27-11-2012, 11:55 AM
A Modified Chaos-Based Joint Compression and Encryption Scheme
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Abstract
An approach for improving the compression performance
of an existing chaos-based joint compression and encryption
scheme is proposed. The lookup table used for encryption is
dynamically updated in the searching process. Once a partition
not matched with the target symbol is visited, this and other partitions
mapped to the same symbol are reallocated to a nonvisited
symbol. Therefore, the target symbol eventually associates with
more partitions and fewer number of iterations are needed to find
it. As a result, expansion of the ciphertext is avoided, and the
compression ratio is improved. Simulation results show that the
proposed modification leads to a better compression performance,
whereas the execution efficiency is comparable. The security of the
modified scheme is also analyzed in detail.
INTRODUCTION
THE EFFICIENCY and security requirements of information
transmission lead to a substantial amount of research
work in data compression and encryption. In order to improve
the performance and the flexibility of multimedia applications,
it is worthwhile to perform compression and encryption in a
single process [1]–[5]. In general, there are two distinct research
directions in this area. One of them embeds key-controlled
confusion and diffusion in source-coding schemes, whereas
another incorporates compression in cryptographic algorithms.
Some attempts of introducing key-controlled operations in
entropy coding can be found in [1]–[4]. The approach based
on multiple Huffman tables [1] simultaneously performs encryption
and compression by a key-controlled swapping of the
left and right branches of the Huffman tree. Some approaches
such as randomized arithmetic coding [2] and key-based interval
splitting [3], [4] were proposed to embed cryptographic
features in arithmetic coding. The compression capability of
these algorithms is close to that of traditional entropy coding.
However, security and efficiency problems were found [6]–[8],
and further modifications are required.
PERFORMANCE OF THE MODIFIED SCHEME
Efficiency Analysis
Due to the hitting of nontarget symbols, the required number
of iterations in the original scheme [5] is large, and the
compression performance is thus limited. In order to avoid the
chaotic map trajectory falling into the partitions corresponding
to the irrelevant symbols again, those partitions are released in
the modified scheme by replacing the nontarget symbol visited
in the previous step by another symbol. This process is similar
to the model of sampling without replacement, and it follows
the hypergeometric distribution for successfully obtaining a
sample from a finite population at the kth time without replacement.
Equation (4) is an expression of the probability of
finding symbol A from the lookup table without replacement,
where N is the total number of partitions, andM is the number
of partitions corresponding to A.
Simulation Results
The proposed algorithm is implemented in C++ programming
language running on a personal computer with an Intel
Core 2 2.00-GHz processor and 2-GB memory. We follow the
choices in [5] so as to make a fair comparison with its results.
The logistic map is chosen as the underlying chaotic map. The
parameter b is set to 3.999999991, whereas the initial condition
is chosen as 0.3388. The maximum number of iterations is set
to 15.
To test the compression capability of the proposed scheme,
the standard files from the Calgary Corpus [10] are used. There
are 18 distinct files of different types, including text, executable
geophysical data, and picture. Two simulation configurations
are chosen.
SECURITY ANALYSES
Key Space, Key, and Plaintext Sensitivities
The key of the proposed scheme is composed of the control
parameter b and the initial value x0 of the logistic map,
together with the 32-bit initial cipher block. In the software
implementation, b and x0 are represented by double precision
format using 52 bits. The total key space can reach 136 bits
and is comparable to 128-bit AES. However, b should be
carefully chosen to avoid the nonchaotic regions with a negative
Lyapunov exponent [11], [12]. A plot of the Lyapunov exponent
against b is shown in Fig. 3. In the chaotic region with a positive
Lyapunov exponent, the period of the output sequence can be
considered as infinity. However, the actual period is limited by
the computation precision. A constructive scheme for finding
the maximum period of a chaotic map under limited precision
is described in [13]. It is employed to find the period of the
logistic map. The results show that all the periods in the chaotic
region are far longer than 107 in double precision computation,
which are sufficient for practical ciphers [13].
CONCLUSION
The existing approach of embedding compression in the
chaos-based cryptosystem suffers from the drawback of low
compression performance. This can be explained by the model
of sampling with replacement. A modified scheme based on the
model of sampling without replacement has been proposed. As
a result, the number of chaotic map iterations wasted for visiting
irrelevant symbols is reduced. The compression capability is
improved, which is close to that of conventional entropy coding.
Moreover, the lookup table can be realized by memory chips or
field programmable gate array, and therefore, the proposed joint
compression and encryption scheme is easy to be implemented
by hardware circuits.