A NEW CHAOS-BASED WATERMARKING ALGORITHM
Christophe Guyeux and Jacques M. Bahi
Computer Science Laboratory, University of Franche-Comt´e (LIFC)
Rue Engel-Gros, BP 527, 90016 Belfort Cedex, Belfort, France
Keywords:
Watermarking, Chaotic iterations, Topological chaos, Information hiding.
Abstract:
This paper introduces a new watermarking algorithm based on discrete chaotic iterations. After defining some
coefficients deduced from the description of the carrier medium, chaotic discrete iterations are used to mix the
watermark and to embed it in the carrier medium. It can be proved that this procedure generates topological
chaos, which ensures that desired properties of a watermarking algorithm are satisfied.
1 INTRODUCTION
Information hiding has recently become a major se-
curity technology, especially with the increasing im-
portance and widespread distribution of digital media
through the Internet. It includes several techniques,
among which is digital watermarking. The aim of dig-
ital watermarking is to embed a piece of information
into digital documents, like pictures or movies for ex-
ample. This is for a large panel of reasons, such as:
copyright protection, control utilization, data descrip-
tion, integrity checking, or content authentication.
Digital watermarking must have essential characteris-
tics including imperceptibility and robustness against
attacks. Many watermarking schemes have been pro-
posed in recent years, which can be classified into
two categories: spatial domain (Wu et al., 2007) and
frequency domain watermarking (Cong et al., 2006),
(Dawei et al., 2004). In spatial domain watermark-
ing, a great number of bits can be embedded without
inducing too clearly visible artifacts, while frequency
domain watermarking has been shown to be quite ro-
bust against JPEG compression, filtering, noise pollu-
tion, and so on. More recently, chaotic methods have
been proposed to encrypt the watermark, or embed it
into the carrier image for security reasons.
In this paper, a new watermarking algorithm is
given. It is based on the commonly named chaotic
iterations and on the choice of relevant coefficients
deduced from the description of the carrier medium.
This new algorithm consists of two basic stages: a
mixture stage and an embedding stage. At each of
these two stages, the proposed algorithm offers addi-
tional steps that allow the authentication of relevant
information carried by the medium or the extraction
of the watermark without knowledge about the origi-
nal image.
This paper is organized as follows: firstly, some
basic definitions concerning chaotic iterations is re-
called. Then, the new chaos-based watermarking al-
gorithm is introducedin Section 3. Section 4 is consti-
tuted by the evaluation of our algorithm: a case study
is presented, some classical attacks are executed and
the results are presented and commented on. The pa-
per ends by a conclusion section where our contribu-
tion is summarized, and planned future work is dis-
cussed.
2 BASIC RECALLS: CHAOTIC
ITERATIONS
In the sequel S
n
denotes the n
th
term of a sequence S,
V
i
denotes the i
th
component of a vector V and f
k
=
f ◦ ... ◦ f denotes the k
th
composition of a function
f. Finally, the following notation is used: J1;NK =
{1, 2, . . . , N}.
Let us consider a system of a finite number N of
cells, so that each cell has a boolean state. Then a se-
quence of length N of boolean states of the cells corre-
sponds to a particular state of the system. A sequence
which elements belong in J1;NK is called a strategy.
The set of all strategies is denoted by S.
Definition 1. Let S ∈ S. The shift function is defined
by σ : (S
n
)
n∈N
∈ S −→ (S
n+1
)
n∈N
∈ S and the initial
function i is the map which associates to a sequence,
its first term: i : (S
n
)
n∈N
∈ S −→ S
0
∈ J1;NK.
455
Guyeux C. and M. Bahi J. (2010).
A NEW CHAOS-BASED WATERMARKING ALGORITHM.
In Proceedings of the International Conference on Security and Cryptography, pages 455-458
DOI: 10.5220/0002978404550458
Copyright
c
 SciTePress

Definition 2. The set B denoting {0, 1}, let f :
B
N
−→ B
N
be a function and S ∈ S be a strategy.
Then, the so-called chaotic iterations are defined by
x
0
∈ B
N
and ∀n ∈ N
∗
, ∀i ∈ J1;NK,
x
n
i
=

x
n−1
i
if S
n
6= i

f(x
n−1
)

S
n
if S
n
= i.
(1)
3 A NEW CHAOS-BASED
WATERMARKING
ALGORITHM
3.1 Most and Least Significant
Coefficients
Let us first introduce the definitions of most and least
significant coefficients of an image.
Definition 3. For a given image, the most significant
coefficients (in short MSCs), are coefficients that al-
low the description of the relevant part of the image,
i.e. its most rich part (in terms of embedding informa-
tion), through a sequence of bits.
For example, in a spatial description of a grayscale
image, a definition of MSCs can be the sequence con-
stituted by the first three bits of each pixel.
Definition 4. By least significant coefficients (LSCs),
we mean a translation of some insignificant parts of a
medium in a sequence of bits (insignificant can be un-
derstand as: “which can be altered without sensitive
damages”).
The LSCs are used during the embedding stage:
some of the least significant coefficients of the carrier
image will be chaotically chosen and replaced by the
bits of the (possibly mixed) watermark.
The MSCs are only useful in case of authentica-
tion, mixture and embedding stages will then depend
on them. Hence, a coefficient should not be defined
at the same time both as a MSC and a LSC: the LSC
can be altered, while the MSC is needed to extract the
watermark (in case of authentication).
3.2 Stages of the Algorithm
Our watermarking scheme consists of two classical
stages: the mixture of the watermark and its embed-
ding into a cover image.
3.2.1 Watermark Mixture
For security reasons, the watermark can be mixed be-
fore its embedding. A common way to achieve this
stage is to use the bitwise exclusive or (XOR), for
example, between the watermark and a logistic map.
In this paper, we will introduce a mixture scheme
based on chaotic iterations. Its chaotic strategy will
be highly sensitive to the MSCs, in case of an authen-
ticated watermark (Bahi and Guyeux, 2010). For the
details of this stage see the Paragraph 4.1.2 in Section
4.
3.2.2 Watermark Embedding
This stage can be done either by applying the logical
negation of some LSCs, or by replacing them by the
bits of the possibly mixed watermark.
To choose the sequence of LSCs to be changed,
a number of integers, less than or equals to the num-
ber N of LSCs, corresponding to a chaotic sequence

U
k

k
, is generated from the chaotic strategy used in
the mixture stage and possibly the watermark. Thus,
the U
k
− th least significant coefficient of the carrier
image is either switched, or substituted by the k
th
bit
of the possibly mixed watermark. In case of authenti-
cation, such a procedure leads to a choice of the LSCs
which are highly dependent on the MSCs.
On the one hand, when the switch is chosen, the
watermarked image is obtained from the original im-
age, whose LSCs L = B
N
are replaced by the result of
some chaotic iterations. Here, the iterate function is
the vectorial boolean negation, defined by f
0
: B
N
−→
B
N
, (x
1
, . . . , x
N
) 7−→ (x
1
, . . . , x
N
), the initial state is L
and strategy is equal to

U
k

k
. In this case, it is pos-
sible to prove that the whole embedding stage satis-
fies topological chaos properties (Bahi and Guyeux,
2010), but the original medium is needed to extract
the watermark.
On the other hand, when the selected LSCs are
substituted by the watermark, its extraction can be
done without the original cover. In this case, the se-
lection of LSCs still remains chaotic, because of the
use of a chaotic map, but the whole process does not
satisfy topological chaos (Bahi and Guyeux, 2010):
the use of chaotic iterations is reduced to the mixture
of the watermark. See the Paragraph 4.1.3 in Section
4 for more details.
3.2.3 Extraction
The chaotic sequence U
k
can be regenerated, even in
the case of an authenticated watermarking: the MSCs
have not been changed during the stage of embedding
watermark. Thus, the altered LSCs can be found. So,
SECRYPT 2010 - International Conference on Security and Cryptography
456

in case of substitution, the mixed watermark can be
rebuilt and “decrypted”. In case of negation, the result
of the previous chaotic iterations on the watermarked
image, is the original image.
If the watermarked image is attacked, then the
MSCs will change. Consequently, in case of authen-
tication and due to the high sensitivity of the embed-
ding sequence, the LSCs designed to receive the wa-
termark will be completely different. Hence, the re-
sult of the decrypting stage of the extracted bits will
have no similarity with the original watermark.
4 A CASE STUDY
4.1 Stages and Details
4.1.1 Images Description
Carrier image is the famous Lena, which is a 256
grayscale image and the watermark is the 64 × 64
pixels binary image depicted in Fig. 1a. The embed-
ding domain will be the spatial domain. The selected
MSCs are the four most significant bits of each pixel
and the LSCs are the three following bits (a given
pixel will at most be modified by four levels of gray
by an iteration). The last bit is then not used. Lastly,
LSCs of Lena are substituted by the bits of the mixed
watermark.
(a) Watermark. (b) Watermarked Lena.
Figure 1: Watermark and watermarked Lena.
4.1.2 Mixture of the Watermark
The initial state x
0
of the system is constituted by the
watermark, considered as a boolean vector. The iter-
ation function is the vectorial logical negation f
0
and
the chaotic strategy (S
k
)
k∈N
will depend on whether
an authenticated watermarking method is desired or
not, as follows. A chaotic boolean vector is generated
by a number T of iterations of a logistic map ((µ, U
0
)
parameters will constitute the private key). Then, in
case of unauthenticated watermarking, the bits of the
chaotic boolean vector are grouped six by six, to ob-
tain a sequence of integers lower than 64, which will
constitute the chaotic strategy. In case of authentica-
tion, the bitwise exclusive or (XOR) is made between
the chaotic boolean vector and the MSCs and the re-
sult is converted into a chaotic strategy by joining its
bits as above. Thus, the mixed watermark is the last
boolean vector generated by the chaotic iterations.
4.1.3 Embedding of the Watermark
To embed the watermark, the sequence (U
k
)
k∈N
of
altered bits taken from the M LSCs must be defined.
To do so, the strategy (S
k
)
k∈N
of the mixture stage is
used as follows

U
0
= S
0
U
n+1
= S
n+1
+ 2× U
n
+ n (mod M).
(2)
To obtain the result depicted in Fig. 1b.
Remark that the map θ 7→ 2θ of the torus, which
is a famous example of topological Devaney’s chaos
(Devaney, 2003), has been chosen to make (U
k
)
k∈N
highly sensitive to the chaotic strategy. As a conse-
quence, (U
k
)
k∈N
is highly sensitive to the alteration
of the MSCs: in case of authentication, any signifi-
cant modification of the watermarked image will lead
to a completely different extracted watermark.
4.2 Simulation Results
To prove the efficiency and the robustness of the
proposed algorithm, some attacks are applied to our
chaotic watermarked image. For each attack, a sim-
ilarity percentage with the watermark is computed,
this percentage is the number of equal bits between
the original and the extracted watermark.
4.2.1 Zeroing Attack
In this kind of attack, some pixels of the image are
put to 0. In this case, the results in Table 1 have been
obtained. We can conclude that in case of unauthen-
tication, the watermark still remains after a cropping
attack: the desired robustness is reached. In case of
authentication, even a small change of the carrier im-
age leads to a very different extracted watermark. In
this case, any attempt to alter the carrier image will be
signaled.
4.2.2 Rotation Attack
Let r
θ
be the rotation of angle θ around the center
(128, 128) of the carrier image. So, the transforma-
tion r
−θ
◦r
θ
is applied to the watermarked image. The
good results in Table 2 are obtained.
A NEW CHAOS-BASED WATERMARKING ALGORITHM
457

Table 1: Zeroing attacks.
UNAUTHENTICATION AUTHENTICATION
Size (pixels) Similarity Size (pixels) Similarity
10 99.08% 10 89.81%
50 97.31% 50 54.54%
100 92.43% 100 52.24%
Table 2: Rotation attacks.
UNAUTHENTICATION AUTHENTICATION
Angle Similarity Angle Similarity
5 94.67% 5 59.47%
10 91.30% 10 54.51%
25 80.85% 25 50.21%
4.2.3 JPEG Compression
A JPEG compression is applied to the watermarked
image, depending on a compression level. Let us no-
tice that this attack leads to a change of the represen-
tation domain (from spatial to DCT domain). In this
case, the results in Table 3 have been found. A good
authentication through JPEG attack is obtained. As
for the unauthentication case, the watermark still re-
mains after a compression level equal to 10. This is
a good result if we take into account the fact that we
use spatial embedding.
4.2.4 Gaussian Noise
Watermarked image can be also attacked by the addi-
tion of a Gaussian noise, depending on a standard de-
viation. In this case, the results in Table 4 have been
found.
5 DISCUSSION AND FUTURE
WORK
In this paper, a new way to generate watermarking
methods is proposed. The new procedure depends on
a general description of the carrier medium to water-
mark, in terms of the significance of some coefficients
we called MSC and LSC. Its mixture and also the se-
lection of coefficients to alter are based on chaotic
iterations, which generate topological chaos in the
sense of Devaney. Thus, the proposed algorithm pos-
sesses expected desirable properties for a watermark-
Table 3: JPEG compression attacks.
UNAUTHENTICATION AUTHENTICATION
Ratio Similarity Ratio Similarity
2 82.95% 2 54.39%
5 65.23% 5 53.46%
10 60.22% 10 50.14%
Table 4: Gaussian noise attacks.
UNAUTHENTICATION AUTHENTICATION
Standard dev. Similarity Standard dev. Similarity
1 74.26% 1 52.05%
2 63.33% 2 50.95%
3 57.44% 3 49.65%
ing algorithm. For example, strong authentication of
the carrier image, security, resistance to attacks, and
discretion.
The algorithm has been evaluated through attacks
and the results expected by our study have been ex-
perimentally obtained. The aim was not to find the
best watermarking method generated by our general
algorithm, but to give a simple illustrated example to
prove its feasibility. In future work, other choices of
iteration functions and chaotic strategies will be ex-
plored. They will be compared in order to increase
authentication and resistance to attacks. Lastly, fre-
quency domain representations will be used to select
the MSCs and LSCs.
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