146
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 2, FEBRUARY 2010
Simultaneous Arithmetic Coding and Encryption
Using Chaotic Maps
Kwok-Wo Wong, Senior Member, IEEE, Qiuzhen Lin, and Jianyong Chen
Abstract—Based on the observation that iterating a skew tent
map reversely is equivalent to arithmetic coding, a simultaneous
compression and encryption scheme is proposed in which the
chaotic map model for arithmetic coding is determined by a secret
key and keeps changing. Moreover, the compressed sequence is
masked by a pseudorandom keystream generated by another
chaotic map. This two-level protection enhances its security level,
which results in high key and plaintext sensitivities. The compression performance of our scheme is comparable with arithmetic
coding and approaches Shannon’s entropy limit.
Index Terms—Arithmetic coding, chaotic map, simultaneous
compression and encryption.
I. I NTRODUCTION
T
RADITIONALLY, source coding and encryption are performed one after another to reduce the data volume while
maintaining information secrecy. A typical example is the
compression of a private photo using JPEG format and then
the encryption of the compressed file using the Advanced
Encryption Standard. However, there is an increasing interest
in simultaneous compression and encryption [1]–[7]. This can
be achieved by either embedding compression into encryption algorithms or adding cryptographic sense in compression
schemes. An attempt using the first approach was reported
in [1]. The allocation of plaintext symbols in the dynamic
lookup table used in a chaos-based cryptographic scheme is
determined by the plaintext statistics. The resultant ciphertext
is shorter than the plaintext, and so compression is achieved
while performing encryption.
There were more reports based on the second approach,
particularly the introduction of cryptographic sense in entropy
coding methods. However, most of these schemes are found
insecure or inefficient. The multiple Huffman table approach
[4] was cryptanalyzed in [8] and [9]. The arithmetic coding
method using key-based interval splitting [5] suffers from
known-plaintext attack [9]. The secure arithmetic coding [6]
was broken in [10]. Randomized arithmetic coding [7] is not
cryptanalyzed but is considered inefficient when compared with
the traditional compress-then-encrypt approach [9].
In recent years, some approaches [11]–[13] utilizing chaotic
systems for source coding, particularly arithmetic coding, have
been suggested. However, they did not investigate the relationship between these two areas, but just employed chaotic
systems as pseudorandom bitstream generators. Moreover, a
design fault of the chaos-based adaptive arithmetic coding
scheme [11] has been recently found, and a modified approach
was suggested [14]. Unfortunately, the modified version is also
vulnerable to chosen-plaintext attack [14]. The relationship
between arithmetic coding and chaotic maps was studied in
[15] and [16]. In [15], arithmetic coding is found equivalent
to finding the best initial condition for iterating a chaotic map
to generate a symbolic sequence corresponding to the source
message. In [16], source coding and chaotic systems are related
to each other by treating messages emitted by independent and
identically distributed sources as symbol sequences of a chaotic
nonlinear dynamical system known as the generalized Luroth
series (GLS). It is proven that GLS achieves Shannon’s entropy
bound and is a generalization of arithmetic coding [16].
Inspired by [15] and [16], here, we propose a scheme for
the simultaneous compression and encryption of message sequences with multiple symbols. Compression is achieved by
iterating a multisegment piecewise linear chaotic map, whereas
encryption is realized by changing the chaotic map model
continuously using a secret key, without affecting the essence
of arithmetic coding. The use of a chaotic map for this purpose
is better than existing schemes based on traditional arithmetic
coding [5]–[7]. This is because the position and the direction
of the linear segments in the map are governed by the secret
key. To further enhance the security, the compressed sequence
is masked by a pseudorandom keystream generated by another
chaotic map, as suggested in [10].
The rest of this brief is organized as follows. In Section II,
the concept of arithmetic coding using a skew tent map is
illustrated. The proposed simultaneous arithmetic coding and
encryption scheme is described in Section III. The performance
of our scheme is reported in Section IV, whereas conclusions
are drawn in Section V.
II. A RITHMETIC C ODING U SING S KEW T ENT M AP
Manuscript received September 16, 2009. First published February 8, 2010;
current version published February 26, 2010. This work was supported by a
grant from Research Grants Council of the Hong Kong Special Administrative
Region, China (Project CityU 122308). This paper was recommended by
Associate Editor Y. Horio.
K.-W. Wong is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong (e-mail: itkwwong@
cityu.edu.hk).
Q. Lin and J. Chen are with the Department of Computer Science and
Technology, Shenzhen University, Shenzhen 518060, China.
Digital Object Identifier 10.1109/TCSII.2010.2040315
Here, the concept of arithmetic coding using a skew tent map
is illustrated. The map is defined as [17]
x/p,
0≤x<p
f (x) =
(1)
(1 − x)/(1 − p), p ≤ x ≤ 1
where p is the point of partition.
Suppose that there is a binary message sequence M =
1001000101. The probability of occurrence of symbol 0 is 0.6,
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WONG et al.: SIMULTANEOUS ARITHMETIC CODING AND ENCRYPTION USING CHAOTIC MAPS
147
a piecewise linear chaotic map, as defined by (3), is constructed
[15], i.e.,
⎧
x ∈ I1
x/P1 ,
⎪
⎪
⎪ (x − P1 )/P2 ,
x ∈ I2
⎨
(3)
f (x) = .. .
⎪
n−1
⎪
⎪
⎩ x−
P
P , x∈I
i
n
n
i=1
where intervals Ii (i = 1, 2, . . . , n) are determined by
Fig. 1.
Skew tent map.
while that of symbol 1 is 0.4. By setting the skew tent map
parameter as p = 0.6, the interval [0, 1] is split into two partitions: [0,0.6) represents symbol 0, while [0.6, 1] corresponds to
symbol 1, as plotted in Fig. 1.
The compression is performed using the reverse interval
mapping method [15], [16]. The reverse function of the skew
tent map is given by
f
−1
(I) =
0.6 ∗ I,
symbol = “0”
1 − 0.4 ∗ I, symbol = “1”
(2)
where I is the interval. The initial value of I is [0, 1]. Start from
the last symbol “1”; the mapping is from the interval [0, 1]
to [0.6, 1]. The second last symbol is “0,” and so the interval
[0.6, 1] maps to [0.36, 0.6]. The third last symbol is “1,” and
the mapping is from the interval [0.36, 0.6] to [0.76, 0.856].
The remaining symbols are encoded in the same way until the
first symbol is encountered. The final interval is obtained, in
which the initial value for iterating the skew tent map lies. We
can choose any real value within this final interval and represent
it in binary format as the compressed sequence. To perform
decoding, we just need to iterate the skew tent map from
the initial value and determine which partition each iteration
output falls in. Then, the original binary sequence M can be
reconstructed correctly. This method is proven to be Shannon
optimal [15], [16].
III. P ROPOSED A PPROACH
A block diagram of our approach is shown in Fig. 2. Arithmetic coding and encryption are performed simultaneously by
a piecewise linear chaotic map governed by the statistics of
the plaintext. To enhance the security level, the compressed
sequence is masked by a pseudorandom keystream generated
by an integer skew tent map whose parameter depends on both
the key and the plaintext. Details of the proposed approach are
described as follows.
A. Simultaneous Arithmetic Coding and Encryption Using a
Piecewise Linear Chaotic Map
The plaintext sequence is partitioned into a number of
multiple-bit symbols. Assume that there are n distinct symbols
S1 , S2 , . . . , Sn with probabilities
of occurrence P1 , P2 , . . . , Pn ,
respectively, where ni=1 Pi = 1. Based on this information,
I1 = [0, P1 ]
⎤
⎡
i−1
i
Ii = ⎣
Pj ,
Pj ⎦ ,
j=1
(4)
i = 2, 3, . . . , n.
(5)
j=1
The intervals Ii (i = 1, 2, . . . , n) associate with the plaintext symbols Si (i = 1, 2, . . . , n), respectively, as shown in
Fig. 3(a). This is the public mode of the piecewise linear
chaotic map.
To change the piecewise linear chaotic map without affecting
its arithmetic coding ability, we can use a secret key KC ∈
{0, 1, . . . , n − 1} to cyclic-shift the position of the line segments and another secret key KS ∈ {0, 1} to determine their
directions. An example is given in Fig. 3(b). The cyclic shift
key is chosen as KC = i, where i ∈ {1, . . . , n − 1}, while the
slope key is KS = 1.
B. Pseudorandom Keystream Generation
In our approach, an integer skew tent map [18] defined by
(6) is adopted to generate the pseudorandom keystream for
masking the compressed sequence
⎧ n
⎨ g xn ,
if 0 < 2Lx+1
<p
pL
xn+1 = f (xn ) =
(6)
⎩ g 2 +1−xn , otherwise
1−p
where p ∈ [0, 1] and xn ∈ {1, 2, . . . , 2L }. The initial value is
x0 , whereas the approximation function g(·) can be a floor, a
ceiling, or a round function. The form of this map is similar
to that shown in Fig. 1, but the range now extends to [0, 2L ].
In [18], it was pointed out that the keystream generated by
the integer skew tent map under finite precision does not
possess sufficient randomness. To overcome this problem, we
only extract a few last bits of xn and limit the value of p to
p ∈ [0.25, 0.75]. The randomness of the generated keystream is
reported in Section IV-A.
C. Key Management
The key length of our cipher is 544 bits, represented by
k1 k2 , . . . , k544 . The first 512 bits, i.e., key1 = k1 k2 , . . . , k512 ,
are used to select the mode of the piecewise linear chaotic
map. The last 32 bits, i.e., key2 = k513 k514 , . . . , k544 , are taken
as the initial value of the integer skew tent map for generating
the pseudorandom keystream. Suppose that there are n distinct
plaintext symbols. The number of key bits used to determine the
mode of the piecewise linear chaotic map is m = log2 (n)+ 1.
The number of symbols to be encrypted by the secret mode of
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148
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 2, FEBRUARY 2010
Fig. 2. Model of the proposed approach.
Fig. 3. Piecewise linear chaotic map in (a) public mode KC = 0, KS = 0.
(b) Secret mode with keys KC = i, KS = 1.
the piecewise linear chaotic map is l = 512/m. The remaining symbols in the same plaintext block are compressed using
the public mode. The cyclic shift keys used for the l symbols
are KC1 , KC2 , . . . , KCl , respectively, whereas the slope keys
used for these l symbols are KS1 , KS2 , . . . , KSl , respectively,
where KCi = k(i−1)∗m+1 k(i−1)∗m+2 , . . . , ki∗m−2 and KSi =
ki∗m−1 (i = 1, 2, . . . , l). To make every bit in key1 of the same
importance, there are two rounds of key transformation before
encryption. The first round of key transformation is performed
as follows:
KCi+1 = (KCi+1 + KCi ) mod n,
i = 1, 2, . . . , l − 1 (7)
KSi+1 = (KSi+1 + KSi ) mod 2,
i = 1, 2, . . . , l − 1. (8)
The second round of key transformation is given by
KC1 = (KC1 + KCl ) mod n
(9)
KS1 = (KS1 + KSl ) mod 2
(10)
KCi+1 = (KCi+1 + KCi ) mod n, i = 1, 2, . . . , l−1 (11)
KSi+1 = (KSi+1 + KSi ) mod 2, i = 1, 2, . . . , l−1.
(12)
D. Encoding Procedures
The encoding procedures are described as follows.
1) Read the plaintext sequence in bytes and divide it into
W blocks. Each of the first (W − 1) blocks contains 128
symbols, while the last block possesses K (128 ≤ K <
256) symbols. This ensures that the minimum length of
each block is 1024 bits, which is sufficiently large to resist
brute force search attack. Moreover, the long last block
prevents attacks targeting at short plaintext blocks. An
“End” character is added to the end of the plaintext, which
is then scanned to find out the number of distinct symbols
and the corresponding probabilities of occurrence.
2) Based on the plaintext statistics, construct the piecewise
linear chaotic map using (3)–(5). Then, start to process
the first plaintext block. The 128 symbols in this block
are encoded using reverse interval mapping [15], [16],
starting from the last symbol. The last (128 − l) symbols
are compressed using the public mode of the piecewise
linear chaotic map plotted in Fig. 3(a). The first l symbols are compressed and encrypted with the secret mode
shown in Fig. 3(b). The keys to select the mode for the
ith symbol are KCi and KSi , i = 1, 2, . . . , l. After all
128 symbols are processed, a final interval [begin, end]
is obtained, and a real value 0.t1 t2 , . . . , tk is chosen
within this interval. The bits t1 t2 , . . . , tk form the k-bit
compressed sequence, which should be sufficiently long
to resist brute force search attack. If k is smaller than
128, more bits than necessary will be extracted to make
the compressed sequence at least 128 bits. The sequence
is then masked by a pseudorandom keystream generated
by the integer skew tent map. The initial value x0 is
determined by key2, while the parameter p is determined
by the cyclic shift key KC1 for the first plaintext symbol
of this block. Suppose that the value of this cyclic shift
key is j. Then, the intervals Ij+1 and Ij+2 are used to
calculate the value of q using the following formula:
q=
Ij+1
.
Ij+1 + Ij+2
(13)
To make p ∈ [0.25, 0.75], we set p = 0.25 + q ∗ 0.5.
Moreover, to ensure p = (xn /(2L + 1)), p is modified
to (2p(2L + 1) − 1)/(2(2L + 1)). If the resultant value
is 0.5, its value will be replaced by (2p(2L +1)+1)/
(2(2L +1)) [18]. To enhance the randomness, the integer skew tent map is first iterated 250 times from x0 .
This is performed only once for the whole plaintext
sequence. Then, the skew tent map is further iterated
k/16 times to obtain k/16 integers. The last 16
bits of each integer are extracted to form the pseudorandom keystream, as represented by r1 r2 , . . . , rk . The final
ciphertext sequence c1 c2 , . . . , ck is obtained by XORing
the compressed sequence and the keystream, i.e.,
ci = ti ⊕ ri , i = 1, 2, . . . , k.
3) The keys need to be updated before processing the next
plaintext block. The values of KC1 and KS1 are modified by some bits extracted from the current output of the
piecewise linear chaotic map, as given by
KC1 = KC1 ⊕ {t9 , t10 , . . . , t9+m−2 }
(14)
KS1 = KS1 ⊕ {t9+m−1 }.
(15)
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WONG et al.: SIMULTANEOUS ARITHMETIC CODING AND ENCRYPTION USING CHAOTIC MAPS
TABLE I
C OMPRESSION R ATIO
Then, a round of key transformation is performed by
KCi+1 = (KCi+1 + KCi ) mod n,
i = 1, 2, . . . , l−1 (16)
KSi+1 = (KSi+1 + KSi ) mod 2,
i = 1, 2, . . . , l−1. (17)
149
Moreover, the initial value of the integer skew tent map
is updated with x0 = xk/16+1 . Then, the next plaintext
block is encrypted in a similar way. The whole plaintext
sequence is compressed and encrypted by processing
each block sequentially until the last block.
For the decoder to correctly identify the ciphertext bits
of each block, a forbidden symbol composing of 1 zero
and 8 ones (011111111) is inserted between two consecutive ciphertext sequences. To distinguish this forbidden
symbol from ordinary ciphertext corresponding to seven
consecutive ones, we add a “0” after the seventh “1.”
E. Decoding Procedures
The decoding procedures are described as follows.
1) Read the header of the compressed file to find out the
distinct plaintext symbols and their corresponding probabilities. Then, construct the public mode of the piecewise
linear chaotic map. Scan the ciphertext sequence until
the first forbidden symbol (011111111) is encountered.
The ciphertext corresponding to the first block is then
obtained. Check this segment of ciphertext and remove
the next 0 if seven consecutive ones are encountered.
2) Based on the secret key, determine the values of KCi
and KSi (i = 1, 2, . . . , l) and start to decompress and
decrypt the ciphertext for the first block. Set the initial
value x0 as key2 and iterate the integer skew tent map for
250 times. Then, further iterate it for k/16 times to regenerate the pseudorandom keystream r1 r2 r3 , . . . , rk
and unmask the ciphertext by ti = ci ⊕ri (i = 1, 2, . . . , k).
Convert the binary sequence t1 t2 t3 , . . . , tk to a real value
0.t1 t2 t3 , . . . , tk , which is the initial value for iterating the
piecewise linear chaotic map. Select the secret mode of
the map according to KCi and KSi (i = 1, 2, . . . , l) to
decode the ith symbol. After l symbols are decrypted, the
public mode of the piecewise linear chaotic map is used
instead to decode the remaining (128 − l) symbols in this
block.
3) Update the values of KCi and KSi (i = 1, 2, . . . , l)
using (14)–(17), as performed in the encoder. The initial
value of the integer skew tent map is also updated as
x0 = xk/16+1 . With these new secret keys, continue to
decompress and decrypt the next block of ciphertext until
the “End” character is encountered in the last block.
IV. S IMULATION R ESULTS
The compression performance and the security of the proposed simultaneous arithmetic coding and encryption scheme
are evaluated using 18 standard test files from the Calgary
Corpus [19]. The 544-bit secret key is randomly generated. All
the simulations are executed in a personal computer with an
Intel dual-core 3.3-GHz processor and 8-GB memory. The test
results are reported as follows.
A. Randomness Test
The randomness of the keystream generated by the integer
skew tent map is evaluated using the statistic test suite designed
by the National Institute of Standards and Technology [20].
In particular, 300 sequences, each of length 1 000 000 bits, are
generated. If the successful percentage of any test is lower than
97.28%, the sequences are considered as not sufficiently random. All the keystreams generated using the 18 test files have
passed the tests as the percentage falls between 97.33% and
100%. This implies that they possess sufficient randomness.
B. Compression and Speed Performance
The compression ratio R of our algorithm is defined as
R=
ciphertext length
× 100%.
plaintext length
(18)
The ciphertext length includes the header information, which
contains the plaintext symbols (8 bits each) and their probabilities (16 bits each). The compression performance for the
18 test files can be found in Table I. For comparison, the best
ratio (BR) calculated based on the entropy and the ratio using
standard arithmetic coding (RAC ) are also listed. The results
show that our compression ratio is slightly higher than the
BR by 1.04%–4.69%. This is because forbidden symbols are
inserted to separate the ciphertext of consecutive blocks, and the
header information is included. However, the compression ratio
is very close to the BR when the plaintext file is sufficiently
large. This ratio can be further improved using a larger block
size, i.e., more than 128 symbols per block.
The compression time Tc and the decompression time Td
are listed in Table II. The compression speed lies between
1.2 and 3.4 MB/s. The decompression speed is lower than the
compression one, from 0.72 to 2.3 MB/s. This is mainly due
to the time-consuming division operation in the decompression
process. In fact, the compression and decompression efficiency
can be improved if the probability of occurrence of the symbols
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TABLE II
O PERATION S PEED
existing schemes based on traditional arithmetic coding
[5]–[7]. This is because both the position and the direction
of the line segments in the piecewise linear chaotic map
are controlled using a secret key. Simulation results reveal
that the compression performance is comparable to arithmetic
coding and approaches Shannon’s entropy limit. The security
of the scheme is supported by its high key and plaintext
sensitivities.
R EFERENCES
is quantized to 2−i (i = 1, 2, 3, . . . , 16). As a result, multiplication and division operations are performed simply by bit
shifting, and the computation time is greatly reduced. The
corresponding compression time Tc and decompression time
Td can be found in Table II. By using this fast approach,
the compression speed ranges from 1.6 to 5.7 MB/s, while
the decompression speed falls between 1.6 and 4.6 MB/s. The
tradeoff is that the compression ratio R is further increased
by 1%–3% in most cases, as listed in the rightmost column of
Table I.
C. Key Sensitivity and Plaintext Sensitivity
To test the sensitivity of key1, one bit is changed arbitrarily,
and the two resultant ciphertext sequences generated from the
same plaintext block are compared bit-by-bit. These operations are performed 100 times. The average percentage of bits
changed in the ciphertext sequences varies from 46.13% to
49.96% and is very close to the ideal value (50%). The results
indicate the high key sensitivity of the proposed scheme.
The plaintext sensitivity of our scheme is evaluated by randomly toggling one bit in each plaintext block and encrypting
using the same key. The two resultant ciphertexts are compared
bit-by-bit. The average percentage of bits changed in 100 runs
ranges from 49.27% to 50.11% for different test files. This
justifies the plaintext sensitivity of our scheme.
The sensitivity of key2 is supported by the randomness test
results found in Section IV-A. As the compressed sequence is
further masked by a keystream with sufficient randomness [10],
our scheme possesses two levels of protection and can resist
known attacks described in [8]–[10].
V. C ONCLUSION
A simultaneous arithmetic coding and encryption scheme
utilizing chaotic maps has been proposed. It is better than
[1] K. W. Wong and C. H. Yuen, “Embedding compression in chaos-based
cryptography,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 11,
pp. 1193–1197, Nov. 2008.
[2] X. Liu, P. Farrell, and C. Boyd, A Unified Code—IMA-Crypto & Coding’99, vol. 1746. Lect. Notes Comput. Sci. Berlin, Germany: SpringerVerlag, 1999, pp. 84–93.
[3] N. Nagaraj and P. G. Vaidya, “Applications of dynamical systems to joint
source cryptographic-channel coding,” in Nonlinear Dynamics, M. Daniel
and S. Rajasekar, Eds. New Delhi, India: Narosa Publishing House,
2009, pp. 393–396.
[4] C. Wu and C. Kuo, “Design of integrated multimedia compression and
encryption systems,” IEEE Trans. Multimedia, vol. 7, no. 5, pp. 828–839,
Oct. 2005.
[5] J. Wen, H. Kim, and J. Villasenor, “Binary arithmetic coding with
key-based interval splitting,” IEEE Signal Process. Lett., vol. 13, no. 2,
pp. 69–72, Feb. 2006.
[6] H. Kim, J. Wen, and J. Villasenor, “Secure arithmetic coding,” IEEE
Trans. Signal Process., vol. 55, no. 5, pp. 2263–2272, May 2007.
[7] M. Grangetto, E. Magli, and G. Olmo, “Multimedia selective encryption
by means of randomized arithmetic coding,” IEEE Trans. Multimedia,
vol. 8, no. 5, pp. 905–917, Oct. 2006.
[8] J. Zhou, Z. Liang, Y. Chen, and O. C. Au, “Security analysis of multimedia encryption schemes based on multiple Huffman table,” IEEE
Signal Process. Lett., vol. 14, no. 3, pp. 201–204, Mar. 2007.
[9] G. Jakimoski and K. Subbalakshmi, “Cryptanalysis of some multimedia
encryption schemes,” IEEE Trans. Multimedia, vol. 10, no. 3, pp. 330–
338, Apr. 2008.
[10] J. Zhou, O. C. Au, and P. H. Wong, “Adaptive chosen-ciphertext attack
on secure arithmetic coding,” IEEE Trans. Signal Process., vol. 57, no. 5,
pp. 1825–1838, May 2009.
[11] R. Bose and S. Pathak, “A novel compression and encryption scheme
using variable model arithmetic coding and coupled chaotic system,”
IEEE Trans. Circuits Syst I, Reg. Papers, vol. 53, no. 4, pp. 848–857,
Apr. 2006.
[12] B. Mi, X. Liao, and Y. Chen, “A novel chaotic encryption scheme based
on arithmetic coding,” Chaos Solitons Fractals, vol. 38, no. 5, pp. 1523–
1531, Dec. 2008.
[13] H. Li and J. Zhang, “A secure and efficient entropy coding based on arithmetic coding,” Commun. Nonlinear Sci. Numer. Simul., vol. 14, no. 12,
pp. 4304–4318, Dec. 2009.
[14] J. Zhou and O. C. Au, “Comments on ‘A novel compression and encryption scheme using variable model arithmetic coding and coupled chaotic
system’,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 55, no. 10,
pp. 3368–3369, Nov. 2008.
[15] M. B. Luca, A. Serbanescu, S. Azou, and G. Burel, “A new compression method using a chaotic symbolic approach,” in Proc. IEEE Commun. Conf., Bucharest, Romania, Jun. 3–5, 2004. [Online]. Available:
http://www.univ-brest.fr/lest/tst/publications/
[16] N. Nagaraj, P. G. Vaidya, and K. G. Bhat, “Arithmetic coding as a
non-linear dynamical system,” Commun. Nonlinear Sci. Numer. Simul.,
vol. 14, no. 4, pp. 1013–1020, Apr. 2009.
[17] G. Alvarez and S. Li, “Some basic cryptographic requirements for chaosbased cryptosystems,” Int. J. Bifur. Chaos, vol. 16, no. 8, pp. 2129–2151,
2006.
[18] H. S. Kwok and W. K. S. Tang, “A fast image encryption system based
on chaotic maps with finite precision representation,” Chaos Solitons
Fractals, vol. 32, no. 4, pp. 1518–1529, May 2007.
[19] [Online]. Available: ftp://ftp.cpsc.ucalgary.ca/pub/projects/text.compression.
corpus
[20] NIST Special Publication 800-22rev1. [Online]. Available:
http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html
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