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Adv Comput Math (2009) 31:267–281

DOI 10.1007/s10444-008-9096-1

A cryptographic watermarking technique

for multimedia signals

Jian Ren

Received: 17 November 2007 / Accepted: 9 May 2008 /

Published online: 20 August 2008

Abstract Digital watermarking has been widely used in digital rights manage-

ment and copyright protection. In this paper, new cryptographic watermark

schemes are proposed. Compare to the existing watermarking techniques, our

proposed watermark schemes combine both security and efﬁciency that none

of the existing schemes can do. We ﬁrst develop an algorithm to randomly

generate the watermark indices based on the discrete logarithm problem

(DLP) and the Fermat’s little theorem. Then we embed watermark signal into

the host image in both time domain and frequency domain at the indices. Our

security analysis and simulation demonstrate that our proposed schemes can

achieve excellent transparency and robustness under the major security attacks

and common signal degradations. The novel approaches provided in this paper

are ideal for general purpose commercial digital media copyright protection.

Keywords Data hiding ·Information embedding ·Watermarking ·

Cryptography

Mathematics Subject Classiﬁcation (2000) 68W99

1 Introduction

Advances in computer networking and high speed computer processors have

made duplication and distribution of multimedia data easy and virtually cost-

Communicated by Lixin Shen and Yuesheng Xu.

J. Ren (

)

Department of Electrical and Computer Engineering,

Michigan State University, East Lansing, MI 48824, USA

e-mail: renjian@egr.msu.edu

268 J. Ren

less, and have also made copyright protection of digital media an ever urgent

challenge. As an effective way for copyright protection, digital watermarking,

a process which embeds (hides) a watermark signal in the host signal to be

protected, has attracted more and more research attention [1–8]. The host

signals are usually images, video or audio signals that are distributed in digital

format. The watermark is desired to reside in the host signal permanently to

provide copyright protection, license restriction, owner identiﬁcation, content

authentication, and traitor tracing [1].

The framework of watermarking can be depicted as in Fig. 1 [9, 10], in which

a watermark signal w is embedded in the host signal h. The embedding is per-

formed using a cryptographic key or the side information about the host signal.

The resulted watermarked data may be subjected to both channel distortion

and security attacks that attempt to remove or destroy the embedded water-

mark. Watermark extraction is the inverse process of watermark embedding.

It can be performed either with or without access to the original signal, the

later case is called blind watermark extraction [1, 11] and is especially useful

when the host signal is not available. Compare to its non-blind counterpart,

blind watermark extraction is generally more challenging and often has limited

extraction capability, hence limits the amount of embedded data.

The digital watermark embedding system needs to satisfy the following

three requirements:

1. Perceptual transparency or ﬁdelity The embedded signal should be “hid-

den” within the host signal, and causes no serious degradation to its host.

2. Robustness The embedded signal should be robust to common degra-

dations of the host signal, including benign processing and transmission

degradation as well as malicious attacks.

3. Security The watermarking procedure should be performed in a way such

that an unauthorized user is unable to detect the presence of the embedded

signal, let alone remove the embedded data.

Existing watermark embedding algorithms can be roughly split into two

different classes: time domain (also know as spatial domain) watermarking and

frequency domain watermarking. In time domain approaches, the watermark

information is embedded into the individual pixels selected under certain

criteria. It only results in alterations to individual pixels. The “localized”

alteration to the individual pixels tend to create highlighted spots that are

Embedding

Public Channel

Extraction

Host signal:

Watermark: w

Noise

Blind and non-blind

For non-blind only

Attack

Side informtion: k

Recovered

watermark:

Fig. 1 Block diagram of watermark embedding and extraction processes

A cryptographic watermarking technique for multimedia signals 269

potentially detectable. While in frequency domain approaches, the watermark

signal is embedded into the spectrum of the host image generated through

the Fourier transform. The embedded watermark is therefore distributed to

the entire host image following the inverse Fourier transform. As a result, the

embedded watermark generally has a high transparency and is hard to be

perceptually detected as long as the embedded watermark power is small

relative to that of the host image.

Most of the existing watermarking schemes are based on signal processing

techniques [1, 5, 7]. In [2], a watermarking scheme based on the RSA public-

key scheme [12] was introduced for time domain. However, the scheme is

very inefﬁcient and cannot be applied to frequency domain straightforwardly.

In an effort to increase the ﬁdelity, robustness and security of the digital

watermarking, in this paper we propose novel watermarking schemes based

on the discrete logarithm problem (DLP) and the Fermat’s little theorem

for general purpose digital rights management. The watermark embedding

includes two basic steps: the watermark index generation and the watermark

embedding. Our proposed approach can be applied to both time domain

and frequency domain with high transparency and robustness under common

signal degradations such as Gaussian noise and random multiplicative noise.

The proposed watermark techniques are also secure under major watermark

attacks such as cropping and least signiﬁcant bit (LSB) destruction.

The rest of this paper is organized as follows. In Section 2, crypto-

graphic watermark index generation based on discrete logarithm is introduced.

Section 3 and Section 4 are devoted to time domain and frequency domain dig-

ital image watermarking approaches, respectively. Security analysis is provided

in Section 5. Simulation results are presented in Section 6, and we conclude in

Section 8.

2 Preliminary

In this section, we will introduce some preliminary for this paper.

One-way function One-way function plays an important role in modern cryp-

tography. Mathematically, a one-way function can be characterized as a rela-

tion f : X → Y that is easy/efﬁcient to calculate, but very difﬁcult/inefﬁcient

or even impossible to inverse. More speciﬁcally,

1. For any given x ∈ X, y = f(x) can be efﬁciently computed (or computed

in polynomial time).

2. Given y = f(x) for some x ∈ X, it is computationally difﬁcult, or infeasible

to ﬁnd x.

Discrete Logarithm Problem (DLP) Let p be a prime number, and α a

primitive element of

(i.e., a generator of Z

∗

), where Z

is the integer

ﬁeld modulo p. The function f

α,p

: Z

∗

→ Z

∗

deﬁned by f

α,p

(x) = α

mod p

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