Structural Properties of Cryptographic Sequences - Computational Intelligence in Security for Information Systems - page 194

( − μ ) 1

⎧

2 μ x + ; 0 ≤ x ≤

⎪
⎨

n n

()

2 2

f x =

(6)

μ n

( − μ ) 1

( )

⎪
⎩

− 2 μ x − 1 + ; ≤ x ≤ 1

n n

2 2

Parameter μ ∈ [0, 1] and the initial condition x 0 are known but arbitrary selected.

Thus, and x 0 could be considered a key in the encryption process. Figure 3 shows

the bifurcation diagram for a family of orbits {

μ

℘ ℵ } for a control parameter μ ∈

[0.799, 0.801].

Fig. 3. Zoom on the bifurcation diagram for the family of orbits {℘ ℵ } with m = 1, 2, 3, …,
when the tent map was not scaled and discretized and μ

∈ [0.799, 0.801]

2.5 Chaos Annulling

When the 1-D chaotic maps are used in a computer, there are problems that must be
considered. There is at least one initial condition for which chaos is not only sustained
but also collapsed; this is an interesting result, and it can be noticed, that at the pa-
rameter value for which this occurs (

= 4, x 0 = 0.5 for logistic map), generates a
condition called “ chaos annulling trap ” or CAT. CAT condition can be achieved as a
result of the rounding off done by the computer; due to it, IEEE recommends calcula-
tions in double precision. In similar way, there are other x 0 values that produce a simi-
lar condition, which is called “chaos fixed trap” or CFT. There are two methods to
determine the initial condition that cause either a CAT or a CFT condition for chaotic
map: Analytic and brute force method [9]. Additionally, when a 1-D chaotic map is
intended to be used as noise function in stream or block cryptosystems, a scaled and
discretized version of the chaotic map must be considered, which is not necessarily a
chaotic map, and therefore this condition must be analyzed.

μ