A New Security Level for Elliptic Curve Cryptosystem Using Cellular Automata Rules

International Journal of Computer Applications Technology and Research
Volume 5–Issue 11, 708-713, 2016, ISSN:-2319–8656
www.ijcat.com 708
A New Security Level for Elliptic Curve Cryptosystem
Using Cellular Automata Rules
Fatima Amounas
R.O.I Group, Computer Sciences Department
Moulay Ismaïl University,
Faculty of Sciences and Technics,
Errachidia, Morocco.
El Hassan El Kinani
A.A Group, Mathematical Department
Moulay Ismaïl University,
Faculty of Sciences and Technics,
Errachidia, Morocco.
Abstract: Elliptic curve cryptography (ECC) is an effective approach to protect privacy and security of information. Encryption
provides only one level of security during transmission over the channel. Hence there is a need for a stronger encryption which is very
hard to break. So, to achieve better results and improve security, information has to pass through several levels of encryption. The aim
of this paper would be to provide two levels of security. First level comprises of plaintext using as security key compressed block to
encrypt text based ECC technique and the second level comprises of scrambling method with compression using 2D Cellular rules. In
particular, we propose an efficient encryption algorithm based ECC using Cellular automata and it is termed as Elliptic Curve
Cryptosystem based Cellular Automata (ECCCA). This paper presents the implementation of ECCCA for communication over
insecure channel. The results are provided to show the encryption performance of the proposed method.
Keywords: Cryptography, Elliptic curve, Cellular automata, Matrix, Scrambling technique, Encryption, Decryption.
1. INTRODUCTION
Security is the important factor in the public network and
cryptography play an important role in this field.
Cryptography is an old art of sending secret messages
between sender and receiver. With the advancement of
internet technologies, cryptography becomes a crucial
aspect for secure communications to protect important
data from eavesdroppers. In fact, cryptography is the
science of devising methods that allow information to be
sent in a secure form in such a way that the only person able
to retrieve this information is the intended recipient.
Cryptography is broadly divided into two categories
depending upon the key, which is defined as the rules used to
convert an original text into encrypted text: - Symmetric Key
Encryption and Asymmetric Key Encryption. Symmetric
Key Encryption uses the same key for encryption and
decryption processes. This technique is simple yet
powerful but key distribution is the chief problem that
needs to be addressed Whereas, Asymmetric Key
Encryption use two mathematically associated keys: Public
Key & Private Key for encryption. The public key is available
to everyone but the data once encrypted by public key of any
user can only be decrypted by private key of that
particular user.
Elliptic curve cryptography is effective security solution to
provide secure communication. Elliptic curve cryptography
transforms a mathematical problem in to an applicable
computer algorithm. Intractable problems are the center of
public key cryptography and bring computationally
demanding operations into a cryptosystem. Elliptic curve
cryptography (ECC) is based upon the algebraic structure of
elliptic curves over finite field. Elliptic curve cryptography is
the most efficient public key encryption scheme based on
elliptic curve concepts that can be used to create faster,
smaller, and efficient cryptographic keys. As result
researchers are engaged to develop different cryptographic
techniques based ECC to enhance network security [1, 2, 3].
Recently, more applications propose to use the elliptic curve
in encryption process and improve their efficiency using
cellular automata [4, 5]. In our previous works [6, 7, 8, 9], we
have proposed cryptographic algorithm for text encryption
using elliptic curve. Basically this paper is proposing a new
encryption algorithm based ECC using the concept of cellular
automata. Finally, expected results are showing the
performance of the proposed algorithm.
The rest of the paper is structured as follows. Section 2 gives
detailed description of commonly employed security concepts
and terminology. In particular, we present basic idea of
elliptic curve cryptography. Section 3 a detailed description of
Cellular automata is presented. In section 4, the proposed
method is introduced. A detailed example is presented that
outlines the working procedure of the proposed method in
section 5. Section 6 presents an implementation of ECCCA
for encryption/decryption process, using Visual Basic as the
implementation tool. Section 7 concludes the paper.
2. CRYPTOGRAPHIC TERMINOLOGY
In this section, we introduce some basics security
terminologies and concepts connected with cryptography. A
message present in a clear form, which can be understood by
any casual observer, is known as the plaintext. The encryption
process converts the plaintext to a form that hides the meaning
of the message from everyone except the valid
communicating parties, and the result is known as the cipher
text. Decryption is the inverse of encryption. The processes
of encryption and decryption are controlled on a quantity
known as the key, which is ideally known only to the valid
users. Strength of a security scheme depends on the secrecy of
the keys used.
A security protocol formally specifies a set of steps to be
followed by communicating parties, so that the mutually
desired security objectives are satisfied. The four main
security objectives include:

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- Confidentiality: This means that the secrecy of the data
being exchanged by the communicating parties is maintained,
i.e., no one other than the legitimate parties should know
the content of the data being exchanged.
- Authentication: It should be possible for the receiver to
ensure that the sender of the message is who he claims to be,
and the message was sent by him.
- Integrity: It provides a means for the receiver of a
message to verify that the message was not altered in
transit. It checks originality of message.
- Non-repudiation: The sender of a message should not be
able to falsely deny later that he sent the message, and
this fact should be verifiable independently by an
independent third-party without knowing too much about
the content of the disputed message(s).
Security protocols realize the security objectives through the
use of appropriate cryptographic algorithms. Security
objectives thus provide trust on the Web. They are realized
through the use of cryptographic algorithms which are
divided into two categories depending on their characteristics:
Symmetric algorithms and Asymmetric algorithms.
2.1 Elliptic Curve Cryptography
Elliptic Curve Cryptography (ECC) was first
introduced by Victor Miller and Neil Koblitz in 1985. The
principal attraction of ECC compared to RSA is that it offers
equal security for a far smaller key size, thereby
reducing processing overhead [10]. The advantage of ECC
over other public key cryptography techniques such as RSA
is that the best known algorithm for solving ECDLP the
underlying hard mathematical problem in ECC takes the
fully exponential time and so far there is a lack of sub
exponential attack on ECC. ECC is based on the
Discrete Logarithmic problem over the points on an
elliptic curve [11].
2.2 Mathematics Background of ECC
Let E be an elliptic curve over Fp, given by an affine
Weierstrass equation of the form:
E: y2
= x3
+ax+b (1)
with coefficients a, b Fp such that 4a3
+27b2
 0. We recall
that the set E(Fp) of points of any elliptic curve E in affine Fp-
valued coordinates form an Abelian group (with a point at
infinity denoted by  as the neutral element).
To encrypt a message, Alice and Bob decide on an
elliptic curve and take a affine point (P) that lies on the
curve. Plaintext M is encoded into a point PM. Alice chooses
a random prime integer nA and Bob chooses a random
prime integer nB. nA and nB are Alice and Bob’s private
key respectively. To generate the public key,
Alice computes,
PA= nAP
and Bob Computes.
PB= nBP
To encrypt a message point PM for Bob, Alice chooses
another random integer named k and computes the
encrypted message PC using Bob’s Public key (PB). PC is a
pair of points:
PC= [(kP), (PM+kPB)]
Alice Sends PC to Bob as a cipher message. Bob, receiving
the encrypted message PC and using his private key, nB,
multiplying with kP and add with second point in the
encrypted message to compute PM, which is corresponding
to the plaintext message M,
PM= (PM+kPB) -[ nB (kP)]
Addition operation for two points P( 1x , 1y ) and Q
( 2x , 2y ) over an elliptic group, if P+Q= ( 3x , 3y ) is
given by (2) and (3) and the parameter s is calculated by
(4):
pxxsx mod21
2
3  (2)
pyxxsy mod)( 1313  (3)
12
12
xx
yy


if PQ
s= (4)
1
2
1
2
3
y
ax  if PQ
Multiplication kP over an elliptic group is computed by
repeating the addition operation k times [12, 13]. The
strength of an ECC-based cryptosystem is depends on
difficulty of finding the number of times that P is added to
itself to get Q=kP. Reverse operation known as Elliptic Curve
Discrete Logarithm Problem (ECDLP).
3. CELLULAR AUTOMATA
Cellular Automata (CA) is a discrete computing model which
provides simple, flexible and efficient platform for simulating
complicated systems and performing complex computation
based on the neighborhood’s information. CA consists of two
components 1) a set of cells and 2) a set of rules. Researchers,
scientists and practitioners from different fields have
exploited the CA paradigm for modelling different
applications [14, 15].
A cellular automaton consists of a graph where each node is a
cell. The state of each cell is updated simultaneously at
discrete time steps, based on the states in its
neighborhood at the preceding time step. The algorithm
used to compute the next cell state is referred to as the
CA local rule.

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Figure 1. Model Diagram of Encryption Technique.
For 2-state 3-neighborhood cellular automata there are 23
=8
distinct neighbourhood configurations and 28
=256 distinct
mappings from all these neighbourhood configurations to the
next state, each mapping representing a CA rule [16].
A cellular automaton (CA) is a dynamic system defined by the
following 4-tuple: dimension, set of finite states,
neighborhood and set of rules. Dimension defines number of
cells. Cells are updated accordingly to some rule. Such rule is
based on the state of the cell and the neighborhood [17, 18].
Figure 2 shows two typical neighborhood options (a) Von
Neumann Neighborhood (b) Moore Neighborhood.
Figure 2. (a) Von Neumann Neighborhood (b) Moore Neighborhood
By applying the transition rule the current state of CA moves
to new state by considering the neighborhood states.
For example:
- Rule 90:
- Rule 153:
4. PROPOSED METHOD
A new encryption method based ECC using cellular
automata is presented in Figure 1. This method tries to use
some asymmetric algorithm to encrypt or decrypt data using
elliptic curve. The proposed scheme noted ECCCA combines
the advantage of Automata theory and asymmetric encryption
based ECC into a total scheme. The overall module design
shows the different levels of security used (Figure 1).
In the proposed system, the first level of security starts with
ECC technique, where the plaint text is used as the set of
points to encrypt [19, 20, 21]. In this work, the resulting
output is sent to the next process based cellular automata. In
fact, the encrypted message is scrambled using the principle of
spiral rotation. The proposed method explains the usage of
second level of security using cellular automata since one
level of security is not enough.
Let CA be the Cellular Automata, which used to scramble
secure key applying the Local Rule.
4.1 Encryption procedure
The encryption is done through the following steps:
Step 1: start
Step 2: Divide the plain text into blocks of characters and
embed the characters in into points on elliptic curve.
Step 3: Generate randomly one number d. Then, Compute Kd
and Ke, which serve as secure keys (e is the x-coordinate of
Kd).
Step 4: Generate a cellular automata rule and convert the
compressed key to binary form.
Step 5: Select b=bit (j), where j is bit position (LSBMSB),
which decides which operation has to be performed.
If b = 0  compute Qi=Mi + Kd
If b=1  Compute Qi= Mi + Kd + Ke.

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Step 6: Convert the result blocks into binary sequence and
generate a compressed blocks by using CA technique.
Step 7: Arrange the first bit of all the blocks in the first row
and second bits of all block in the second row and continuing
this process arrange the remaining bit of all the blocks in the
corresponding row of matrix.
Step 8: Apply spiral technique to scramble the data matrix and
to get the cipher text.
Step 9: Stop
4.2 Decryption procedure
The decryption process involves converting the encrypted data
back to its original form for the receiver’s understanding. The
cipher text is decrypted using the reverse process of the
technique explained in encryption algorithm. The steps in
decryption algorithm are as follows:
Step 1. start
Step 2. Divide the cipher text to blocks and the bits are
arranged into a square matrix.
Step 3. Generate a reversible transition rule. Convert the
compressed blocks to a normal form using reversible CA rule.
Step 4. Apply the corresponding reversible principle of spiral
process.
Step 5. Defuse it to get the encrypted points on elliptic curve.
Step 6. Find the equivalent characters by decrypting each
point.
Step 7. Accumulate characters to form the secret message.
Step 8. Stop.
5. EXPLANATION WITH EXAMPLE
We have following example on which we have applied our
new encryption algorithm ECCCA, the explanation has been
provided below.
For the system parameters, we used the following data:
- p and n: two prime numbers (p=29 , n=31).
- E29 (-1, 16) an elliptic curve defined on finite field F29.
- P (5, 7): a point on elliptic curve E with order n.
- Key values:
k= 19 and PB=(16, 6).
d=13 and PA=(7, 27).
Kd = (5,22) and Ke= (6, 20).
- CA Rule chosen: ‘90’
Phase 1:
Plain Text: “ENCRYPTION”
Charact
er
Point on
EC
Bit selected
Encrypted
point Qi
E (6,20) 1 (23, 3)
N (1, 4) 0 (7, 27)
C (18, 1) 1 (2, 14)
R (7, 25) 0 (1, 25)
Y (13,24) 1 (28, 25)
P (0, 25) 0 (0, 4)
T (14, 7) 0 (2, 15)
I (23, 3) 1 (21, 11)
O (0, 4) 1 (16, 6)
N (1, 4) 0 (7, 27)
Phase2:
Therefore, the final encrypted text is compressed as follow:
01011011000001110100111001100111010001.
This algorithm compresses the data to reduce its length
without compromising the compression efficiency and the
information security.
1 0 0 0 1 0 0 1 1 0
0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 0 0 1 0 1
1 1 1 0 0 0 1 0 0 1
1 1 0 1 0 0 0 1 0 1
0 1 0 1 1 0 0 0 0 1
0 1 1 1 1 0 1 1 0 1
0 0 1 0 0 1 1 0 1 0
1 1 1 0 0 0 1 1 1 1
1 1 0 1 1 0 1 1 0 1
1 0 0 0 1 0 0 1 1 0
0 1 1 1 1 1 0 1 1 0
1 1 0 1 1 0 1 1 1 0
0 0 1 1 1 0 0 0 0 1
0 0 0 0 0 0 0 0 0 1
1 1 1 0 0 0 1 1 0 1
1 1 1 1 0 0 1 0 0 1
0 1 0 1 0 1 1 0 0 1
1 0 0 1 0 0 0 1 0 0
1 0 1 1 1 1 0 0 0 1

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6. RESULTS
In this section we proceed with our implementation using
Visual Basic as tool. Winsock Control has been used for
connecting two systems. Messages are transmit from user A to
user B when a socket is created. In our implementation, we
have used the curve E29 (-1, 16) in entire process.
Figure 3. Shows the Key generated using CAT
Figure 4. Encryption process
Figure 5. Decryption process
7. CONCLUSION
In this paper, we introduced the concept of cellular
automata as a promising approach to enhance the security
of the elliptic curve cryptosystem. By using two different
levels of security, the transmitted message is much secure as
compared to simple encryption method. From the above
results it is clearly found that the security against few
Attacks have been enhanced. Thus the proposed work of
joining the elliptic curve cryptography and Cellular
Automata has desirably increased the security level of the
encrypted data. Our algorithm, being based on concept of
CA, helps scrambling process due to rule-90. The Strength of
the algorithm due to the difficulty level used in secure key
generated. In fact, Cellular Automata is the strengthen
method to generate strong keys. Also integration of elliptic
curve cryptosystems and the concept of cellular automata has
improved the security level provided by ECCCA. Therefore, it
can be consider as a good alternative to some applications. In
future, we are interested to extend the proposed system to
image encryption and multimedia encryption.
8. REFERENCES
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[3] Ikshwansu Nautiyal, Madhu Sharma, “Encryption using
Elliptic Curve Cryptography using Java as
Implementation tool”, International Journal of Advanced
Research in Computer Science and Software Engineering
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