Upload
dinhminh
View
224
Download
2
Embed Size (px)
Citation preview
www.ijaret.org Vol. 2, Issue I, Jan. 2014 ISSN 2320-6802
INTERNATIONAL JOURNAL FOR ADVANCE RESEARCH IN
ENGINEERING AND TECHNOLOGY WINGS TO YOUR THOUGHTS…..
Page 37
Design of a Cryptosystem Using Two-Level Hill Cipher
Sarla Dewangan
1, Mrs. Shikha Pandey
2, Mohammad Imroze Khan
3
1M-Tech scholar, 2Assistant Professor, 3Assistant Professor 1,2Rungta College of Engineering & Technology, Bhilai (C.G.), 490024
3National Institute of Technology, Raipur (C.G.), 492010 1 [email protected], [email protected],[email protected]
Abstract: Pioneered in the field of polygraph substitution cipher, operation of Hill Cipher is based upon the linear
algebraic equation by working on more than three symbols simultaneously. Its high speed of processing is advantageous
and so it has inherent property of opposing the analysis based upon frequency letter. The advantage of faster processing
time is due to the mode of operation, in which matrices are multiplied linearly. But due to its linearity, it’s very difficult to
decrypt any encrypted message as it consists of all matrices of a given dimension that are invertible over Zm. This is the
same reason why it is considered to be difficult for decrypting image of same shade over a large space, which makes it
useless to withhold any specific aspect out of an image, which might be pointing out any impression over it. The main aim
of this paper is to use a technique for image encryption, which is different from the conventional one.
Keywords: Cryptosystem, Encryption, Decryption, Hill Cipher
1. INTRODUCTION With the advancement of network technology, there is a
crucial problem regarding the security of the information.
Increasing network transmission ability and popular
multimedia technology application gradually leads us to
retrieve information clearly and directly from images.
Therefore, the security of data is very complicated and
imperative issue now [3]. Cryptography, which is the
science of encryption, plays an important role in
communication like mobile phones, pay-TV, e-commerce,
private email transformation, financial information
transmission, ATM cards security, computer passwords
etc are those examples which touches on many things in
our daily lives [2, 9]. Science is the study of principles,
rules and laws, whereas art is the study of methods to do
something. Hence, we consider Cryptography as the art
and science, because it includes the principles and
methods of transforming an intelligible message
(plaintext) into one that is unintelligible (Cipher text) and
then retransforming the message into its original form [1,
2, 6].
Now-a-days, Cryptography is also compared with both
mathematics and computer science and it has also the
affiliation to information theory, computer security and
engineering [2]. Substitution cipher is one of the most
basic components of the classical cipher. It is an
encryption method, which substitutes the units of plaintext
with cipher text according to particular system. The units
may be in the form of single letters, pairs of letters, triplets
of letters, mixture of the above and so forth. By
performing the inverse substitution, the receiver can
decipher the text [5, 8, and 10]. The units of plaintext are
retained in the similar sequence as in the cipher text, but
the units themselves are changed. There are different types
of substitution cipher. The cipher is called as simple
substitution ciphers, when the cipher works on single
letters. The cipher is said to be polygraphic, when it works
on group of letters. Cipher can be classified in terms of
mono alphabetic and poly alphabetic. A fixed substitution
is used by the mono alphabetic cipher over the entire
message, whereas a number of substitutions are used by
the poly alphabetic cipher at different times in the
message. Hill cipher is one of the mono alphabetic
polygraphic substitution cipher. It is a block cipher which
has many advantages like distinguishing letter frequencies
of the plaintext, its simplicity because of the use of the
multiplication of matrices, and inversion for enciphering
and deciphering, its high speed and high throughput [1, 3,
and 6]. In our paper, we encrypt gray scale as well as color
images using the pseudorandom key matrix.
2. PSEUDO-INVERTIBLE KEY
MATRIX The pseudo-inverse of a m x n matrix A is a matrix that
generalizes to arbitrary matrices the notion of inverse of a
square, invertible matrix. The pseudo-inverse can be
expressed from the singular value decomposition (SVD)
of A, as follows.
www.ijaret.org Vol. 2, Issue I, Jan. 2014 ISSN 2320-6802
INTERNATIONAL JOURNAL FOR ADVANCE RESEARCH IN
ENGINEERING AND TECHNOLOGY WINGS TO YOUR THOUGHTS…..
Page 38
Let the SVD of A be
A = U S 00 0
VT
Where U, V are both orthogonal matrices and S is a
diagonal matrix containing the (positive) singular values
of A on its diagonal.
Then the pseudo-inverse of A is the n x m matrix defined
as
A† = V S−1 00 0
UT
Here, A† has the same dimension as the transpose of A.
The pseudo-inverse matrix has the following properties:
AA† = (AA†)*
A†A = (A†A)*
AA† A = A
A† AA† = A†
3. HILL CIPHER Hill cipher is an application of linear algebra to
cryptology. It was developed by the mathematician Lester
Hill. The Hill cipher algorithm takes m successive
plaintext letters and substitutes them by m cipher text
letters. The substitution is determined by m linear
equations in which each character is assigned a numerical
value (a=0, b=1… z = 25) [4, 8, 10]. Let m be a positive
integer, the idea is to have m linear combinations of the m
alphabetic characters in one plain text element and
produce corresponding m characters in one cipher text
element. Then, a m × m matrix A as shown below in
equation (1) is used as a key of the system such that A is
invertible modulo 26 [1, 2].
Let aij be the entry of A.
A =
a11 a12 … a1m
a21 a22 … a2m
… … … …am1 am2 … amm
(1)
For the plain text block, the numerical equivalents of m
letters are given by Equation (2) as follows.
x = (x1, x2… xm) (2)
and a key matrix A, the corresponding cipher text block
are given by Equation (3) as follows.
y = (y1, y2,….,ym) (3)
Thus the Encryption will be computed as follows in
Equation (4)
(y1, y2,…., ym)) = (x1, x2,…., xm)A (mod 26) (4)
The cipher text is obtained from the plain text by means of
a linear transformation.
Thus the Decryption will be computed as follows in
Equation (5)
(x1, x2,….,xm) = (y1, y2,….,ym) A−1 (mod26) (5)
Where A-1
is given by Equation (6)
A−1 =
a11 a12 … a1m
a21 a22 … a2m
… … … …am1 am2 … amm
−1
(6)
Since the block length is m, there are 26 different m letters
blocks possible, each of which can be regarded as a letter
in a 26 letter combination. Hill’s method amounts to a
mono-alphabetic substitution on this alphabet.
A11 is a 1 x 1 matrix and is given by the Equation (7)
A11 = a11 , (7)
A12 is a 1 x (n-1) matrix and is given by the Equation (8)
A12 = a12 a13 … a1n (8)
A21 is a (n-1) x 1 matrix and is given by the Equation (9)
A21 =
a21
a31
…an1
, (9)
A22 is a (n-1) x (n-1) matrix and is given by the Equation
(10)
A22 =
a22 a23 … a2n
a32 a33 … a3n
… … … …an2 an3 … ann
(10)
So, A12 A21 = I - A112 = I - a112 (11)
And A12 (a11I + A22) = 0. [1,7] (12)
4. TWO-LEVEL HILL CIPHER As we know that both grayscale and color images can be
encrypted using Hill cipher technique [6], they can also be
encrypted using our proposed Two-Level Hill Cipher
algorithm, the modulus will be 256(the no. of levels is
considered as the no. of alphabets). On working with color
images on encryption side, firstly the color image is
decomposed into R-G-B components. Secondly, each
component is encrypted by the algorithm individually. At
last, the encrypted components are combined together to
obtain the encrypted color image.
In a Two – Level Hill Cipher, Hill-Cipher technique is
applied twice onto the blocks. Once, it is applied onto the
www.ijaret.org Vol. 2, Issue I, Jan. 2014 ISSN 2320-6802
INTERNATIONAL JOURNAL FOR ADVANCE RESEARCH IN
ENGINEERING AND TECHNOLOGY WINGS TO YOUR THOUGHTS…..
Page 39
temporary block. Then, the resultant matrix is transposed
and Hill cipher is again applied to this matrix. The
algorithms are given below and the block diagram for the
encryption and decryption processes is shown in Figure 1
and Figure 2 respectively.
4.1 Sender Side:
At the Sender’s side, we implement encryption process.
Figure 1: Sender’s End Flowchart
Encryption:
Generate a Pseudorandom key image of order m
x m (P2).
Convert original image P in RGB2Gray (P1).
Divide P1 into m x m symmetric blocks.
The ith
pixels of each block are brought together
to form a temporary block.
a) Hill cipher technique is applied onto the
temporary block.
b) the resultant matrix is transposed and Hill
cipher is again applied to this matrix.
The final matrix obtained is placed in the ith
block of encrypted image.
Finally encrypted image of length L is obtained.
4.2 Receiver Side:
At the Receiver’s side, we implement decryption process.
Figure 2: Receiver’s End Flowchart
Decryption: Encrypted image of length L is placed such that it
forms an m x m matrix.
Inverse Hill cipher is applied to m x m matrix.
The resultant matrix is reverse transposed and
inverse Hill cipher is again applied to resultant
matrix to form a temporary block.
Temporary block is restored to form a m x m
symmetric blocks by replacing ith
pixel in each
block
Encrypted Image
of Length L
Place Resultant Image
of Length L in ith
Block to form m x m
matrix
Restore Temporary
Block to form a m
x m matrix by
replacing ith pixel
of each block
Apply
Inverse Hill
Cipher
Apply
Inverse
Hill Cipher
Apply
Inverse
Transpose
Two Level Hill Cipher
Pseudo invertible
key matrix image
of order m x m
(P)
Separate Pseudo
invertible Key to
obtain m x m
Symmetric
Blocks
Find Image
Resolution
Original
Image
Original
image
(P)
Rgb2Gray
(P1)
Pseudo invertible
Key Matrix
image of order m
x m (P2)
Construct ith
temporary
block from
ith pixel of
each block
Divide P1 in m
x m symmetric
Blocks
(provide
padding)
Apply
Hill
Cipher
Apply
Transpose
Place Resultant
Image in ith block of
Encrypted Image
Encrypted Image
of Length L
Apply
Hill
Cipher
Two Level Hill Cipher
www.ijaret.org Vol. 2, Issue I, Jan. 2014 ISSN 2320-6802
INTERNATIONAL JOURNAL FOR ADVANCE RESEARCH IN
ENGINEERING AND TECHNOLOGY WINGS TO YOUR THOUGHTS…..
Page 40
Separate pseudorandom key (P2) from m x m
symmetric blocks to restore original image.
5. RESULT Hill cipher is a block cipher that has several advantages
such as disguising letter frequencies of the plaintext, its
simplicity because of using matrix multiplication and
inversion for enciphering and deciphering, its high speed,
and high throughput. However, Hill cipher succumbs to a
known plaintext attack and can be easily broken with such
attacks [2,3,]. Although the algorithm presented in this
paper aims at image encryption, it is not just limited to this
area and can be widely applied in other information
security fields such as video encryption.
(a)
(b)
(c)
(d)
(e)
Figure 3: Encryption Using Two-Level Hill Cipher
(a) Original Image, (b) Histogram of Original Image, (c)
Key Image, (d) Histogram of Key Image,
(e) Encrypted Image
We have taken one image as original image, another as the
key image, which are shown in Figure 3(a) and 3(c)
respectively. The histograms of both the images are shown
in Figure 3(b) and 3(d) respectively. They are encrypted
them using the Two-level Hill cipher algorithm. The result
is shown in Figure 3(e) as the encrypted image.
Figure 4 shows the result of decryption process. Here, we
have decrypted the image shown in Figure 3(a).
(a)
www.ijaret.org Vol. 2, Issue I, Jan. 2014 ISSN 2320-6802
INTERNATIONAL JOURNAL FOR ADVANCE RESEARCH IN
ENGINEERING AND TECHNOLOGY WINGS TO YOUR THOUGHTS…..
Page 41
(b)
Figure 4: Decryption Using Two-Level Hill Cipher
(a) Decrypted Image, (b) Histogram of Decrypted Image
6. CONCLUSIONS The competent method of encryption of an illustration is
presented in this paper. Two-Level Hill Cipher is a fast
encryption technique which can provide satisfactory
results against the normal hill cipher technique. It can
easily resist to the infamous “Known Plain Text Attack”
(KPTA) for secure transmission of graphic encoded
information while taking over various limitation in the
aspect of efficiency and security of various encryption
technologies of previous generation. Hence this hybrid
system proves to be a very sound technique for
transferring messages from sender to the receiver,
achieving confidentiality as well as message
authentication.
REFERENCES [1] Bibhudendra Acharya, Saroj Kumar
Panigrahy, Sarat Kumar Patra, and Ganapati
Panda.”Image Encryption Using Advanced
Hill Cipher Algorithm”, International Journal
of Recent Trends in Engineering, Issue. 1, Vol.
1, May 2009.
[2] Bibhudendra Acharya, Girija Sankar Rath,
Sarat Kumar Patra, Saroj Kumar Panigrahy.
“Novel Methods of Generating Self-Invertible
Matrix for Hill Cipher Algorithm”,
International Journal of Security, Vol 1, Issue
1, 2007, pp. 14-21.
[3] S.K.Muttoo, Deepika Aggarwal, Bhavya
Ahuja. ”ASecure Image Encryption Algorithm
Based on HillCipher System” , Buletin Teknik
Elektro dan Informatika (Bulletin of Electrical
Engineering and Informatics) Vol.1, No.1,
March 2012, pp. 51~60 ISSN: 2089-3191
[4] Rushdi A. Hamamreh, Mousa Farajallah, “
Design of a Robust Cryptosystem Algorithm
for Non-Invertible Matrices Based on Hill
Cipher”, International Journal of Computer
Science and Network Security, 2009.
[5] Bibhudendra Acharya, Mohammad Imroze
Khan, S K Patra, G Panda. “Implementation
of Hybrid Cryptosystem Using Non-Invertible
Matrices Based on Hill Cipher and RSA
Algorithm”
[6] Saroj Kumar Panigrahy, Bibhudendra
Acharya, Debasish Jena” Image Encryption
Using Self-Invertible Key Matrix of Hill
Cipher Algorithm” 1st International
Conference on Advances in Computing,
Chikhli, India, 21-22 February 2008
[7] Bibhudendra Acharya, Debasish Jena, Sarat
Kumar Patra, and Ganapati Panda.
“Invertible, Involutory and Permutation
Matrix Generation Methods for Hill Cipher
System”, International Conference on
Advanced Computer Control, 2009.
[8] W. Stallings, Cryptography and Network
Security, 4th
edition, Prentice Hall, 2005
[9] Blakley G.R., “Twenty years of cryptography
in the open literature”, Security and Privacy
1999, Proceedings of the IEEE Symposium, 9-
12 May 1999
[10] A. J. Menezes, P.C. Van Oorschot, S.A. Van
Stone, “Handbook of Applied
Cryptography”,CRC press, 1996