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
Z
p
(i.e., a generator of Z
∗
p
), where Z
p
is the integer
field modulo p. The function f
α,p
: Z
∗
p
→ Z
∗
p
defined by f
α,p
(x) = α
x
mod p
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