LOSSLESS RECONSTRUCTION OF SECRET IMAGE USING THRESHOLD SECRET SHARING AND TRANSFORMATION

International Journal of Network Security & Its Applications (IJNSA), Vol.4, No.3, May 2012
DOI : 10.5121/ijnsa.2012.4307 111
LOSSLESS RECONSTRUCTION OF SECRET IMAGE
USING THRESHOLD SECRET SHARING AND
TRANSFORMATION
P. Devaki1
, Dr. G. Raghavendra Rao2
1
Department of Information Science & Engineering, NIE, Mysore
p_devaki1@yahoo.com
2
Department of Computer Science & Engineering, NIE, Mysore
grrao57@gmail.com
ABSTRACT
This paper is proposed to provide confidentiality of the secret image which can be used by multiple users
or to store on multiple servers. A secret sharing is a technique to protect the secret information which
will be used by multiple users. The threshold secret sharing is more efficient as it is possible to
reconstruct the secret with the threshold number of shares. Along with Shamir’s secret sharing method
we propose to use the radon transformation before dividing the image in to shares. This transformation is
used so that the shares will not have the original pixel intensity. The run length code is used to compress
the image after the transformation. Then apply secret sharing technique. The reconstruction of the image
results in original image by applying the operations in the reverse order.
KEYWORDS
Threshold secret sharing, Confidentiality, Transformation, Compression.
1. INTRODUCTION
Handling secret information in a public network like internet is not easy. It is not safe to store
the secret information on a system which can be accessed by multiple users. Even if the secret
information stored in a system which is protected with a password also, there may be a single
point of failure. It is not completely protected because multiple users will have the password. It
may be misused or compromised by one or multiple users with out the knowledge of other
users. This compromises the confidentiality of the information. If the secret information needs
to be transmitted over a network, it can not be sent as it is over a network. There are 2 reasons
for this restriction. One reason is the bandwidth required to send the information. Second reason
is the issue about maintaining the confidentiality of the secret information during the
transmission over the network.
To overcome these problems of handling secret information, an efficient method proposed by
Shamir [4] and Blakley [5] can be used. This method is nothing but instead of allowing all the
authorized users to have the accessibility to the secret information or transmitting the whole
information as a single unit of data, divide the information in to number of blocks.
Each block is called as a share. The number of shares depends on the number of authorized
users of that information. Each user will get a share of the information instead of the whole
information. This ensures that even if a user misuses or compromises his/her share with the
unauthorized user, it’s of no use for the unauthorized user.

112
With a single share, the unauthorized user will not be able to guess the information. When the
information is required by any of the users, that user can request and collect the shares from the
other users and reconstruct the information.
This method is referred to as secret sharing. At no point of time any authorized user will not get
the information about the length of the information and also the shares belonging to other users.
In this conventional secret sharing method, all the shares are required to reconstruct the
information. Even if one share is not available during the reconstruction process, it will not be
possible to reconstruct the original information.
Threshold secret sharing [4] [5] [6] is an extension of the conventional secret sharing method.
Threshold secret is nothing but, it is not necessary to collect all the shares to reconstruct the
information. It is sufficient to collect the threshold number of shares to reconstruct the
information. (k, n ) indicates k out of n number of shares is sufficient to reconstruct the
information. Here k is the threshold value, and n is the total number of shares of the
information. With even k-1 number of shares it is not possible to reconstruct the information. It
is flexible compared to the conventional secret sharing method, at the same time it is secured
also. The information can be text, audio, video or image.
While reconstructing the secret information, the authorized users will come to know about the
user who has requested for the secret shares. During this time the users can verify whether the
requester is an authorized user or not.
In some applications which deal with the images, the reconstruction of the secret image can
tolerate some loss. Even with the loss of some data also it is possible to recognize the image.
Only the image needs to be visually recognizable. This is referred to as visual cryptography.
Many researchers have worked in this area [9][10]. Here the quality of the reconstructed image
will be degraded due to either bits transformed in to frequency domain or division of pixels. But
in some applications it is necessary to reconstruct the image in such a way that there should not
be any loss of data. Examples: medical, military, business documents, images taken by the
satellite etc for which any loss of data is undesirable. Many researchers have been working
towards this in providing security to the secret images [11][12].
For larger amount of information, it is necessary to compress the data before performing the
sharing operation and distributing the shares. Compression must be lossless. Compression also
provides a first level of confidentiality to the information. Since, it is not possible to recognize
the compressed data. Then sharing can be made. This provides second level of confidentiality to
the information. Before compressing the Image it is necessary to transform the image.
The transformations can be frequency or spatial based. There are various transformation
methods available. By performing inverse transformation some methods result in the original
image reconstruction, where as some result in some loss in the image but visually it can be
recognized. The loss may be due to quantization error. But for applications like medical and
military, it is necessary to reconstruct the original image rather than the image with loss of data.
In this paper we propose to use the threshold secret sharing based on Shamir. Radon
transformation is used with run length coding to compress the image. The rest of the paper has
been organized as follows. Concept of secret sharing is explained Section 2 , radon
transformation is explained in section 3, compression using run length coding in section 4,
proposed work in section 5, conclusion in section 6.
2. Threshold secret sharing
Shamir’s threshold secret sharing[4] is based on the polynomial interpolation. He uses
Lagrange’s interpolation for reconstruction of the key. When it is necessary to distribute the
secret information among multiple users, this secret sharing method can be used to generate the
shares using the polynomial of the order m-1 for (m,n) secret sharing. m is the threshold value
and n is the total number of shares to be done based on the number of authorized users.
The polynomial is

113
f(x) = S + Cx + Cx2
+ …….Cx m-1
Where S is the secret , to be shared among n users. C1, C2, C3 are the coefficients whose values
are randomly selected. Using this polynomial and selecting unique values for x, the dealer
divides the secret in to number of shares.
f(x1) = y1
f(x2) = y2
….
F(xn) = yn
Where y1, y2,…… yn are the shares generated. Now the shares along with the x values will be
distributed to the authorized users. The dealer is not responsible for reconstruction of the key.
So each user i will get a pair of values xi and yi.
When it is necessary for a user to reconstruct the information, he requests other users to send
their shares. After receiving at least m number of shares the user can reconstruct the
information. The significance of this secret sharing method is, no single share will reveal any
information about the secret information, and with less than m number of shares also it is not
possible to reconstruct the secret information. Only if a user gets minimum m number of shares
from the authorized users, he can reconstruct the key. This will ensure that even if an attacker
gets a share or few shares which is less than m, the attacker will not be able to reconstruct the
secret information. This gives protection against any attacker getting the secret, and also
protects against any authorized user miss using the secret.
The reconstruction of the secret is performed by using the Lagrange’s interpolation. This is used
to solve for the coefficients especially for S.
The interpolation formula is:
f(x)= (x-x1)(x-x2) y0 (x-x0)(x-x2)y1 (x-x0)(x-x1) y2
----------------------- + --------------- + -----------------------+.................
(x0-x1)(x0-x2) (x1-x0)(x1-x2) (x2-x0)(x2-x1)
Solving for the above will result in S.
This is a perfect secret sharing, because the secret S is reconstructed only with m or more than
m shares. The secret S can not be constructed with less than m number of shares.
As a simple example we have considered (3,5) where 5 is the number of authorized users, and 3
is the threshold. The given secret is 222.
The polynomial is
F(x) = 222 + 3x + 2x2
We can select 5 values for x and calculate F(x).
Y1= F(x1) = 227
Y2 = F(x2) = 236
Y3 = F(x3) = 249
Y4 = f(x4) = 266
Y5 = F(x5) = 287
After obtaining the shares y1 to y5, the shares along with the values of x for each user can be
given to the users as follows.
U1 = (x1, F(x1)) = (1, 227)
U2 = (x2, F(x2)) = (2, 236)
U3 = (x3, F(x3)) = (3, 249)
U4 = (x4, F(x4)) = (4,266)
U5 = (x5, F(x5)) = (5, 287)
When the 5 users get their respective shares, they can store them.

114
3. Radon transformation
The Radon transform is the projection of the image intensity along a radial line oriented at a
specific angle. The radon function computes projections of an image matrix along specified
directions.
R = radon(I, theta) returns the Radon transform R of the intensity image I for the angle theta
degrees. If theta is a scalar, R is a column vector containing the Radon transform for theta
degrees. If theta is a vector, R is a matrix in which each column is the Radon transform for one
of the angles in theta.
[R,xp] = radon(I,theta) returns a vector xp containing the radial coordinates corresponding to
each row of R.
The radial coordinates returned in xp are the values along the x'-axis, which is oriented at theta
degrees counterclockwise from the x-axis. The origin of both axes is the center pixel of the
image, which is defined as floor((size(I)+1)/2).
A projection of a two-dimensional function f(x,y) is a set of line integrals. The radon function
computes the line integrals from multiple sources along parallel paths, or beams, in a certain
direction. The beams are spaced 1 pixel unit apart. To represent an image, the radon function
takes multiple, parallel-beam projections of the image from different angles by rotating the
source around the center of the image. The following figure shows a single projection at a
specified rotation angle.
Fig -1
Projections can be computed along any angle [[THETA]]. In general, the Radon transform of
f(x,y) is the line integral of f (x,y) parallel to the y´-axis
where
The following figure illustrates the geometry of the Radon transform.

115
Geometry of the Radon Transform
Fig – 2
The radial coordinates returned in xp are the values along the x'-axis, which is oriented at theta
degrees counterclockwise from the x-axis. One important feature of this transformation is on
retransformation it results in the original image.
Radon retransformation:
IR = iradon(R,theta);
iradon reconstructs an image from parallel-beam projections. In parallel-beam geometry, each
projection is formed by combining a set of line integrals through an image at a specific angle.
I = iradon(R, theta) reconstructs the image I from projection data in the two-
dimensional array R. The columns of R are parallel beam projection data. iradon assumes that
the center of rotation is the center point of the projections, which is defined as
ceil(size(R,1)/2).
theta describes the angles (in degrees) at which the projections were taken. It can be either a
vector containing the angles or a scalar specifying D_theta, the incremental angle between
projections. If theta is a vector, it must contain angles with equal spacing between them. If
theta is a scalar specifying D_theta, the projections were taken at angles theta =
m*D_theta, where m = 0,1,2,...,size(R,2)-1. If the input is the empty matrix
([]), D_theta defaults to 180/size(R,2).
iradon uses the filtered back-projection algorithm to perform the inverse Radon transform.
The filter is designed directly in the frequency domain and then multiplied by the FFT of the
projections. The projections are zero-padded to a power of 2 before filtering to prevent spatial
domain aliasing and to speed up the FFT.
I = iradon(P, theta, interp, filter, frequency_scaling,
output_size) specifies parameters to use in the inverse Radon transform. You can specify

116
any combination of the last four arguments. iradon uses default values for any of these
arguments that you omit.
interp specifies the type of interpolation to use in the back projection. The available options
are listed in order of increasing accuracy and computational complexity.
4. Runlength Encoding
Run-length encoding is a data compression algorithm that is supported by most bitmap file
formats, such as Tiff, BMP, and PCX. RLE is suited for compressing any type of data
regardless of its information content, but the content of the data will affect the compression ratio
achieved by RLE. Although most RLE algorithms cannot achieve the high compression ratios
of the more advanced compression methods, RLE is both easy to implement and quick to
execute, making it a good alternative to either using a complex compression algorithm or
leaving your image data uncompressed.
RLE works by reducing the physical size of a repeating string of characters. This repeating
string, called a run, is typically encoded into two bytes. The first byte represents the number of
characters in the run and is called the run count. In practice, an encoded run may contain 1 to
128 or 256 characters; the run count usually contains as the number of characters minus one (a
value in the range of 0 to 127 or 255). The second byte is the value of the character in the run,
which is in the range of 0 to 255, and is called the run value.
Ex:Uncompressed, a character run of 15 X characters would normally require 15 bytes to store
XXXXXXXXXXXXXXX
The same string after RLE encoding would require only two bytes
15X
The 15X code generated to represent the character string is called an RLE packet. Here, the first
byte, 15, is the run count and contains the number of repetitions. The second byte, A, is the run
value and contains the actual repeated value in the run.

117
Fig - 3
There can be bit level, byte level and pixel level RLE.
5. Proposed work
If the image is transformed to frequency domain as many researchers have done, then
quantization is required and also processing time is required for the conversion. Due to
quantization the retransformation may not result in the original image. So we are using spatial
domain transformation.
The given gray image is transformed using radon transform. Radon is a spatial domain
transform, where the pixel’s intensity will be oriented in a particular angle. Since the
transformed image is different from the original image the confidentiality will be maintained.
This gives the first level of confidentiality to the secret image. Then the byte runlength code is
applied to get the compressed transformed image. This gives second level of confidentiality.
Then apply Shamir’s secret sharing on this encoded data to get n number of shares. This gives
third level of confidentiality.
The shares will be a sequence of numeric values. These values are represented in Braille form.
The Braille images are sent to the authenticated users. This gives the fourth level of
confidentiality. Here the purpose of using the Braille is to avoid using the cover images to hide
the shares.

118
When it is necessary to obtain the image, atleast m shares which are in the Braille form is
collected and transform Braille to normal sequence of numeric values. Then apply the
Lagrange’s interpolation to obtain the integer value of the compressed image. Then apply the
run length decoding to obtain the transformed image. After these apply the radon retransform to
obtain the original image.
Distribution of the secret image:
Gray Image
Fig - 4
Reconstruction of the image:
Fig - 5
6. Conclusions
The proposed work is fault tolerant, as even if few shares are corrupted also it is possible to
reconstruct the image with m shares. Unlike the visual cryptography, this method allows to
reconstruct the original image. The size of the shares is not the size of the original image, as the
image is compressed. This reduces the bandwidth required to transmit the shares on the network
and also the space required to store the shares. It is highly secured, as the image is transformed,
compressed and then shared among multiple users, represented in Braille. The added advantage
of this work is , almost all the methods use cover images to hide the share while transmitting the
shares, but here we are not using any cover image instead the shares will be represented in
Braille. As it is difficult for an attacker to determine the language in the Braille format. Even if
an attacker gets one or few shares, he can not reconstruct the image, as minimum m number of
shares is required to reconstruct the image. It results in original image when the shares are used
to reconstruct the image. No extra processing is required to improve the quality of the image.
Gray image Radon
transform
Run length
encoding
Share the
secret
Send to n
users
Convert
to Braille
Collect atleast
m shares
Convert from Braille
to numeric integers
Apply Lagrange’s
polynomial
Apply runlength
decoding
Apply Radon
retransformation
Gray image

119
References
[1]. M. Weinberg, G. Seroussi, and G. Sapiro. LOCO-I a low complexity, context-based, lossless image
compression algorithm. In Proc. DCC’96 Data compression conference, pages 140–149,
Snowbird, Utah, Mar. 1996.
[2]. M.Weinberger, J. Rissanen, and R. Arps. Applications of universal context modeling to lossless
compression of gray-scale images. IEEE Transactions on ImageProcessing, IP-5:575–586, Apr.
1996.
[3]. R. Calderbank, I. Daubechies, W. Sweldens, and B.- L. Yeo. Lossless image compression using
integer to integer wavelet transforms. In Proc. ICIP-97, IEEEInternational Conference on Image,
volume 1, pages 596–599, Santa Barbara, California, Oct. 1997.
[4]. A. Shamir, “How to share a secret,” Communications of the ACM, Vol. 22, 1979, pp. 612-613.
[5]. G. R. Blakley, “Safeguarding cryptographic keys,” in Proceedings of the AFIPS National Computer
Conference, Vol. 48, 1979, pp. 313-317.
[6]. C. C. Thien and J. C. Lin, “Secret image sharing,” Computers and Graphics, Vol. 26, 2002, pp. 765-
770.
[7]. C. P. Huang and C. C. Li, “A secret image sharing method using integer multiwavelet transform,” in
Proceedings of IEEE International Conference on Image Processing, 2006, pp. 1969-1972.
[8]. C. P. Huang and C. C. Li, “Secure and progressive image transmission through shadows generated
by multiwavelet transform,” International Journal of Wavelets, Multiresolution and Information
Processing, Vol. 6, 2008, pp. 907-931.
[9] Young-Fu Chen , Yung-Koun Chan, Ching-Chun Huag , ” A Multiple level visual secret sharing
scheme without image size expansion “ , Information Science 177 , 2007.
[10] Jen-Bang Feng, Hsien-Chu Wu , et al, “ Visual secret sharing for multiple secrets” , Pattern
Recognition 41 , 2008.
[11] Chin-Pan Huag, Chin Chang Li , “ A Secret Image Sharing method using Integer Wavelet
Transform” EURASIP Journal on advances in signal processing , Vol 2007.
[12] Jun Kong ,Yanfen Zang et al, “ A Scalable Secret Image Sharing method based on Discret Wavelet
Transformation” Springer-Verlog Berlin Heidelberg , 2007.

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