Information Technology Reference
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long-term predictions of chaotic systems [2]. Examples of discrete, deterministic and
non-linear dynamic systems are the one-dimensional (1-D) chaotic maps. They offer
identically distributed binary sequences and they can be used as deterministic genera-
tors of unpredictable sequences. Therefore, 1-D chaotic maps can be used in crypto-
graphic, steganographic and digital marking systems. A 1-D chaotic map is specified
by the following iterated expression:
()
x
=
τ
x
,
n
=
0
2
"
(1)
n
+
1
μ
n
is a non linear function,
I
denotes the interval in which the function
is defined,
n
represents the actual iteration index of
Where
τ
μ
:
I
→Ι
τ
μ
(),
x
0
is the initial condition and
μ
is the control parameter of the function.
1-D chaotic maps generate orbits, which are defined by,
(){
}{}
∞
=
φ
x
=
x
,, , , ,
n
x
x
""
x
=
x
(2)
0
0
1
2
n
n
0
Note that each generated orbit depends on the initial condition and the control pa-
rameter
of the used map. In this way, it is possible that these orbits tend to one or
more fixed values, called map attractors. However, there are other values that are
repellers and will produce unstable orbits in the map. Control parameter
μ
μ
of the map
τ
μ
() defines that a point is an attractor or repeller.
Works related to the application of chaotic map on cryptographic systems can be
found in [3], [4], [5], [6] and [7]. Therefore, chaos theory is currently used in the
analysis and design of cryptosystems that provide technological solutions for
information protection.
2 Chaotic Block Cryptosystem
2.1 Description
The block cryptosystem presented here, is based on G. Jakimoski and L. Kocarev
proposal [3]. They suggest the use of logistic map in a balanced and dynamic cipher-
ing structure that processes 64 bits blocks, considering 8 sub-blocks of 8 bits each
one. The ciphering process consists of 8 rounds over the same data block
B
j
with
j
=
0, 1, 2, …, 7. The block of the plaintext is B
0
. The output block in a round is the input
block of the following round, except in the last one, in which the output block corre-
sponds to the ciphered block of the cryptogram. G. Jakimoski and L. Kocarev suggest
to use a scaled and discretized logistic map in the noise function of the cryptosystem,
according to the following equations:
(
)
x
=
x
⊕
f
x
,...,
x
,
z
;
k= 1, 2, …, 8
(3)
i
,
k
+
1
i
−
1
k
k
−
1
i
−
1
i
−
1
k
−
1
i
−
1
k
−
1
The values
x
i,0
, x
i,1
, …x
i,j
with
i = 1, 2, …, r
, and
j = 0, 1, 2, …, 7
represent each 8-
bits sub-block that conform the block
B
i
, which is the transformed version of
B
i-1
. In
this way,
B
r
will be the ciphered block of every block
B
0
from the plaintext. In the
other hand, chaotic functions
f
j
:[0, 255]
, receive and transform the sum
of one or more sub-blocks with its corresponding sub key, except
f
0
. The function
→
[0, 255]
∈
ℵ
⊕
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