A New Method for Embedding Secret Data to the
Container Image Using ‘Chaotic’ Discrete
Orthogonal Transforms
Vladimir Chernov
1
, Oleg Bespolitov
2
1
Image Processing Systems Institute of RAS
151 Molodogvardejskaya St., 443001, Samara, Russia
2
Samara State University
Abstract. In this paper a method for embedding the secret data into the
container image is considered. The method is based on specifics of the spectral
properties of ad hoc two-dimensional discrete orthogonal transform. The values
of functions forming the basis of this transform are `chaotically’ distributed.
Two ideas provide the ground for the synthesis of these bases. Firstly, the 1D
M-transforms, that were introduced and investigated in certain particular cases
by H.-J. Grallert. Secondly, the application of introduced by I. Kàtai canonical
number systems in finite fields to numerating the input image pixels.
1 Introduction
Many methods of embedding secret data into the container image are based on the
modification of one or several least-significant bits of digital image pixels [1]-[3]. In
this paper we propose an alternative method, based on modification of spectral
components (components of two-dimensional orthogonal transform spectrum of the
container image) [4]-[5]. Let
12 12
(, ); , 0,1,..., 1xn n n n N
=
− be the container image.
Let the discrete orthogonal transform be given by:

12
12
1
12 12 , 12 12
,0
( , ) ( , ) ( , ); , 0,1,..., 1
N
mm
nn
xm m xn n h n n m m N
−
=
==−
∑
,
(1)
12 12 12 12 11 22
1
,, ,12,12 ,,
0
,(,)(,)
N
uu vv uu vv uv u v
n
hh hnnhnn δδ
−
=
<>= =
∑
.
(2)
Let
{}
12
,0,1,...,1Dmm N⊂= −, let
(
)
{
}
(
)
12 12
ˆ
,0,1;,
s
mm mm D
∈
∈ be the sensitive
(secret) data that is to be embedded into the container
12
(, )
x
nn . Let the spectrum of
the output image (with the secret data embedded) be given by
()
(
)
(
)
12 12 12
ˆˆˆ
,,,
y
mm smm xmm= .
(3)
Chernov V. and Bespolitov O. (2006).
A New Method for Embedding Secret Data to the Container Image Using ‘Chaotic’ Discrete Orthogonal Transforms.
In Proceedings of the 4th International Workshop on Security in Information Systems, pages 115-124
Copyright
c
 SciTePress

Applying the inverse transform, with respect to the transform (1), we obtain the image
()
12
,
y
nn , which contains in its spectrum (3) the secret data
(
)
12
ˆ
,
s
mm .
Unfortunately, there are certain reasons that do not allow for use of ‘classical’ discrete
orthogonal transforms (1)-(2) for the proposed approach. These transforms have
several specific properties, which enable their successful application to video data
encoding and make efficient certain image compressing algorithms. In particular,
when these transforms are applied, the image energy is concentrated in relatively
small fraction of spectral components. Thus, embedding the secret data into the
container image according to the spectral technique described by (3) will result in
addition of the ‘structured’, ‘texturized’, ‘regular’ noise (distortion) to the container.
Note that this type of noise is known to be much more perceptible by human than
random noise.
In the Fig. 2 example of the ‘structured noise’ is displayed. An example of the
‘random’ noise is provided in the Fig. 3.
Fig. 1. Original image.
Fig. 2. Distrortion after
Hartley transform is applied.
Fig. 3. Distrortion after
proposed transform is appli-
ed.
Thus, it is worth considering other types of orthogonal transforms, the transforms that
don’t concentrate the signal energy in few spectral components and allow for
effective removal of the inessential data. In this work, we intent to consider the
discrete orthogonal transform with the basis that is composed of ‘noise-similar’,
‘chaotic’ functions. For these transforms all the spectral components are
‘energetically equivalent’ and the image distortion associated with these transforms is
similar to the additive Gaussian noise. For this transform the corresponding
steganographic process (i.e. embedding of the secrete data to the container) represents
addition of a low-energy Gaussian noise, and this process will be more secure than
any bit-replacement technique. As during the image acquisition process, many
different independent sources of Gaussian noise with varying amplitudes are
superimposed onto the image, this is hard to determine whether the additional
Gaussian noise is due to the channel/sensor properties or steganography [5].
One-dimensional transforms (1) that provide required distribution of the signal energy
were introduced in [6]. The basis functions of these transforms have only two
different values. In the papers [7]-[9] application of these transforms to processing the
video information was considered. Various generalizations for the scheme described
116

in [6] were proposed by one of the authors in the papers [10]-[11] for the functions
()
m
hn with k different values.
The essence of constructing the set of the considered orthogonal transform basis
function is in use of the linear recurrence
1
( ) ( 1) ... ( ); , 0.
rjqr
yn ayn ayn r a a=−++− ∈≠F
(4)
This recurrence is defined over the finite field
q
F that consists of
s
qp
=
elements
(where
р is prime). The period of recurrence (4) is assumed to be maximal:
1
r
Nq=−(in this case, the recurrent sequence (4) is an m-sequence [12],[13]).
While constructing the basis functions of the transform (1) the elements of the
sequence
()
q
yn∈ F are replaced with the real numbers ()
m
hnin such a way, that for
the functions
()
m
hn the orthogonality-constraints (2) are satisfied.
One of the principle obstacles that prevents the results introduced in the cited papers
from being extrapolated to the two-dimensional case is the following: for 2D case a
‘good’ one-dimensional numeration of the two-dimensional array
()
{}
12 12
,;,nn nn∈ Z
is hard to be constructed. In the papers [14], [15] the conception
was introduced of the canonical number systems (CNS) in the ring
(
)
dS of
integers from the quadratic fields
(
)
{
}
;,dzabdab==+ ∈QQ. In terms of
CNS, the elements
()
zd∈S may be represented in a form of the finite sum
()
kz
j
j
jo
zz
α
=
=
∑
,
(5)
where the ‘digits’
j
z are from the certain finite subset ⊂ ZN , and the element
α
(the base of the canonical number system) is an element of the ring
(
)
dS .
In this paper, we define the one-to-one map that takes the elements of the ‘caterpiller’
of the
N-periodic m-sequence (4):
()
(
)
01
(0), , ( 1) , (1), , ( ) ,yyr yyr=−=YY………
(6)
to the elements of the ring
()
dS , represented in a form of r-term sums (5). Using
this map for processing the two-dimensional signals (images) we may construct the
one-dimensional numeration of the points from the two-dimensional integer lattice
2
Z and synthesize the discrete orthogonal transforms (1)-(2) with the ‘chaotic’
distribution of the basis functions
()
m
hn values.
117

2 Mathematical Background
The proof of the facts that are stated below may be found in [12], [13] (linear
recurrences) and in [14]-[15] (canonical number systems).
Recurrent functions in the finite fields
.
Let
q
F be a finite field that consists of
s
qp
=
elements, where p is prime.
Definition 1. The function that satisfies the linear recurrence (4), where
10
,..., , 0, ( (0),..., ( 1))
rq r
aa a y yr
∈
≠= −FY ,
is called a linear recurrent sequence of the order
r with the initial values
( (0),..., ( 1))yyr=−Y . The recurrence (4) of the maximal possible period 1
r
Nq=−
is called an m-sequence. Elementary properties of the m-sequence are stated in the
following Lemma.
Lemma 1 Let the recurrence (4) with non-zero initial values
0
Y be an m-sequence,
then
0.
if the n runs the full period of the sequence (4), that is equal 1
r
Nq
=
− , then
among the generated elements any element
0
q
a
≠
∈F will occur
1r
q
−
times, and
the zero element
0
q
∈F
will occur
1
1
r
q
−
−
times;
1.
in the entire period of the "caterpillar" (6) of the recurrent sequence (4) every non-
zero r-component vector from the space
(
)
r
q
F occurs only once.
Canonical number systems (CNS) in quadratic fields.
Let (){ ;, }dzabdab==+ ∈QQ be a quadratic Q extension field, d be a
square-free integer number. Note that if
0d > , the quadratic field is called real;
if 0d < , it is called imaginary. If the trace
(
)
(
)
() 2zabdabd a
=
++−=∈Tr Z
and the norm
(
)
(
)
22
()zabdabdadb=+ − =− ∈Norm Z of the element
()zabd d=+ ∈Q are integer, then the element z is called the algebraic integer
in
()dQ . Denote by
()
dS the subring of the integers from ()dQ .
Definition 2. The algebraic integer
A
Bd α =+ is called the base of the canonical
number system
in the ring of integers from the field ()dQ , if every integer z in
()dQ can be uniquely represented in a form of the finite sum
()
0
kz
j
j
j
zz
α
=
=
∑
,
()
{
}
0,1,..., 1
j
z α
∈
=−NormN .
The pair
{,}
α
N is called the canonical number system (CNS) in the ring ()dS of
integers from
()dQ . Below there are several examples of canonical number
systems.
1. Let () 2
α
=Norm , then there exist only three imaginary quadratic fields with the
rings of integers where binary canonical number systems exist, namely:
118

(а) the field ()iQ with the base 1 i
α
=
−±; (b) the field 7()iQ with the base
7(1 )2i
α
=−± ; (c) the field 2()iQ with the base 2i
α
=
± .
2. Let
(
)
3 α =Norm , then there exist only three imaginary quadratic fields with the
rings of integers where exist ternary canonical number systems, namely:
(a) the field
2()iQ with the bases 21 i
α
=
−± ; (b) the field 3()iQ with the base
3(3 )2i
α
=−± ; (c) the field 11()iQ with the base
(
)
1112i α =−± .
3 М-transforms
In [1] the orthogonal M-transform (1) was introduced. The M-transform basis
functions
()
m
hnare ‘very similar’ to random noise. Particularly, the functions ()
m
hn
are randomly equal to one of two values, and the relative frequencies of these values
are almost equal. In the paper [1] construction of the set of the basis functions was
grounded on use of the
m-sequence (3) for 2p
=
. For prime 2p
=
the basis
functions of this transform may be constructed using the following scheme.
• In the process of the functions
0
()hn construction, the ()yn sequence elements
are replaced with the real numbers
2
0
2
,if()1 ;
:() ()
,if()0 .
Ayn
yn h n
Byn
ϕ
=∈
⎧
=
⎨
=∈
⎩
F
F

(7)
•
The functions ()
m
hn may be obtained from the function
0
()hnapplying the
circular shift of the argument
(
)
0
() ( ); 0,1, , 1; 2 1
r
m
hn hmnm N N=+= −=−…
.
(8)
•
The numbers A and B are selected so that for the functions
{
}
()
m
hn the condition
on the function orthogonality (2) is satisfied.
The essential technical obstacle is the difficulty to obtain the relations from where A и
B may be easily determined. The following theorem generalizes the results of the
paper [6].
Theorem 1. Let
s
qp= , where p is prime, 1
r
Nq
=
− , the numbers
01
,...,
q
A
A
−
are
such that
()
(
)
1
010
, 1 , ( 0,..., 1)
kq
Ak A q A A k q λλ
−
−
=
+=− − =−. Let the functions
()
m
hn be given by
00
() , if () ; () ( )
km
hn A yn k h n hm n===+. Then there exist
efficiently computable constants
0
A
and
λ
, such that
(а) the set of functions
{}
(); , 0,1, , 1
m
hnmn N
=
−… forms an orthonormal basis;
(b) the constants
0
A
A= and
λ
are the solution of the following system of equations
119

Step 4. Consider the equality
(
)
(
)
{
}
:,
rr
dzvz v d αα
Ω
+=+∈Ω∈=Ω+ΣSS.
It may be easily verified that for the map of sets inducted by the map (8), the
following relation holds
(
)
()
(
)
(
)
**
22*
\\
rr
dd αα Ω+ = = ΣSZSZ. In other
words, the additive shifts of the domain
*
Ω
cover ‘almost’ all the points of the
discrete lattice
2
Z
, with the exception for the points that belong to the set
*
Σ
.
Step 5. We say that the points
(
)
(
)
** 2
12 1 2
,, ,zzzwww==∈Z are congruent
()
mod Σ , if for their prototypes given by (8) the following relation holds: zw−∈Σ.
It can be shown that every point
(
)
*
12
,
N
www
=
∈Δ is congruent
(
)
mod Σ to some
point
()
**
12
,zzz=∈Ω
of the fundamental domain. In their turn, for the points from
the fundamental domain
*
Ω there exists a one-to-one map to the elements of the
set
Ω that are numerated using (9).
Ring ()iS
Ring
(2)iS Ring (7)iS
Fig. 4. Fundamental domains
Ω
, associated with the binary CNS in ()idS .
Step 6. Therefore, summarizing the above-stated facts, a new numeration of the
points from the set
N
Δ may be obtained:
()
()
()
()
()
()
mod .(8)
***
12
.(8) .(9)
,
.
Eq
N
Eq Eq
wzzz
zzn n
Σ
∈Δ ⎯⎯⎯→ = ∈Ω ⎯⎯⎯⎯→
⎯⎯⎯⎯→= ⎯⎯⎯⎯→∈Z
(11)
The above-constructed functions
()
m
hn generate the basis functions
12
(, )
m
H
ν
ν , that
are defined in the two-dimensional domain
N
Δ
. In fact, consider, for example,
() ()
0
hn h n= . Similar to (10), we obtain:
(
)
(.(9)) (.(12))
(.(12)) * (mod)
12 12
() (), ()
(, ) (, )
Eq Eq
Eq
N
nznznzk
nn vv
Σ
⎯
⎯⎯⎯→↔ ⎯⎯⎯⎯→
⎯⎯⎯⎯→∈Ω⎯⎯⎯→∈Δ
Rat Irr
and the assume that
0012
() ( , )hn H
ν
ν = . The examples of these basis functions are
provided in the Fig.5.
121

Fig. 5. M-transform basis functions, 1d
=
5 The Results of the Experiments
In the Figs 1-3 the typical experimental results are displayed. Into the container
‘Lena’ image (256х256 pixels, Fig. 1) using the relation (1) we embedded the same
secret data using three different discrete orthogonal transforms:
•
two-dimensional discrete Hartley transform;
•
two-dimensional discrete Hadamard transform;
•
M-transform in the version, described in this work.
In the array of the container image spectrum about 20% of spectral components were
changed. In the Figs.2, the "structured" nature of the decoding error is noticeable. For
the images in the Figs. 3 and 8, the decoding error is similar to the random ‘non-
structured noise’. The images in these figures were obtained using the M-transform
instead of classical orthogonal transforms.
The structure of the distortion, that is revealed in the decoded image when certain
subset of the M-transform spectral components
ˆ
()
x
m
are "lost"/"modified", this
structure becomes more clear, if certain probabilistic interpretation is used (the
authors do not claim the absolute mathematically correctness of this interpretation).
Let the basis functions
()
m
hn be interpreted as random variables. These random
values are not correlated as the transform has its orthogonality property. For the input
signal
(
)
x
n
and for the distorted signal
(
)
*
x
n
the following relations hold
()
(
)
(
)
(
)
(
)
(
)
ˆ
*,
T
x
nxn xhnxn nT
τ
τ
τξ
∈
=+ =+
∑
,
where T is a set of the indices corresponding to the lost spectral components. The
random value
()
,nT ξ is linear combination of the random values (the values that has
the same distribution) and therefore for practical tasks
(
)
,nT ξ may be interpreted as
a Gaussian noise with the parameters that may be easily calculated.
122

Fig. 6. Error-field for discre-
te Hartley transform.
Fig. 7. Error-field for discre-
te Hadamard transform.
Fig. 8. Error-field for discre-
te M-transform.
In the figures 9-11 the autocorrelation functions (autocorrelation of the field of errors
for the used transform) is displayed.
Fig. 9. Autocorrelation fun-
ction of the field of errors for
Hartley transform.
Fig. 10. Autocorrelation fun-
ction of the field of errors for
Hadamard transform.
Fig. 11. Autocorrelation fun-
ction of the field of errors for
M-transform .
6 Conclusion
The major contributions of this article arise from development of the mathematical
fundamentals for application of canonical number systems and discrete orthogonal
transforms to the tasks of steganography. The proposed approach is based on a new
mathematical technique, namely on the theory of canonical number systems that so
far has not been applied to this tasks of digital signal processing. The empirical and
theoretical results are provided that if the proposed transform is used for embedding
the secrete data, then to the container image additive random noise is added, that is
much less perceptible than ‘regular’, ‘structured’ noise typical for classical discrete
orthogonal transforms.
123

Acknowledgements
This research was financially supported by the RF Ministry of Education and U.S.
Civilian Research & Development Foundation (CRDF Project SA-014-02) as part of
the joint Russian-American program "Basic Research and Higher Education"
(BRHE), and by Russian Foundation for Basic Research (Project # 06-01-00722)
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