procedure [32], [33].

Chaotic encryption schemes are provided superior level of security than conventional ciphers [34]. A chaotic system is also used with ECG signal to provide higher speed and security with encryption [35]. The hardware implementation of chaotic system details over Xilinx Virtex - 6 FPGA is provided [36]. Logistic map has high potential to be used to design a stream cipher for real-time embedded systems [37], [38]. Fu’s chaotic cipher is secure and the information leak of chaos map is discovered [39].

A new scheme is used which performs lossless compression is based on the arithmetic coding (AC) as well as encryption of data is based on a pseudo random bit generator (PRBG). The standard logistic map based PRBG and the Engel Continued Fraction (ECF) map to generate a key stream with both chaotic and algebraic characteristics. The effectiveness of the BAC in lossless data compression and the compensation of chaotic theory in data encryption to offer a method used in many applications such as multimedia applications and medical Imaging [39], [40].

• III. Proposed methodology

It has been proposed a Symmetric key encryption technique which is used one or more than one key for encryption and decryption process. But these keys are similar for encryption and decryption process at any instant of time. It means different keys are used for different messages for enhancing the security. Firstly generates the number of keys by using the chaos logistic function (logistic map) providing only initial condition [40].

For n=1 to j

Xn+1 = {A * Xn (Xn -1)}MOD256

A= any integer (1, 2, 3,......)

X = initial value of chaotic function which is 2, 3,..........

j = Number of keys

These keys are controlled by providing some satisfactory condition. So the whole letters of the message are encrypted with these keys. Using multiple keys are enhanced the security and it is not necessary that any repeated letters in the message are encrypted by the same key [40].

It has been provided multiple different keys for different messages and also applied some digital circuit system concept of the keys to increase the complexity of the keys and speed of the encryption and decryption so the intruder is not found that how keys are coming [41].

The following notations are used in the key generation encryption and decryption scheme.

P = Plain text

C = Cipher text

E ( P )= Encryption the plaintext P .

D ( C ) = Decryption the ciphertext C

For n=1 to j

Xn+i = {A * Xn (X„-1)}MOD256

A= any integer (1, 2, 3,......)

X = initial value of chaotic function which is 2, 3,4,…………… j = Number of keys

Xn+1 = keys K1, K2, K3,..............K,  (after applying gray code on Xn+1)

• A.    Key generation Scheme

• 1.    Firstly generates the pseudo random numbers by using the logistic map function at both ends (sender and receiver) by using an equation.

• 2.    Then generate different multiple values of X (used for keys after applying gray code on these values of X ) and fixed the random numbers n +1 ⁷

• 3.    Complexity of keys is increased by applying a

• 4.    The keys are shown in 8 bit binary form.

For n=1 to j

Xn+1 = {A * Xn (Xn-1)}MOD256

by using some specific condition j.

gray code on X so keys are independent with each other.

• B.    Encryption Scheme

Each character is shown in ASCII character format. Then ASCII characters are converted into 8 bit binary numbers with respect to their decimal numbers after that these characters are encrypted by using a digital logic XOR function. Then perform XOR operation on each character by a single binary coded key. Keys are also repeated for encrypting the whole message.

P = ASCII (character1)

P converted into 8 bit binary numbers.

E,( P) = C1

P = ASCII (character 2)

P is converted in 8 bit binary numbers.

E k ( P ₂) = C 2

P = ASCII (character i)

P is converted in 8 bit binary numbers.

E m ( P ) = C i

Where m = 1 to j

• C.    Decryption Scheme

The cipher texts are decrypted (converted into plain text) by using the reverse process of encryption scheme .

P = D k . ( C i )

P is converted into ASCII( P ) with respect to its decimal value.

Character1 = ASCII( P )

P 2 = D_k 2 ( C 2 )

P is converted into ASCII( P ) with respect to its decimal value.

Character2 = ASCII( P )

P = D k m ( C i )

Where m= 1 to j	K	75	01001011
P is converted into ASCII( P ) with respect to its	U	85	01010101
decimal value.	R	82	01000001
	A	65	01000001

Character i = ASCII( P_i )                  A_. K_ey G_enerati_on

• D.    Algorithm for key generation

• 1.    Decide the values of parameter (M, A, X ).

• 2.    Generate the pseudo random numbers from the equation.

• 3.    Apply gray code on these random numbers which are generated from X to build up the keys K , K , K ,.............. K .

• 4.    Keys are represented in 8 bit binary form.

For n=1 to j

Xn+1 = {A * Xn (Xn-1)}MOD256

• E.    Algorithm for encryption

• 1.    Each character is shown in ASCII character, P = ASCII(character i).

• 2.    ASCII character P is converted into 8 bit binary form.

• 3.    Using the equation E_k ( p ) = C. for all i>0, and m = 1 to j for encryption, Where E ( P ) is bit wise XORing on plaintext P with single key K m .

• 4.    Find the 1`s complement of ciphertext ( C ).

• F.    Algorithm for decryption

• 1.    Find the 1`s complement of receiving ciphertext ( C i ).

• 2.    Using the equation p = D_k ( C ) . Where m=1 to j for decryption, where D (C ) is bit wise XORing on cipher text C with single key K .

• 3.    Plain text P is converted into ASCII( P ) with respect to its decimal value.

• 4.    Then Character i = ASCII ( P ).

Xn+1 = {A * Xn (Xn -1)}MOD256

Let A=2, X =3, j=4

Xn+1 = 12, 8, 112, 32.

The keys are generated by applying a gray code on random numbers X .

12 = 00001100

Gray code (0,0 XOR 0,0 XOR 0, 0 XOR 0, 0 XOR 1, 1

XOR 1, 1 XOR 0, 0 XOR 0) =00001010= 10

8=00001000 gray code = 00001100 = 12 112=01110000 gray code = 01001000 = 72 32= 00100000 gray code = 00110000 = 48

Keys ( K = 10, K =12, K =72, K = 48)

• B.    Encryption Algorithm

Encryption algorithm converts the plain text into cipher text by using multiple keys.

Table 3. Cipher Text For Giving Plain Text

Plain text (ASCII)	XOR	Key K j	Cipher Text	1`s complemented Cipher Text
A 65 01000001	XOR	10 00001010	01001011	10110100 180 H
N 78 01001110	XOR	12 00001100	01000010	10111101 189 ^
K 75 01001011	XOR	72 01001000	00000011	11111100 252 n
U 85 01010101	XOR	48 00110000	01100101	10011010 154
R 82 01010010	XOR	10 00001010	01011000	10100111 167 º
A 65 01000001	XOR	12 00001100	01001101	10110010 178

• IV. Example

Given plain text - ANKURA

Letter	ASCII	Binary Number
A	65	01000001
N	78	01001110

Ciphertext - H ^”U |

• C.    Decryption Algorithm

Decryption algorithm converts the cipher text into plain text by using the same key.

Cipher Text: ^^"U ^

Table 4. Plain Text For Cipher Text

Received Cipher Text	1`s complemente d cipher text	XOR	Key K j	Plain text (ASCII)
^ 180 10110100	01001011	XOR	10 00001010	01000001 65 A
^ 189 10111101	01000010	XOR	12 00001100	01001110 78 N
“ 252 111111100	00000011	XOR	72 01001000	01001011 75K
154 10011010	01100101	XOR	48 00110000	01010101 85 U
º 167 10100111	01011000	XOR	10 00001010	01010010 82 R
178 10110010	01001101	XOR	12 00001100	01000001 65 A

Plain text - ANKURA

• V.    Analysis of security of keys

To find all the values (M, A, X ) is hard to compute. In this section we show the sensitivity of the secret key with negligible difference in the key parameters.

Xn+i = {A * Xn (Xn —1)}MOD256

Plain text: - ANKURA

Parameter (j, A, X )

• A.    Sensitivity of number of keys j

It has been claimed that if a number of keys are changed, then the cipher texts are also completely changed from one another for same plain text.

Table 5. Sensitivity Of Number Of Keys j

J	A	X n	X n + 1	KEYS ( K j )	CIPHER TEXT	1`s Complem ented cipher Text
1	2	3	12	10,10,10,10 ,10,10	KDA_XK	Ъ^Ч
2	2	3	12,8	10,12,10,12 ,10,12	KBAYX M	^”i
3	2	3	12,8,11 2	10,12,72,10 ,12,72	KBETX_^ TAB	-^иГ
4	2	3	12,8,11 2,32	10,12,72,48 ,10,12	KBETXe XM	^^°u°i
5	2	3	12,8,11 2,32,19 2	10,12,72,48 ,160,10	KBETXe> =K	^•|'('R

In table 5 negligible differences in the j (number of keys) produced different cipher text.

• B.    Sensitivity of constant A

It is claimed that negligible changes in constant A generated totally different keys so cipher text are also different from each other.

Table 6. Sensitivity Of Constant A

J	A	X n	X n + 1	KEYS ( K j )	CIPHER TEXT	1`s Compleme nted cipher Text
4	2	3	12,8,112 ,32	10,12,72,4 8,10,12	KBETXe XM	V°u°i
4	3	3	18,150,2 34,238	27,221,15 9,153,27,2 21	Z6^^I£	Nl+3^c
4	4	3	24,160,1 28,0	20,240,19 2,0,20,240	U 1UF	-At-^N
4	5	3	30,254,3 0,254	17,129,17, 129,17,12 9	P^Z ^CL	»0N+4?
4	6	3	36,136,8 0,32	54,204,12 0,48,54,20 4	wé3edì	6}^U0r

In table 6 it has been represented that small difference in the values of constant A generated different cipher text.

• C.    Sensitivity of Initial condition Xn

It has been represented that the small changes in initial condition Xn generated totally different keys, so cipher texts are totally different, it is called confusion.

Xn+i = {A * Xn (Xn -1)}MOD256

Plain text: - ANKURA

Parameter (j, A, X )

Keys (K1, K2, K3,..............Kj)

Table 7. Sensitivity Of Initial Condition X _n

J	A	X n	X n + 1	KEYS ( Kj )	CIPHER TEXT	1`s Complemen ted cipher Text
4	2	3	12,8,112 ,32	10,12,72,48, 10,12	KBETXe XM	H^°u°i
4	2	4	24,80,96 ,64	20,120,80,9 6,20,120	U6;5N9	-p-^ h
4	2	5	40,48,16 0,192	60,40,240,1 60,60,40	}f^Jnl	éÖDLF æî
4	2	6	60,168,4 8,160	34,252,40,2 40,34,252	cicNJ^	£M£Z ÅB
4	2	7	84,120,1 44,224	126,68,216, 144,126,68	?LF6a,E NQ	LJ1SUB

In table 7 it has been represented that small difference in values of initial variable X generated totally different ciphertext.

• VI.    Cryptanalysis

Cryptanalysis is used to check the security of algorithm by breaking the codes and find the possible keys and plain text as well.

• A.    Ciphertext only attacks

Parameter (j=4, A=2, X =3)

Keys (10,12,72,48) Given.

C 1 = E k 1 (P 1 ), C 2 = E k ₂ ( P 2 ),................, C i = E k m ( P i )

Where m = 1 to j.

Deduce: - Either P , P , P ,..................... P ;

K 1 , K 2 , K 3 , K 4 ;

Or an algorithm to infer P

From C i + i = ^Ek_m ⁽ P i ₊ i )

Keys (10,12,72,48)

Example

P 1 =N then C i = E ^ ( P i ) = E io (N)= ^

P ₂ =NN then C ₂ = Е_ц ( P 2 ) = Е 1о _Л2 (NN)= ^

1,2

Р з =NNN then C ₃ = E^ _з ( P ₃) = E^ (NNN)= ^^J'

P ₄ =NNNN then C 4 = E   ( P 4 ) = E ^ (NNNN) =

1,2,3,4

^•U

P =NNNNN then    C, = E,    ( P ) = E_{10 12 72 48 10}

• 5                                             5        k _{1 2 3 4 1}     5        10,12,72,48,10

(NNNNN) = ^ -U^

From the above example, it would be said that if any character is repeated many times, the cipher text is not same for same letter N. The Cipher text of character N occuring as the first letter is different from N occuring as nth character in plaintext.

• B.    Known plain text attack

Parameter (j=4, A=2, X =3)

Keys (10, 12, 72, 48)

Given-

where m = 1 to j

Deduce: - Either K , K , K , K ;

Or an algorithm to infer P

From ^Ci + i = ^Ek m ^{( P}i + i )

Keys (10, 12, 72, 48)

Example

P =M then C i = E ^ ( P i ) = E io (M)= =|

P 2 =MM then C 2 = E ( P 2 ) = E io^ (MM)= ^

1,2

P ₃ =MMM then C₃ = E_k  ( P ) = E _{lo 12 72} (MMM)= ^^

3                          3         _1,2,3     3        10,12,72

P 4 =MMMM then C 4 = E^ ₃₄ ( P 4 ) = E io,i2,72,48 (MMMM) = ^•e

P₅   =MMMMM then   C₅ = E,    ( P ) = E_{lo 12 72 48}

• 5                                             5        k _1,2,3,4,1     5        10,12,72,48,10

(mmmmm) = ^ •e^

From the example it can be said that there are several cipher text for several plaintexts. It is difficult to deduce the key or the algorithm for decrypting plaintexts encrypted with the keys. Keys are created by very controlling and sensitive parameters. Cipher text of character M occuring as first letter is not same as M occuring as nth character in the plaintext.

• C.    Chosen plaintext attacks

Parameter (j=4, A=2, X =3)

Keys (10,12,72,48)

Given-

Where the cryptanalysis gets to choose P , P , P ,..................... P and m= 1 to j.

Deduce: - Either P , P , P ,..................... P ;

Or an algorithm to infer P

From C i + i = E k_m ( P i + i )

Keys (10,12,72,48)

Example

P = XY then ciphertext C = E ^ ₂ ( P ) = E ₁₀₁₂ (XY)= -

P = YX then ciphertext C₂ = E_k ( P ₂) = E ₁₀₁₂ (YX) =

2                                            2         _1,2     2        10,12

½¼

It is hard to deduce the key or the algorithm to decrypt the plaintext encrypted with the same keys.

• D.    Chosen ciphertext attack

Parameter (j=4, A=2, X =3)

Keys (10,12,72,48)

Given-

C 1 , P 1 = D k ( C 1 ), C 2 , P 2 = D k ( C 2 ),.........., C i , P i = D k_m ( C i )

where m = 1 to j

Deduce: - K , K , K , K ;

Example

C = ¡ then plaintext P = D ( C ) = D (RU) = XY

1                                            1            _1,2       1           10,12

C = ½¼ then plain text P = D ( C ) = D (ST) = 2                                             2         _1,2     2         10,12

YX

Keys are generated by two different parameters X and A which are very sensitive and different. So it is difficult to compute the keys by knowing the cipher text and its decrypted plaintext.

• VII.    Conclusion

There is an enormous literature of chaotic function based cryptographic techniques. In this paper a new chaotic encryption algorithm is developed using the properties of a chaotic map (sensitivity of parameters) like constant A and initial condition X . Cryptosystem is also depending on the number of keys which are generated by the use of logistic map. It has been also shown that completely different keys are generated when parameters are slightly changed. This enhanced the security of keys. Cryptanalysis is showing that there is a negligible modification of parameters is generated great confusion and diffusion. The algorithm is also worked and analyzed against all four cryptanalysis attack cipher text only, known plain text, chosen plaintext and chosen cipher text attacks and fast computing. Time complexity is reduced by using digital circuits ie. Gray code, XOR gate, and 1`s complement etc. Fast computing is also achieved by digital logics.

Acknoledgement

We grateful to Dr. Piyush Kumar Shukla and Dr. Sanjay Silakari for stimulating discussions.