INTRODUCTION
The digital information revolution has brought about changes in our
society and our lives. The many advantages of digital information have
also generated new challenges and new opportunities for innovation.
Every few years, computer security has to re-invent itself. New
technology and new applications bring new threats and force us to invent
new protection mechanisms [6]. The fractals theory has proved to be
suitable in many fields and particularly interesting in various
applications of complex systems. Recently, some researchers developed
cryptosystem based on fractals, since one of the fractal properties was
having extremely high visual complexity while having low information
contents, which can make simple cryptographic and Steganography methods
very complex [4].
In most applications, image data is two-dimensional data;
therefore, an image can be considered as two-dimensional memory. Fractal
archiving is based on image representation in compact form by means of
iterated function system coefficients. First important advances are due
to Barnsley [1], who introduces for the first time the term of Iterated
Function Systems (IFS), based on the self-similarity of fractal sets.
Barnsley's work assumes that many objects can be closely
approximated by self-similarity objects that might be generated by the
use of IFS simple transformations.
The natural question may appear: "Can we use IFS to
approximate images?" The seminal research by Jacquin [3], then a
Ph.D. student of Barnsley at Georgia Tech, provided the basis of
block-based fractal image coding which is still used today.
Jacquin's research launched an intensive activity in fractal image
compression [2,7]. From this assumption, the IFS can be seen as a
relationship between the whole image and its parts, thus exploiting the
similarities that exist between an image and its smaller parts. At that
point, the main problem is how to find these transformations or, (what
is the same) how to define the IFS. There is, in fact, a version of the
IFS theory, the local iterated function systems theory that minimizes
the problem by stating that the image parts do not need to resemble the
whole image but it is sufficient for them to be similar to some other
bigger parts in it. It was Arnaud Jacquin [3], who developed an
algorithm to automate the way to find a set of transformations giving a
good quality to the decoded images.
MATERIALS AND METHODS
The major concepts and results of IFS and their application to the
study of functions are presented. A more detailed review of the topics
was given in [1,5,6]. IFS Theory: Let's consider a metric space
([pounds sterling], d) where d is a given metric. A Hausdorff space H
([pounds sterling]) is defined to be the space of all compact subset of
[pounds sterling] with the Hausdorff distance h. A contractive
transformation is defined by: [beta]: [pounds sterling] [right arrow]
[pounds sterling], that satisfies:
d([beta](x),[beta](y))[less than or equal to] sd(x,y), x,y [member
of] [pounds sterling], 0 [less than or equal to] s [less than or equal
to] 1
We write Con ([pounds sterling], d) for the set of all contractive
maps [beta]: [pounds sterling] [arrow right] [pounds sterling]. An IFS
consists of a complete metric space ([pounds sterling], d) and a number
of contractive mappings [[beta].sub.i] defined on [pounds sterling]. The
fractal transformation associated with IFS is defined by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where, E is any element of the space H of non-empty compact subsets
of [pounds sterling]. If [[beta].sub.i] is contractive for every i, then
B is contractive and there exist a unique fixed point for which:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
or
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
A is called the attractor of B. If B is continuous then A is called
a fixed point of B. The fundamental result upon which the entire theory
of iterated function systems is founded is the Banach Contraction
Mapping Principle (BCMP) or Fixed Point Theorem, which state that, if
([pounds sterling], d) is a complete metric space and [beta] [member of]
Con([pounds sterling], d) with contractivity factor s, then [beta] has a
unique fixed point A [member of] [pounds sterling]. Furthermore, A is
the attractor of B.
The transformations B are usually chosen to be affine. For the two
dimensional case the affine transformations have the following forms:
[x.sub.n+1] = [ax.sub.n] + [by.sub.n] + e
[y.sub.n+1] = [cx.sub.n] + [dy.sub.n] + f
The coefficients a, ..., f are the IFS "code". This also
be written in the affine form as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[beta] is said to be linear, if e = f = 0.
Now, suppose that we are given [alpha] [member of] [pounds
sterling]. A natural question that was first asked in IFS theory is
whether or not it is always possible to find a contractive operator B
[member of] Con([pounds sterling], d) whose fixed point is [alpha]. We
expect that, in general, this is not possible and that one must be
satisfied in finding fixed points [[alpha].sub.i] of contractive
operations [[beta].sub.i] that are approximations to [alpha]. Even in
this case, however, we are faced with the problem of finding such fixed
points [[alpha].sub.i]. This problem is called the Inverse Problem. It
is generally stated as follows. Given (Y; [d.sub.Y]) a metric space, y
[member of] and [epsilon] > 0, can we find a non constant [beta]
[member of] Con(Y; [d.sub.Y]) such that [d.sub.Y](y; [y.sub.[beta]])
<[epsilon]?
Before commenting on this question, an additional question that
arises is, "given y [member of] Y and [beta] [member of] Con(Y;
[d.sub.Y]), how close is y to [y.sub.[beta]]"? The following
proposition lends an answer. Let y; Y and [beta] be as above. Then:
[d.sub.Y] (y, [y.sub.[beta]]) [less than or equal to] 1/1 - s
[d.sub.Y] (y,[beta](y))
This is often called the collage theorem. It is important in
helping to identify the functions to use in an IFS in order to
approximate the attractor. The Collage Theorem is fundamental to the
theory of IFS because it states that if [beta](y) is close to y, then
[y.sub.[beta]] is also close to y. Of course, if s [congruent to] 1, the
right hand side of the inequality might not be very small. Thus, this
gives some insight into finding our desired function. We should find
[beta] [member of] Con(Y; [d.sub.Y]), which takes y close to itself. We
recall from the BCMP that [y.sub.[beta]] is the attractor of [beta] if Y
is complete. Hence we can iterate [beta] to retrieve [y.sub.[beta]] and
get the desired approximation to y. Therefore, the Inverse Problem is
often formulated as follows: Let (Y; [d.sub.Y]) be a complete metric
space and let y [member of] Y. Given [epsilon]>0, can we find a non
constant [beta] [member of] Con(Y; [d.sub.Y]), such that
[d.sub.Y](y;[beta](y)) < [epsilon]?
A formal solution to this problem was given in [12] in the case of
IFS on grey-level maps. This will be important in our study of
approximations of images. Once [beta] is determined, it is easy to get
the decoded image by making use of the BCMP, the transformation B is
applied iteratively on any initial point until the succession of images
does not vary significantly. However, given a set M, how to find a
contractive transformation B such that its attractor A is close to M? To
answer this question in symbols is to apply the collage theorem.
For a set M and a contraction B with attractor A:
h(M,A) [less than or equal to] h(M,B(M))/1 - s
where, h is the Hausdorff Distance. That is to say that M and A
will be sufficiently close, if M and B(M) are made close enough in terms
of [[beta].sub.i] and combining the following two expressions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Which implies:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
So, M can be partitioned as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then, [m.sub.i] can be closely approximated by applying a
contractive affine transformation [[beta].sub.i] on the whole M:
[m.sub.i] = [[beta].sub.i](M)
From IFS to IFSM fractal Transform: The concepts of IFS, first
developed by Barnsley and Demko [5] and IFS on grey-level maps (IFSM),
was introduced by Forte and Vrscay [9]. We continue with a discussion of
the inverse problem for IFSM. The main idea of a fractal based image
coder is to determine a set of contractive transformations to
approximate each block of the image (or a segment, in a more general
sense), with a larger block. More details and explanation can be found
in [12].
The Collage Theorem tells us that in order to find an IFS whose
attractor looks like a given set, we must find a set of contractive
transformations on a suitable space, in which the given set lies, such
that the distance between the given set and the union of the
transformations is small. In other words, the union of the
transformations is close to, or looks like, the given set. The IFS which
satisfies this may be a good candidate for reproducing the given set, or
image, by the attractor of the IFS. Thus this image can be stored using
much less space [14].
Consider applying this theory to images, (i.e., computer images).
One can think of an image as being a compact subset of [R.sup.n]. One
can model a computer screen by [pounds sterling] = [[0,1].sup.2] or
[R.sup.2] and define an image on the screen to be a set A in [pounds
sterling], with points being screen pixels. If x [member of] A, the
associated pixel is plotted white. If x [not member of] A, leave the
pixel black. Hence a white screen represents A [subset or equal to]
[[0,1].sup.2]. Since the world is not black and white. What is needed is
an IFS-type method, which allows for, greys, i.e., maps, which move
pixels around and then scale their grey-levels. These thoughts lead to
IFSM theory. There is however a fundamental difference between the IFS
and IFSM. The IFS works with measure and a set of probabilities
[p.sub.i] associated with the [[beta].sub.i] which acts as
multiplicative weight. The IFSM work with function u:[pounds sterling]
[right arrow] [0,1] and function [phi]:[0,1] [right arrow] [0,1] which
are composed with the u. From the viewpoint of image processing the
value u(x) may be interpreted as a nonnegative gray level or brightness
value at the point (or pixel) x [member of] [pounds sterling] [13].
Let us consider a compact subset A of [R.sup.2] to stimulate some
ideas. Formulate a definition of A being a grey-scale image is to think
of the image as a function, rather than a set. That's mean
formulate IFS method on functions from sets to grey-levels in the form
of A = {([x.sub.i], [y.sub.i], u([x.sub.i], [y.sub.i])), i = 1, ..., N},
where u ([x.sub.i], [y.sub.i]) represent the grey level value of the set
([x.sub.i], [y.sub.i]) [10].
These developments of IFS give a necessity to define a complete
metric space of these functions. A local metric for the gray level maps
with respect to an element u [member of] [pounds sterling] was
contracted and the continuity of attractor [u.sub.k] with respect to
[[phi].sub.i] maps was then established. Let [OMEGA] ([pounds sterling])
= {[beta]:[pounds sterling] [right arrow] R | [[beta].sup.-1](r) [member
of] H([pounds sterling]), [for all]r [member of] R}, this set is defined
as the set of grey level maps on [pounds sterling]. Now a metric on this
space must be defined such as:
D(u, v) = Sup h([u.sup.-1](r),[v.sup.-1](r)) [for all]u, v [member
of] [OMEGA] ([pounds sterling]), [for all]r [member of] R
If ([pounds sterling], d) is complete metric space then ([OMEGA]
([pounds sterling]),D) is also. The operator T on [OMEGA]([pounds
sterling]), is defined by, Tu(x) = max u([[beta].sup.-1](x)), [for
all]u[member of][OMEGA]([pounds sterling]), x [member of] [pounds
sterling]. Thus, to find an IFS whose attractor is 'close to'
or 'looks like' a given image, one must find a set of
contraction mappings such that the union, or collage, of the given set
under the transformations is 'close to' or 'looks
like' the given image. This leads to the next result. Let ([pounds
sterling], d) be a metric space and let B = {[[beta].sub.n]: n = 1,2,
..., N} is contractive with contractivity factor [s.sub.n]. Then T is
contractive with contractivity S = max{[s.sub.n]: n = 1, 2, ..., N}.
Also T has a unique attracting fixed point p [member of] T, T (p) = p.
Since u took only two values, modify this new operator to the grey level
values, so a grey level component is added.
The IFSM operator [T.sub.u] (x) = max [phi] (u([[beta].sup.-1](x)))
[for all] x [member of] [pounds sterling], [phi]: R [right arrow] R.
where [phi] is defined by,. [for all]u [member of] [OMEGA] ([pounds
sterling]), [phi](u([[beta].sup.-1](x))) = [alpha] u([beta].sup.-1](x))+
[xi]. Therefore define the operator T on u[member of][OMEGA]([pounds
sterling]) by [T.sub.u](x) = [summation over (i)] [[alpha].sub.i]
u([[beta].sub.i.sup.-1](x)) + [[xi].sub.i] [for all] x [member of]
[pounds sterling], where [summation] indicates that the sum runs over
the all indices i with x [member of] [[beta].sub.i]([pounds sterling]).
The diagram in Fig. 1 shows these operations.
Proposed approach: There are many types of cryptography, in which
there are "double enciphering" and "double
deciphering" processes, that make the codes more difficult to crack
and to analyze. For enciphering, firstly, one of the classical
Cryptographic methods are used to convert message letter into integer
numbers, secondly arranging the resulting code in a chosen manner of
affine IFS transformation and the resulting enciphering code is the
attractor of the IFS system. For deciphering, the receiver of the
attractor A retrieves affine IFS transformation B using "Inverse
Problems" techniques to perform the first level of deciphering
method, then some algebraic calculation applied to obtain the plain
text. To illustrate the method some algebraic facts are recalls. Let m
be a positive integer, the idea is to take m linear combination of the n
alphabetic characters in one plaintext element thus producing the m
alphabetic characters in one ciphertext element. An m x m matrix K =
([k.sub.i,j]) is taken as a first key. Let X = ([x.sub.1], [x.sub.2],
..., [x.sub.m]) and k [member of] K (the set of all m x m invertible
matrices), we compute y = eK (X) = ([y.sub.1], [y.sub.2], ...,
[y.sub.m]). We say that the ciphertext is obtained from the plaintext by
means of a linear transformation and [K.sup.-1] is used for deciphering
as X = [YK.sup.-1] [8]. A matrix K has an inverse if and only if its
determinant is non-zero. [Z.sub.n] denotes the ring of integer's
modulus n. [Z.sub.n] is Galois field if and only if n is prime number.
So we assume that our language has n-letter, n is prime, enciphering and
deciphering m units of messages of length 1 at a time. K represents an m
x m matrix whose entries belong to [Z.sub.t] for which t = [n.sup.m], D
represents the det(K). The relevant result for our purpose is that a
matrix K has an inverse modulo n if and only if GCD(det(K),n) = 1 [11].
Theorem: [beta](X) = AX + b could be used as a secret key to
encipher p messages of length m at a time in n-letter alphabet if and
only if GCD(D, [n.sub.m]) = 1.
[FIGURE 1 OMITTED]
Proof: If B is secret key then B is one to one map from [Z.sub.t]
to [Z.sub.t] where t = [n.sup.m] and hence onto and so invertible. Thus
GCD(D, [n.sup.m]) = 1. Conversely if GCD (D, [n.sup.m]) = 1, then A is
invertible and hence [beta] is one to one.
The sender arranges each unit of length m in entries with value one
in the affine IFS transformation. The elements of the B maps constructed
from ([C.sub.ij]/[n.sup.m]) where Cij = [p.sub.1] x [n.sup.m] +
[p.sub.2] [n.sup.m-1] + ... + [p.sub.m].
Affine IFS maps: An IFS is a standard way to model natural objects.
The intuitive key for deriving IFS that models any given object is
self-tiling (similarity). One can always view an object as the union of
several sub-objects. Let the sub-objects be actually scaled-down copies
of the original object. Each of these subjects is called a tile. In
particular, each sub-object is obtained by applying an affine
transformation to the entire object. Now consider the original object
with two or more affine transformed copies of itself. The tiling scheme
should completely cover the object, even if this necessitates
overlapping the tiles. Each transformation used to "create" a
tile corresponds exactly to one map in the IFS. In order to create an
IFS, one first specifies a finite set of contractive affine
transformations {[[beta].sub.i]; i = 1, ..., n} in [R.sup.2]. In
general, a contractive affine transformation [beta] in [R.sup.2] is of
the form: [beta](X) = AX + b, which could be used as a secret key to
produce an enciphering code. There are different possibilities to
arrange element in IFS invertible maps, therefore, for abbreviation,
binary sequences of 0's and 1's used to represent all
possibilities for element arranging in the [[beta].sub.i] maps, as
follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
All the above orders are for linear affine transformation. Now for
non- linearity order each one of the above maps is extended to three
forms by adding the translation part b. For example, for [beta] =
111000, we have:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
RESULTS
Conversion of the plain-text message to the unreadable format is
known as enciphering of the message. Similarly, conversion of the
enciphered message back to the human readable form through the reversal
of the encryption algorithm is known as deciphering of the message [8].
Encryption method: Let's assume that there are two parties
(sender and receiver) in two far places that need to communicate
secretly in a way that a third person (intruder) won't figure or
recognize that they are exchanging information between them. However,
the alphabetic, the classical encryption method and the order of the
affine IFS maps must be agreed upon between sender and receiver.
Enciphering algorithm: In this algorithm an alphabet of n = 29
character is chosen:
* The message characters are given a numbers as it appear in Table
1, show the length of the message
* Divide the message of length l into units of length m = 3,
represented by [p.sub.i][p.sub.i+1] [p.sub.i+2]
* Calculate the numeric value of each unit using the polynomial C =
[p.sub.i] [n.sup.2] + [p.sub.i+1] n + [p.sub.i+2], or matrices operation
to perform first level of the proposed method
* The contraction factor used is r = 1/nm
* The elements of the chosen affine IFS transformations
[[beta].sub.i] are calculated by [[beta].sub.i] = [r.sup.*]C.
Notice that B = {[[beta].sub.1], [[beta].sub.2], ...
[[beta].sub.i]} called a (hyperbolic) IFS
* The attractor A is generated using Random Iterated Algorithm [1]
* The enciphering code is the picture represents the Attractor A
Example: To encrypt the message, "We will attack at dawn
through the left flack", the sender and the receiver agreed on an
alphabet mentioned in Table 1. The message is divided into units of
three characters and used as inputs to the affine transformations after
applying the polynomial C = [p.sub.i] [n.sup.2] + [p.sub.i+1] n +
[p.sub.i+2], the enciphering code is shown in Table 2.
If the affine mappings, 111001, 101110, 111000, 100111 are chosen,
then the IFS for example 1 are constructed as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Applying the random iterated algorithm, the attractor of these
transformations is shown in Fig. 2.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Decryption method: The main idea to automate the searching of local
IFS relies on a partition of the image into N non-overlapping blocks of
a fixed size, called Range Blocks. Each range block [R.sub.i], for i
[member of] 1, ..., N}, is coded independently by matching it with a
bigger block [D.sub.i] in the image, called Domain Blocks. This match
defines a transformation [[tau].sub.i] and the global fractal code is
then given by the union [tau] = [union][[tau].sub.i] of local transforms
as shown in Fig. 3. Moreover, each local code [[tau].sub.i] restricted
to consist of a reduction, a discrete isometric and an affine
transformation on the luminance. Hence, [[tau].sub.i] can be modeled by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where, [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i], [t.sub.i,1],
[t.sub.i,2] represent the geometric transforms and [s.sub.i], [o.sub.i]
the grey-levels transform; x, y are the pixel coordinates and z the
corresponding luminance value [14].
Deciphering algorithm:
* 1-Upon the receipt of the attractor (picture) A, the receiver
retrieves B using "Inverse Problems" techniques. Let A denote
the image we want to encode. Let also [A.sub.r] denote a partition of A
to n x n blocks referred to as Range Blocks ([R.sub.b]). Similarly,
[A.sub.d] will denote another partition of A, this time to 2 x 2 n
blocks or Domain Blocks ([D.sub.b]) in steps of n x n pixels. The goal
of the deciphering algorithm is to establish a relationship between
[A.sub.r] and [A.sub.d] in such a way that any [R.sub.b] can be
expressed as a set of transformations to be applied on a particular
[D.sub.b]. This algorithm is illustrated by the flowchart in Fig. 4 [6]
* The receiver then modifies the entries of the retrieved IFS
system B to get [[beta].sub.i] as they agreed on before
* By multiplying each entry in the affine IFS map by [n.sup.m] and
rounding them to the nearest integer we perform the first level of
decrypting method
[FIGURE 4 OMITTED]
The second level is performed by applying some algebraic
calculation to find [p.sub.1], [p.sub.2], [p.sub.3] in each cipher unit,
as follows:
* [p.sub.1] = int(C/n)
* R = C mod [n.sup.2]
* [p.sub.2] = int(R/n)
* [p.sub.3] = R mod n
DISCUSSION
The theory of IFS was extended to local IFS where each part of the
image is approximated by applying a contractive affine transformation on
another part of the image: [m.sub.i] = [[beta].sub.i] ([D.sub.i]).
[D.sub.i] is the bigger part from which mi is approximate. The main idea
of a fractal based image coder is to determine a set of contractive
transformations to approximate each block of the image (or a segment, in
a more general sense), with a larger block. In this paper we propose a
new Cryptographic method using the fractals theory (more precisely the
IFS theory). For enciphering, firstly, one of the classical
Cryptographic methods are used to convert message letter into integer
numbers, secondly arranging the resulting code in a chosen manner of
affine IFS transformation and the resulting enciphering code is the
attractor of the IFS system. For deciphering, the receiver of the
attractor A retrieves affine IFS transformation B using "Inverse
Problems" techniques to perform the first level of deciphering
method, an algorithm based on Jacquin's work is used, then some
algebraic calculation applied to obtain the plain text.
CONCLUSION
The proposed approach employs double enciphering and double
deciphering process. The fractal image generation through the given
parameters, needs a great amount of iterations to converge into an
attractor, but at the same time, it provides non uniform randomness and
it is independent of the image size [9]. In the proposed method the IFS
(B(X) = AX + b) could be used as a secret key to encipher p units
messages of length m at a time in n-letter alphabet if and only if the
GCD (D, [n.sub.m]) = 1, then generates the fractals associated with the
IFS. The receiver can recover the message using the collage theorem and
simple algebraic calculations.
The proposed fractal encryption technique gives the possibility to
hide maximum amount of data in an image that represent the attractor of
the IFS without degrading its quality. The other advantage of using
fractal as an encryption technique is to make the hidden data robust
enough to withstand image processing technique which does not change the
appearance of image. For better results images should be in 24-Bit bit
map (bmp) format and much better results are obtained by using larger
size image (512 x 512).
ACKNOWLEDGEMENT
The researchers would like to acknowledge the Institute for
Mathematical Research (INSPEM), University Putra Malaysia (UPM) for its
continuous support.
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Function Systems for still Image Processing. IWISP-96, Manchester, UK.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi= 10.1.1.42.623
Nadia M.G. Al-Saidi and Muhammad Rushdan Md. Said
Institute for Mathematical Research, University Putra Malaysia,
43400, Serdang, Selangor, Malaysia
Corresponding Author: Nadia M.G. Al-Saidi, Institute for
Mathematical Research, University Putra Malaysia, 43400, Serdang,
Selangor, Malaysia Tel : +60162144183/+03 8946 6878 Fax: +03 89423789
Table 1: English Alphabet used for encryption
English letters with
with integer values
A = 0 B = 1 C = 2 D = 3
E = 4 F = 5 G = 6 H = 7
I = 8 J = 9 K = 10 L = 11
M = 12 N = 13 O = 14 P = 15
Q = 16 R = 17 S = 18 T = 19
U = 20 V = 21 W = 22 X = 23
Y = 24 Z = 25 $ = 26 . = 27 ? = 28
Table 2: Message units and their enciphering code
M. unit Value M. unit Value
we$ 18644 hro 6394
wil 18745 ugh 17291
1$a 10005 $th 22424
tta 16530 e$l 4129
ck$ 1998 eft 3528
at$ 577 $fl 22022
daw 2545 ack 68
n$t 11706 -- --