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Public Key Cryptosystem Technique Elliptic Curve Cryptography
with Generator g for Image Encryption
ABSTRACT This paper present goal of cryptography is the
secure communication through insecure channels
with the help of an algorithm ‘Elliptic Curve
Cryptography with generator g for Image
Encryption’. ECC is an efficient technique of
transmitting the image securely. It has been shown
though the image encryption by ECC to transmits
the image secretly and efficiently recovers the same
at the receiver end. The scheme comprises of the
important algorithms namely encryption algorithm
is used to create every 2-D image pixels of the
original image into the ECC points in a finite
abelian group over mGF 2 )( or E a bm
( , ).2
These
ECC points convert into cipher image pixels at
sender side and decryption algorithm is used to get
original image within a very short time with a high
level of security at the receiver side.
Keywords: Security, Elliptic Curve Cryptography
(ECC), The Generator g, RGB color Model, image
encryption algorithm at sender side process and
Image decryption algorithm at Receiver side process.
1. INTRODUCTION In fact, most of today‟s application of cryptography
asks for authentication and secrecy of the data. Secret
transmission of data is an important task to preserve
the data from the immune to attacks, threats and
misuse. The text or data can be encrypt but it is not
reliable due to brute force techniques, choose
ciphertext attacks etc. but an image cannot be easily
decrypt by attackers. Even data can be transmitted
more securely by converting it into an image. The
most of hardware and software products and
standards that use public key technique for
encryption and decryption, authentication etc. are
based on RSA cryptosystem by using non
Conventional algorithms among RSA [1], [2] and
ECC. The main attraction of ECC is that it can
provide better performance and security for small key
size, in comparison of RSA cryptosystem. In ECC a
160-bit key provides the same security as compared
to the traditional crypto system RSA with a 1024-bit
key, thus in this way it can reduced computational
cost or processing cost.ECC was proposed by Miller
and Koblitz [1].The security of ECC depends on the
difficulty of finding K for the given P and KP. The
security level for difference key size of RSA and
ECC is given table1. ECC is not easy to understand
by attackers. So provides better security through
insecure channels.
Time to break
in MIPS yearRSA Key
size
ECC
Key sizeECC/RSA Key
Size
104
109
1011
1020
1079
512
768
1024
2048
21000
106
132
160
210
600
5:1
6:1
7:1
10:1
35:1
ECC was proposed by Miller and Koblitz
RSA was proposed by Rivest, Shamir and Adleman
Table1. Equivalent Security of ECC and RSA for some key size
Rest of the paper is organized as follows. Section 2
presented the related work. The Elliptic Curve
Cryptosystem has been described in Section 3. In
Section 4 and Section 5, we presented a generator g
of finite field F and elliptic curve over mGF 2 ).( In
Section 6 and Section 7, we presented RGB colour
model and image encryption and decryption
techniques .paper is concluded in Section 8.
Vinod Kumar Yadav1
Dr. A.K. Malviya2 D.L. Gupta
3 Satyendra Singh
4 Ganesh Chandra
5
Dept.of CSE Associate Professor Assistant Professor Dept.of CSE Dept.of CSE
Kamla Nehru Institute Kamla Nehru Institute Kamla Nehru Institute Kamla Nehru Institute Kamla Nehru Institute
Of Engineering Technology of Engineering Technology of Engineering Technology of Engineering Technology of Engineering Technology vinodrockcsit@gmail.com anilkmalviya@yahoo.com dlgupta2002@gmail.com Satyendra.cse@gmail.com ganesh.iiscgate@gmail.com
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298
ISSN:2229-6093
2. RELATED WORK The use of elliptic curves in public key cryptography
was independently proposed by Koblitz and Miller in
1985 [3] and since then, a lot of work has been done
on elliptic curve cryptography. Various techniques
have been proposed in the literature, many authors
have tried to exploit the features of ECC field to
deploy for security applications. We have outlined
some of the highlights of the relevant work in this
section. M.Ayodos, et al. [4] has presented an
implementation of ECC over the field GF (p) on an
80 MHZ, 32 bit RAM microprocessor along with
results. Kristin Lauter has provided an overview of
ECC for wireless security [5]. To achieve higher
security of digital image RSA scheme with MRF and
ECC proposed for image encryption [6]. This paper
proposed first encrypt original image with XOR
concealed image that generate with MRF using seed
and generate secret image using Elliptic Curve
Cryptography. XORing message again encrypted by
RSA scheme. In [7] ECC scheme was Proposed and
which is based on binary finite GF [2m].This work
describe the basic design principal of ECC protocol
like EC, Diffie-Hellman, EC Elgamal and ECDSA
protocol. The value encryption algorithm was
proposed in [8].The VEA could be applied using
polynomial inversable function defined on GF (2m).
An image encryption for secure internet Multimedia
application was proposed in [9]. This paper presents a
join image compression and encryption scheme for
internet multimedia application. C J. Mclvor , et al.
[10] introduces a novel hardware architecture for
ECC over GF(p).The work presented by Gang Chen
presents a high performance EC cryptography
process for general curves over GF(p) [11]. A simple
tutorial of ECC concept is very well documented and
illustrated in the text authored by Williams Stallings,
et al. [12]. The Mixed image element encryption
using elliptic curve cryptography has been proposed
in [13]. This work proposed highly secured image
element because it gives two level encryption.
Kamlesh Gupta, et al. [14] has been proposed An
Ethical way for Image Encryption using ECC.
Kamlesh Gupta, et al. [15] has been proposed
Performance Analysis for Image Encryption using
ECC. Kamlesh Gupta and Sanjay Silakari [14], [15]
technique was based upon a prime curve over ZP, use
a cubic equation in which the variables and
coefficients all take on values in the set of integers
from 0 through p-1 and in which calculations are
performed modulo p.It was based on equation(1)
2 3y (modp) x ax b(modp)= + + .With the condition
3 24a +27b 0¹ ........................................ (1)
The major drawback in applying equation (1) for
image encryption is described below.
In this scheme the mod taken over GF (p), n
p
with n>1, Operation modulo n
p do not produce a
field. In this paper we use Elliptic curve over mGF 2 )( with
np
elements.
3. ELLIPTIC CURVE CRYPTO-
SYSTEM The most of the hardware and software products and
standards that use public key technique for
encryption, decryption are based on RSA
cryptosystem. The increment in the key length can
increase the security of the RSA cryptosystem, but on
the other hand it requires extra computational,
computational cost. This Extra cost has ramifications,
especially for those commerce sites which conduct
large numbers of secure transactions. Keeping in
view, in the recent year, a new public key
cryptosystem has shown his competency to challenge
RSA. This cryptosystem is Elliptic curve
cryptosystem. The main attraction of ECC is that it
can provide better performance and security for a far
smaller key size, in comparison of RSA
cryptosystem. In this way can reduced computational
cost or processing cost. The mathematics of ECC is
more complex than RSA cryptosystem.
3.1 MATHEMATICS OF ECC In this part, we explain elliptic curves and graph of
the cubic curves. Elliptic curves are cubic equation,
of the form
y x ax bx c2 3 2= + + + ............................. (2)
If we have two points on an elliptic curves and draw
a line through both of them, the line will intersect the
curve on unique third point. In equation (1) a, b, c are
real numbers and x, y are real variable. For our
explanation the following form is sufficient
y x ax b2 3= + + .............................. (3)
3.2 GRAPH OF THE CUBIC CURVE
y x ax b2 3= + +
For the fix value of a, b and for every value of x we
get a value of y. As, we know that if all powers of y
are even then, the curve will symmetric about x-axis,
hence the cubic curve equation (2) , together with a
point O, called zero point or point at affinity. For
notation E (-1, 0) denotes the set of all points,
Vinod Kumar Yadav et al ,Int.J.Computer Technology & Applications,Vol 3 (1), 298-302
IJCTA | JAN-FEB 2012 Available online@www.ijcta.com
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ISSN:2229-6093
E (-1, 0) = {(0,0),(-1,0),(1,0),(2,√6)............}.which
satisfy the equation y x x2 3= - .Fig.1(a) represent
graph of this cubic curve equation.
-2 -1 2 41 30 5
6
4
2
-2
-4
0
-(P+Q)
(P+Q)
P
Q
Fig.1 (a) 2 3
y x x= -
Similarly, E (1, 1) denotes the curve2 3y x x 1= + + .
Fig.1 (b) 2 3
y x x 1= + +
4. GENERATOR g OF FINITE FIELD F An equivalent technique for defining a finite field of
the form mGF 2 )( Using same irreducible
polynomial, is sometimes more convenient. To begin,
we need two definitions: A generator g of a finite
field F of order q (contains q elements) is an element
whose first (q-1) powers generate all the nonzero
elements of F. That is, the elements of F consist of 0 1 q 2
0,g ,g ,........,g-
.Consider a field F defined by
a polynomial f(x). An element b contained in F is
called a root of the polynomial if f (b) = 0. It can be
shown that a root g of an irreducible polynomial is a
generator of the finite field defined on that
polynomial. The Generator of the finite field defined
on that is given table 2.
Power
Representation
Polynominal
RepresentationBinary
RepresentationDecimal(Hex)
Representation
0
g0
g1
g2
g3
0
1
g
g2
0
1
2
4
3
g4
g5
g6
g +1
g2
g
g2
++
+
g2
+ 1
1
g
000
001
010
100
011
110
111
101
6
7
5
Table 2 .Generator for mGF 2 )( using
3 1x x+ +
In general, for mGF 2 )( with irreducible
polynomial f(x), determine4
g g 1= + . Then
calculate all of the powers of g from n 1
g+
through n
2 -2g .The elements of field correspond to the powers
of 0
g throughn
2 -2g , plus the value 0. For
multiplication of two elements in the field, use the
equality n
k kmod(2 1)g g
-= for any integer k. [12]
5. ELLIPTIC CURVES OVER mGF 2 )(
A finite field mGF 2 )( consists of 2
m elements,
together with addition and multiplication operations
that can be defined over polynomials. The form of
cubic equation appropriate for cryptography
applications for elliptic curves is somewhat
difference for mGF 2 )( than for GF (p). The form is
in this case shown in equation (4).
y xy x ax b2 3 2+ = + + ............................ (4)
Where it is understood that the variables x and y the
coefficients a, b are elements of mGF 2 )( and that
calculations are performed in mGF 2 )( .The elliptic
curve for image is as follows.
In this section we are try to give a novel proposed
scheme for image encryption with finite field mGF 2 )( , where m is any binary digit numbers and
irreducible polynomial f x x x4( ) 1= + +
.This yields a
generator that satisfies f (g) = 0, with a value of
4g g 1= + , or in binary 0010.we can calculate the
powers of g as follows:
Vinod Kumar Yadav et al ,Int.J.Computer Technology & Applications,Vol 3 (1), 298-302
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ISSN:2229-6093
g0
= 0001
g1
g2
g3
= 0010
= 0100
= 1000
g5
g4
g6
g7
= 0011
= 0110
= 1100
= 1011
g8
g9
g10
g11
= 1110
= 0111
= 1010
= 0101
g15
g14
g13
g12
= 0001
= 1001
= 1101
= 1111
If we want to calculate the value of g5
= g2+g=0110.
Now consider the elliptic curve 2 3 4 2y xy x g x 1+ = + + .In this case a = g
4 and
b=1.One point that satisfies this equation is (g ,g )5 3:
3 2 5 3 5 3 4 5 2(g ) (g ) (g ) (g ) (g )(g ) 1
6 8 15 14g g g g 1
1100 0101 001 1001 1001
1001 1001
+ = + +
+ = + +
+ = + +
=
Similarly others points are shown in fig.2.
Fig.2 Elliptic Curve4
E (g ,1)
24. [12]
6. RGB COLOR MODEL RGB stands for red green blue 24-bit color pixels (8-
bits per color), representing 224
(16, 777, 216)
different colors. RGB Pixel objects have some
special properties and methods.
Fig.3 RGB color model
The RGB color model is an additive color model,
which produces various colors by adding red, green,
and blue light in various ways [16].As shown in fig.3,
colors are represented The RGB color model within a
cubic volume defined by orthogonal Red, Green,
Blue axes [17].
7. IMAGE ENCRYPTION AND
DECRYPTION ECC can be used for encryption and decryption.
Consider the user A want to encrypt a sw image for
the user B, and then the following steps are involved.
Elliptic Curve
Cryptography
with Generator g
sw image Encrypted image
Fig.4 Encryption Model Step1.Take any RGB color image as sw.
Step2. A encodes the sw image as swP (x, y)= =
(g ,g ).5 3 similarly others points are calculated
using equation (4) with generator g.
Step3. A choose a random number K and produce the
ciphertext C [k G,P k P ]sw sw B
= ´ + ´ and
sends this ciphertext swC to B.
Step4. To decrypt the sw image, B computes
Bn k G.´ ´
Step5. B again computes
sw B B sw B BP k P n K G p k(n G) kP+ ´ - ´ ´ = - ´ +
=sw B BP k n k n- ´ + ´ =
swP . In other words, we
can say B picks the first co-ordinate KG ofswC ,
multiply that with his private key and then subtract
this form the second pointsw BP k P+ ´ .
Vinod Kumar Yadav et al ,Int.J.Computer Technology & Applications,Vol 3 (1), 298-302
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ISSN:2229-6093
8. CONCLUSION AND FUTURE
DIRECTION In this paper, we have presented an application of
ECC with Generator g in image encryption. ECC
points convert into cipher image pixels at sender side
and Decryption algorithm is used to get original
image within a very short time with a high level of
security at the receiver side. Elliptic curves are
believed to provide good security with smaller key
sizes, something that is very useful in many
applications. Smaller key sizes may result in faster
execution timings for the image encryption, which
are beneficial to systems where real time
performance is a critical factor ECC can be used into
a security system such as Video Compression, Face
recognition, Voice recognition, thumb impression,
Sensor network, Industry and Institutions.
REFERENCES
[1] Koblitz N., Menezes A.J., and Vanstone S.A. The state
of elliptic curve cryptography. Design, Codes, and Cryptography. Vol 19, Issue 2-3, 2000, page 173-193.
[2] ElGamal, T., “A public key cryptosystem and a
Signature scheme based on discrete logarithm,” IEEE Trans. Informn, Theory, IT-31, no.4, pp 469-472, July
1985.
[3] N.Koblitz, Elliptic Curve Cryptography, Mathematics of Computation, vol.48, 1987, pp-203-209.
[4] M.Ayodos, T.Yanik and C.K.Kog, “High-speed
implementation of an ECC based wireless authentication protocol on an ARM microprocessor,” IEE Proc Common,
Vol.148, No.5, pp.273-279, October 2001.
[5] Kristin Lauter, “The Advantages of Elliptic Cryptography of Wireless Security,” IEEE Wireless
Communications, pp.62-67, feb.2006.
[6] Chaur - Chin Chen “RSA scheme with MRF and ECC for Data Encryption,” 0-7803-8603-5/04 IEEE, 2004.
[7] Kefa Rabah “Elliptic Curve Cryptography over Binary
Finite Field GF (2m)”. Information Technology Journal 5(1) pp. 204-229, ISSN 1812-5638, 2006.
[8] Luminita Scripcariu and Mircea Daniel Frunza, “A
New Image encryption Algorithm based on Iversiable Functions defined on Galois Fields, ” pp. 243-246l, ISSN
0-7803-9029-6/05, IEEE, 2005.
[9] Philip P. Dang and Paul M. Chau, “Image Encryption for Secure Internet Multimedia Application”, IEEE
Transaction on Consumer Electronics, Vol. 46, No.3 pp.
395-403, Aug.2000.
[10] C.J. Mclvor, M.McLoone, and J. V. McCanny,
“Hardware elliptic curve cryptography processor over GF (p),” IEEE Trans. Circuits Syst.-I: Reg.papers, vol.53, no.9,
pp. 1946-1957, sep. 2006.
[11]Gang Chen, Guoqiang Bai, and hongi Chen, “A High-Performance Elliptic Curve Cryptography Processor for
General Curve over GF (p) Based on a Systolic Arithmetic
Unit,” IEEE Trans. Circuits Syst.-II: Express Briefs, vol.
54, on.5, pp.412-416, May.2007.
[12] William Stallings, Cryptography and Network
Security, Prentice Hall, 4th Edition, 2006.
[13] Guiliang Zhu and Xiuaoqiang Zhang, “Mixed Image
Element Encryption System ,” 9th IEEE International
Conference for young computer Scientists ISSN 978-0-7-
3398, pp. 1995-1600,Aug.2008.
[14] Gupta, K., Silakari, S., “Performance Analysis for
Image Encryption Using ECC,” Computational Intelligence
and Communication Networks (CICN), 2010 International Conference on, vol., no., pp.79-82, 26-28 Nov. 2010.
[15] Gupta, K., Silakari, S., Gupta, R.; Khan, S.A., “An
Ethical Way of Image Encryption Using ECC,” Computational Intelligence, Communication Systems and
Networks, 2009. CICSYN „09‟. First International
Conference on, vol., no., pp.342-345, 23-25 July 2009.
[16] R. C. Gonzalez and R E Woods, Digital Image
Processing. NJ: Prentice Hall, 2002.
[17] B. D. Fu, J. C. Yuan, and C. X. Guo, “Distinguished arithmetic for cotton impurity based on RGB color model,”
Beijing Textile Journal, vol. 26, 2005, pp. 48-51.
Vinod Kumar Yadav et al ,Int.J.Computer Technology & Applications,Vol 3 (1), 298-302
IJCTA | JAN-FEB 2012 Available online@www.ijcta.com
302
ISSN:2229-6093