IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR...
Transcript of IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR...
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 1
Network analysis:::::::::::::::::Dynamic
:::::::::::::::::::Analysis
:of dynamics of
chaotic maps in digital domain:::::::::::::Digital
:::::::::::::::::Chaotic
::::::::::::Maps
::::::via
::::::::::::::::::::::::::::::State-Mapping
:::::::::::::::::::::Networks
Chengqing Li, Senior Member, IEEE, Bingbing Feng, Shujun Li, Senior Member, IEEE, Juergen::::::Jurgen
:Kurths,
Guanrong Chen, Life Fellow, IEEE
Abstract—Dynamics of::::::Chaotic
:::::::::dynamics
::is:::::::
widely::::used
:to:::::::
design:::::::::::::pseudo-random
:::::::number
::::::::::generators
::::and
:::for
:::::other
:::::::::applications
::::such
::as
::::::secure
:::::::::::::communications
:::and
::::::::::encryption.
:::This
:::::paper
:::::aims
::to
:::::study
:::the
::::::::dynamics
::of
:::::::::::discrete-time chaotic maps
in an infinite-precision domain have been established completemathematical framework.The case in
::the
::::::digital
:::::(i.e.,
:finite-
precisioncomputer remains to be further explored. The relatedprevious works treated )
:::::::domain.
::::::::Differing
::::from
::::the
::::::::traditional
:::::::::approaches
:::::::treating
:a digital chaotic map as a black box and
gave::::with
:different explanations according to the test results
of the output. Using the Logistic and Tent maps as typicalexamples, we disclose some ,
:::the
:dynamical properties of chaotic
maps in:::such
:::::::chaotic
::::maps
::::are
:::first
::::::::explored
::::with
:a:
fixed-pointarithmeticby studying its
:,:::::using
:::the
::::::Logistic
:::::map
:::and
:::the
::::Tent
:::map
:::as
::::two
::::::::::::representative
::::::::examples,
:::::from
::a::::new
:::::::::perspective
:::with
::::the
:corresponding state-mapping network (SMN), where
:::::::networks
::::::::(SMNs).
:::In
:::an
:::::SMN,
:every possible value
::in
:::the
:::::digital
::::::domain
:is considered as a node and the mapping relation
existing:::::::::relationship
:between any pair of nodes works as
::is
:a
directed edge. The scale-free properties of SMN are quantitativelyproven. The obtained results can be
::the
:::::::Logistic
:::::map’s
::::SMN
:::are
::::::proved.
:::The
:::::::analytic
::::::results
:::are
::::::further extended to the scenario
of floating-point arithmetic and to:::for
:other chaotic maps.
Understanding the real network structure of SMN of a chaoticmapin the digital computer
::’s
::::SMN
:::in
::::::digital
:::::::::computers
:can
facilitate counteracting dynamics degeneration of digital chaoticmaps, which also help evaluate
::the
::::::::::undesirable
:::::::::::degeneration
:of:::::::
chaotic::::::::
dynamics:::
in:::::::::::::
finite-precision::::::::domains,
::::::helping
::::also
::::::classify and improve the randomness of pseudo-random numbersequences generated by iterating chaotic maps.
Index Terms—Chaotic:::::Chaos,
::::::::chaotic
::map, complex
networks::::::network, dynamics degradation,
::::::::fixed-point
:::::::::arithmetic,
floating-point arithmeticdomain, PRNS,::::::::::::pseudo-random
::::::number
:::::::generator
::::::::(PRNG),
:randomness.
This work was supported by the National Natural Science Foundation ofChina (61772447, 61532020); DAAD/K.C. Wong Fellowship (91664078);Royal Society, UK (IE111186); and the Hong Kong Research Grants Councilunder the GRF Grant CityU 11234916.
C. Li is with School of Computer Science and ElectronicEngineering, Hunan University, Changsha 410082, Hunan, China([email protected]).
B. Feng is with College of Information Engineering, Xiangtan University,Xiangtan 411105, Hunan, China.
S. Li is with School of Computing & Kent Interdisciplinary Research Centrein Cyber Security (KirCCS), University of Kent, Canterbury, Kent, CT2 7NF,UK.
J. Kurths is with Potsdam Institute for Climate Impact Research, PotsdamD-14415, Germany.
G. Chen is with Department of Electronic Engineering, City University ofHong Kong, Hong Kong SAR, China.
I. INTRODUCTION
DYNAMICS of chaos in infinite mathematical world:::the
:::::::::continuous
::::(i.e.,
::::::::::::::::infinite-precision)
:::::::domain
:is a funda-
mental research topic in the field of::::fields
:::of
:::::::::::mathematical
chaos theory and even nonlinear sciences . Meanwhile,implementation of any
:::::::nonlinear
::::::::sciences
::::[1]
:.::::
Yet,::::
the
:::::::::::::implementation
::of
::a:
chaotic system in a digital device is
::::::always
:an inevitable problem for its real application [2]
::::::::::withholding
:::its
::::real
:::::::::::applications
:::::::[3], [4]. Under the joint
influence of round-off error and truncation error (algorithmicerror) in the
::::errors
::::and
::::::::::truncation
:::::errors
:::::(i.e.,
::::::::::algorithmic
:::::errors)
:::in
::a
:finite-precision domain (or
:::e.g.
::in
:a finite-state
machine), the obtained digital orbit is biased:a::::::::resultant
:::::digital
::::orbit
::::will
:::be
::::::::::off-tracking
:from the theoretical one [5], [6].
Based on the famous::::::::::well-known
:shadowing lemma, some
researchers::::many
:believed that any pseudo-random number
sequence generated by iterating chaotic map remains:a::::::chaotic
:::map
:::::::retains
:::the
:complex dynamics of the original chaos to
some extents [7], [8]. Meanwhile, some work pointed out thatdynamics of digital chaotic maps
::::::chaotic
::::map
::to
::a::::high
:::::extent
::[7]
:.::::::::However,
:::it
::::was
:::::found
::::that
::::the
:::::::::dynamics
::of
::a::::::
digital
::::::chaotic
::::map
:are definitely degraded to various degrees
::::some
:::::degree
:[9]. In 1988, J. Yorke et al. investigated
::::Yorke
:::et
::al.
::::::::::investigated
::the
:period distribution of an orbit of Ikeda map
::the
::::Ikeda
:::::map,
:starting from a specific initial point under various
roundoff precisions::::::::round-off
:::::::::precisions,
:and found that the
expected number of period orbits is scale:::::::periodic
:::::orbits
::is
:::::scaled to the precision [10]. In [11], a series of objective metric
::set
:::of
::::::::objective
::::::metrics
:were proposed to measure the degree
of dynamics degradation of piecewise-linear chaotic maps. In1991, Prof. Chua proposed
::::Chua
:::::[12]
::::::::suggested
:that a real
digital filter can exhibit a near-chaotic behavior:::::::behaviors
:if
its wordlength is sufficiently large (e.g.:, larger than 16 bits)[12]
. In [3], Kocarev defined digital chaos via convergence of itsdiscrete Lyapunov exponent to a positive number when the sizeof discrete space approach infinity. The controversial issueson relationship between digital chaos and the correspondingoriginal chaos received intensive attentions.
:.:
Intuitively, the seemingly complex dynamics of chaosare very appealing for random number generation[13], [14], [15], [16] and random permutation [17]. In1947, J. von Neumann suggested using Logistic map asa pseudo-random number generator (PRNG) [18]. Sine
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 2
then, a large number of PRNG have been proposed basedon various chaotic maps and their variants: Logisticmap [19], [7], [20], [21], [16]; Tent map [22], [17], [15];Sawtooth map [23]; Renyi chaotic map [24]; Cat map [25]. In addition, chaos was widely used to design hash andencryption schemes. As reviewed in [26], any dynamicsdegradation of digital chaos may facilitate thwarting securityof the supported schemes.
Due to the “pigeonhole principle” and:”
::::and
::a
:limited
possible number of:a:digital state, the orbit of
::::::::generated
::by
iterating a chaotic map from any initial state in digital domaindefinitely enters
:::the
::::::digital
::::::domain
::::will
::::::::definitely
:::::enter a peri-
odic loop after a::the
:transient process, which can be attributed
:::::::referring
:to the general cycle detection problem of periodic
functionsdiscussed in [27]. Some researchers ignored existenceof
::::about
::::::::periodic
::::::::functions,
:::as
:::::::::discussed
::in
:::::::::[27], [28].
::::The
:::::::existence
:::of
:a network relationship among all possible states of
a given chaotic map in a digital domain [29]. They just focused
::the
::::::digital
:::::::domain
:::was
:::::often
::::::ignored
::::[29]
:,::::::instead
::::::having
::::focus
on statistics along an:::the orbit (path) of the actually existing
::on
:::the
:network. When the trajectories of
:::::orbits
::of
:::the
:Logistic
map are computed in 64-bit floating-point precision, reference[5] statistically analyzed how their
::by
::::::::statistical
:::::::analysis
::it
:::was
:::::shown
::in
::::[5]
:::how
:::the
:errors change with respect to
::the
:control
parameters. In [25], period distribution of the generalizeddiscrete Arnold cat map is exactly
::::was
::::::::precisely
:derived.
Furthermore, Liao explicitly presented the maximum period ofthe sequences generated by iterating
::the
:Logistic map over
::the
field Z3n :::was
:::::::derived
::in [30]. In [31], [32]
:::[32], the influences
of different rounding methods on the transient length and cyclelength
::::::lengths
::::and
::::cycle
:::::::lengths were analyzed experimentally.
The influences caused by control parameters were furtheranalyzed in [33]. In [9], period, cycle of Logistic map is
::the
::::::periods
:::and
::::::cycles
::of
:::the
:::::::Logistic
::::map
:::are
:exhaustively calcu-
lated in 32-bit floating point precision using High-ThroughputComputing
:::::::::::::high-throughput
:::::::::computing.
Complex network has become a popular tool to studydynamics of chaotic times series since the pioneering workon it was proposed in [34]. Essentially, some
:::::Some
:earlier works on studying
:::the
:dynamics of digital
chaos by studying::via
::::::::studying
:::the
:state-mapping network
(SMN), composed of relationship::or
::::state
:::::::::transition
:::::::diagram,
::or
:::::::::functional
:::::::graph),
::::::::::composing
:::of
:::::::::::relationships
::between
every pair of states, can be attributed to:::fall
::::into
:the s-
cope of complex network:::::::network
:::::::science
::::::studies. In 1986,
Binder::::[35] drew the state network of
::the
:Logistic map in
::the
:5-bit fixed-point arithmetic domain and reported that the
counterparts of some metrics on::for
:measuring dynamics in
continuous chaos, such as Lyapunov exponent and entropy,can still work as the latter [35]
::::work
::::just
:::as
::::well. In [36],
Binder::he
:further experimentally studied how the number of
limit cycles and the size of::the
:longest cycle change with
the fixed-point precision. Kurths et al. proposed::::Later,
::an
analytical framework:::was
::::::::proposed
:for recurrence network
analysis of chaotic time series [37], [38], [39]::::::::[37], [38]. In
[40], dynamics of series generated by::the
::::::::::complexity
:::of
:::1-D
cellular automata was classified by two parameters of theassociate network.
::its
::::::::::::state-mapping
::::::::network.
:::::::::However,
:::the
::::::validity
::of
:::::such
:::::::::::classification
:::::::method
:::was
::::::::::questioned
::in
::::[41].
In [42], an orbit of SMN is:a
::::::::::::state-mapping
:::::::network
::::::(SMN)
:::was
:transformed into a network via horizontal visibility. It
was found that the network entropy can mimic:::the Lyapunov
exponent of the original map in the:a:
subtle level. In [43],[44], [45], [46],
:a mapping network among sub-intervals was
established to explore::the
:coherence between network pa-
rameters and the::::some
:well-recognized metrics characterizing
chaotic dynamics. In [47], the relative frequency of different4-nodes
:::::4-node
:subgraphs of SMN of conventional
::::some
chaotic maps and flows can discriminate and identify:::was
::::used
::to
::::::::::discriminate
:the underlying chaotic systems.
::In
:::::::::retrospect,
:::the
::::::::::seemingly
::::::::complex
::::::::dynamics
:::of
:::::chaos
:::has
:::::been
:::::very
:::::::::appealing
::::for
:::::::random
::::::::number
:::::::::generation
:::::::::::::[13], [15], [16]
:::and
:::::::random
::::::::::permutation
::::[17].
::In
::::fact,
::in:::::1947,
:::von
::::::::Neumann
:::::::already
:::::::::suggested
:::::using
:::the
:::::::Logistic
::::map
:::as
:a
::::::::::::pseudo-random
:::::::number
:::::::::generator
:::::::(PRNG)
:::::[18].
::::::Since
::::then,
:a:::::
large:::::::
number:::
of:::::::
PRNG:::::
have:::::
been::::::::
proposed::::::
based:::
on
::::::various
::::::chaotic
:::::maps
::::and
::::their
:::::::variants,
::::e.g.,
:::the
:::::::Logistic
::::map
::::::::::::::::::::[19], [7], [20], [21], [16]
:,:::the
:::::Tent
::::map
::::::::::::::[22], [17], [15],
:::the
::::::::Sawtooth
::::map
::::::::[23], [48]
:,:::the
::Re
:::nyi
::::::chaotic
::::map
::::[24]
:,:::and
:::the
:::Cat
::::map
::::[25]
:.::In
::::::::addition,
:::::chaos
::::::theory
::::was
:::::::widely
::::used
::to
:::::design
:::::hash
::::::::functions
:::and
:::::::::encryption
::::::::schemes.
:::::::::However,
:it::
is
:::::::::impossible
:::for
:::any
::::::chaotic
::::map
::to:::::
reach:::the
:::::ideal
::::::chaotic
::::state
::in
:a:::::::::::::finite-precision
::::::digital
:::::::domain.
:::As
::::::::reviewed
::in
::::::::[26], [49]
:,:::any
:::::::::dynamics
:::::::::::degradation
::of
::::::digital
::::::chaos
::::may
::::::::facilitate
::::::::thwarting
:::::::security
::of
:::the
:::::::::supporting
::::::::::encryption
::::::::schemes.
To counteract dynamics degradation, all kinds of methodswere proposed, such as
::::many
::::::::methods
::::have
::::been
:::::::::proposed,
::for
:::::::example adopting higher precision [12], perturbing chaotic s-tates [50], [19], [20], [21], [51]
:::::::::::::::::[50], [19], [20], [21], perturb-
ing control parameter [52], [51]::::::::parameters
::::[52], and cascading
two and multiple chaotic maps [53], switching multiple chaoticmaps [54], [55], and feedback control [56], [16]. Almost allthe
::::Most
:::of
:::::these
:works claimed that the improved discrete
chaotic maps can work as good alternatives to classic PRNGbythe results passing by some randomness test suite. It needs auniform simple tool to compare randomness performances ofthe PRNG based on chaotic maps .
Most researchers in the field of nonlinear sciences donot care about how their computers perform floating-pointarithmetic, and just treat the chaotic system implementedin computer as a black box. The overall structure of thechaotic map implemented in computer remains mysterious.Some
::of
:::the
::::::classic
::::::PRNG,
:::by
:::::::showing
::::that
::::their
::::::results
::::pass
:::::typical
:::::::::::randomness
::::test
::::::suites.
::::The
::::real
:::::::::structures
:::of
::::such
::::::chaotic
:::::maps
::::::::::implemented
:::in
::the
:::::::::computer
::::::remain
:::::::::mysterious,
::in
:::::which
:::::some
:important details occurring with a very low
probability were omitted by limited times:a::::::limited
:::::::number
of random experiments. This paper studied::To
::::::::improve
::::such
:a
:::::::situation,
::::this
:::::paper
::::::studies
:the properties of SMN composed
by iterating any::::::::generated
:::by
:::::::iterating
::a:chaotic map in
::the
digital domain: every representable value in the definitiondomain of
::::::domain
:::of
:::the
:chaotic map is considered as a
node; the direct edge between any ,:::::while
::a::::::::
directed::::edge
:::::::between
::a pair of nodes is built if and only if the former
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 3
node is mapped to the latter one by the::::::chaotic
:map. Using
the Logistic and Tent maps as typical examples, we disclosemyth of some
:::map
::::and
:::the
::::Tent
::::map
:::as
:::::::::illustrative
::::::::examples,
::the
:dynamical properties of chaotic maps in
:::the
:fixed-point
arithmetic domain by studying its:::are
::::::::disclosed
:::by
:::::::studying
::::their corresponding SMN. The scale-free properties of SMNare quantitatively proven
:::::::::::::mathematically
::::::proved. The relation-
ship between::an
:SMN obtained in
:a:floating-point arithmetic
domain and that in:a:fixed-point domain is clarified
::::::::arithmetic
::::::domain
::is
:::::::revealed. Finally, we demonstrate
:it::::will
:::be
:::::shown
that SMN can work as a fingerprint:::::::::fingerprints of chaos-based
pseudo-random number generator::::::PRNG to coarsely evaluate
their randomness.The remainder of this
::the
:paper is organized as follows.
Section II gives network analyses of SMN of Logistic andTent maps in
::::::::performs
:::::::network
:::::::analysis
:::on
:::the
::::::SMNs
::of
:::the
:::::::Logistic
::::map
:::and
::::the
::::Tent
::::map
:::in
:::the
:fixed-point arithmetic
domain. Section III presents the analysis of SMN::an
:::::::analysis
::of
:::the
::::::SMNs of the two maps in
::the
:floating-point arithmetic
domain. Application of SMN in evaluation of pseudo-randomnumber generator
::An
::::::::::application
::of
:::::SMN
::to
:::the
:::::::::evaluation
::of
:::::PRNG
:is discussed in Sec. IV. The last section concludes the
paper:::::::::::investigation.
II. STATE-MAPPING NETWORK OF DIGITAL CHAOTIC MAPSIN
::::THE FIXED-POINT ARITHMETIC DOMAIN
A. Some properties of maps implemented in fixed-pointarithmetic precision
::::First,
:::::the
:::::::::::::state-mapping
:::::::::network
:::of
::::::::chaotic
::::::maps
::::::::::implemented
::::in
::::the
::::::::::fixed-point
::::::::::arithmetic
:::::::::precision
:::is
::::::defined
::::and
:::its
:::::::general
::::::::::properties
:::are
::::::::proved.
::::::Then,
:::the
::::::special
::::::::properties
:::of
:::::SMN
::of
::::the
:::::::Logistic
::::map
::::and
:::the
::::Tent
:::map
::::are
::::::::analyzed.
:
A.::::Basic
::::::::::properties
::of::::::::
chaotic:::::maps
::::::::::::implemented
:::in
:::the
:::::::::fixed-point
::::::::arithmetic
:::::::::precision
Given a map f : [0, 1] → [0, 1] and a computing domainof fixed-point arithmetic precision nwith
:,::::with
::a:
specifiedquantization scheme, its
::::::domain
:::and
::::::range
:::are
::::both
::::::defined
::as
:a:::::::discrete
:::set
:::::::::::::::{x | x = i
2n }2n
i=0.::::The
:associate state-mapping
network Fn can be built by:is:::::
built::in:
the following way:the 2n
:::::2n + 1
:possible states are viewed as 2n
::::::2n + 1 nodes;
every pair of nodes with labels i and j is linked with a directededge if f(i/2n) = j/2nunder the given condition
:,::::::::calculated
::in
:::the
::::::::::quantization
::::::domain. Let fn(i) denote the original value
corresponding to the node with label i in Fn before the finalquantization step, namely,
:
Fn(i) = R (fn(i) · 2n) ,
where R(·) is an integer quantization function, e.g. floor,ceil, and round(In this paper, only .
:::As
:::::::::discussed
::in
::::[11]
:,:a
:::::::different
::::::::::quantization
::::::::function
::::only
::::has
:a:::::slight
::::::effect
::on
:::the
::::::::quantized
:::::value
:::and
::::does
:::not
:::::::::::significantly
:::::::influence
:::the
::::::overall
:::::::structure
::of
::::::SMN.
:::So,
:::::only
:round quantization is considered
.).:::::::::throughout
:::the
::::::paper.
:
With the above notations, one can obtain the followingthree properties on the relationship between
:::::::Property
::1:::::below
:::::::describes
:::the
:::::::::::relationship
:::::::between
:::the
:::i-th
:::::node
::in
:Fn and
::the
:::::(2i)-th
:::::node
::in
:Fn+1.
::::::::::Meanwhile,
::::::::Properties
:::2,
:3::::::::::characterize
:::that
:::::::between
::::the
::::::former
:::and
:::the
::::::::::(2i± 1)-th
:::::node
::in
:::::Fn+1.
:
Property 1. The node with label (2i)::::“2i” in Fn+1 and that
with label:“i
:” in Fn satisfy
Fn+1(2i)− 2Fn(i) =
1 if rn ∈ [0.25, 0.5);
−1 if rn ∈ [0.5, 0.75);
0 otherwise,(1)
wherern = frac(fn(i) · 2n),
frac(x) = x − bxc, i ∈ {0, · · · , 2n}, and bxc returns thegreatest
::::::largest integer less than or equal to x.
AsProof:
::::Since
:Fn+1(2i) = R (2 · fn+1(2i) · 2n) and
fn+1(2i) ≡ fn(i),
the proof of this property is straightforward from:::::using
R(2x) = 2 · R(x) +
0 if 0 ≤ frac(x) < 0.25;
1 if 0.25 ≤ frac(x) < 0.5;
−1 if 0.5 ≤ frac(x) < 0.75;
0 if 0.75 ≤ frac(x) ≤ 1.
Property 2. The node with label (2i+ 1):::::::“2i+ 1”
:in Fn+1
and that with label:“i
:”:in Fn satisfy
|Fn+1(2i+ 1)− 2 · Fn(i)|≤∣∣R ((fn+1(2i+ 1)− fn+1(2i)) · 2n+1
)∣∣+
{2 if rn ∈ [0.25, 0.75);
1 otherwise,(2)
and i ∈ {0, · · · , 2n − 1}.
Proof: According to the triangle::::::::triangular
:inequality, one
has
|Fn+1(2i+ 1)− 2 · Fn(i)| ≤|Fn+1(2i+ 1)− Fn+1(2i)|+ |Fn+1(2i)− 2 · Fn(i)|.(3)
Utilizing the property of::the
:integer quantization function
|R(x)− R(y)| ≤ |R(x− y)|+ 1, (4)
one can get::::::obtains
:
|Fn+1(2i+ 1)− Fn+1(2i)|=∣∣R (fn+1(2i+ 1) · 2n+1
)− R
(fn+1(2i) · 2n+1
)∣∣≤∣∣R (fn+1(2i+ 1) · 2n+1 − fn+1(2i) · 2n+1
)∣∣+ 1 (5)
Incorporating the above inequality and Eq. (1) into inequali-ty (3), the property can be proved.
:is:::::::proved.
:
Property 3. The node with label (2i− 1):::::::“2i− 1”
:in Fn+1
and that with label:“i
:”:in Fn satisfy
|Fn+1(2i− 1)− 2 · Fn(i)|≤∣∣R ((fn+1(2i− 1)− fn+1(2i)) · 2n+1
)∣∣+
{2 if rn ∈ [0.25, 0.75);
1 otherwise,(6)
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 4
and i ∈ {1, · · · , 2n}.
Proof: As the proof is very similar to that of Proper-ty 2and omittedhere. ,
::it
::is
:::::::omitted.
:
B. SMN of::the
:digital Logistic map
In the digital domain with fixed-point precision n, theLogistic map
f(x) = µ · x · (1− x) (7)
becomes
fn(i) = (Nµ/2nµ) · (i/2n) · (1− i/2n), (8)
where Nµ is an odd integer in {0, · · · , 2nµ+2}, µ = Nµ/2nµ ,
and nµ ≤ n. To facilitate description, we draw SMN of::the
::::::::following
:::::::::discussion,
:::::draw
:::the
:::::SMN
::of
:::the
:Logistic map, F ∗n ,
with a fixed control parameter and::in
:::the
:three arithmetic
domains::::::shown in Fig. 1a), b), and c), respectively.
Observing::::From
:Fig. 1, one can notice the following basic
characteristics of SMN of digital map::::the
:::::digital
::::map
::::can
::be
::::::noticed:
:
• The whole SMN is composed of a small number of::::::number
::of
:weakly connected component
::::::::::components, a maximal
subgraph:::::which
:::are
::::::::maximal
:::::::::subgraphs of a directed graph
such that:, for every pair of nodes u, v in the subgraph, there
is an undirected path from:a
::::path
::::::::between
:u to v and a
directed path from:::and
:v to u
::in
:::the
::::::::::underlying
:::::::::undirected
::::::version
::of
:::the
::::::::subgraph.
• Each weakly connected component has one and only oneself-loop (an edge connecting a node to itself) or cycle (asequence of nodes starting and ending at the same node suchthat, for each
::::every
:two consecutive nodes of
:in
:the cycle,
there exists an edge directed from the former node to thelater
::::latter
:one.)
• Any node is linked to the cycle of the associated weaklyconnected component via a transient process.
From Fig
::::From
::::Figs. 1a), b), c), one can further summarize
::::::observe
:a
special property of SMN of::the
:::::SMN
::of
:::the
:Logistic map: one
weakly connected component dominates the whole SMN andthere has
:is
:a clear decreasing order among all the weakly
connected components [57]. More precisely,::the
:size of the
component account:::::::accounts
:for more than half of the size of
the whole network.
Property 4. The nodes of odd label in the state network ofF ∗n+1 ::::::
labeled:::::::number
:::“i”
::in
:::F ∗n:and that in the state network
of F ∗n :::::F ∗n+1::::
with:::::::number
::::::::“2i+ 1”
:satisfy∣∣F ∗n+1(2i+ 1)− 2 · F ∗n(i)
∣∣ ≤ {6 if rn ∈ [0.25, 0.75);
5 otherwise,
where i ∈ {0, · · · , 2n − 1}, and n ≥ 3. Putting
Proof:::By
::::::putting
:Eq. (8) (8) into inequality (2), one can
get:::gets
:∣∣F ∗n+1(2i+ 1)− 2 · F ∗n(i)∣∣
≤∣∣R ((Nµ/2nµ+2
)·(4− (1 + 4i)/2n−1
))∣∣+{2 if rn ∈ [0.25, 0.75);
1 otherwise,
16
30
1814
17
15
5
2226
29
68
103
24
27 11
21 25
9
7 232
3112 32201
13 194 28
0
a)
37
17
0
40
44
7
64
2459
57
52
1 5263 6242 4722
18 1046
32
16 54
45
48
15 49
38
8
25
23
13
50
19
39
41
26
58
56
51
11
436
3634
21
3
53
60
4
1429 28
955
27
12
30
31
35
33
6120
b)
80
48
26
15 113
50
124
7860 115
6367
39 4
1
31
89
97127
46
33
93
82
111
95
119
729
556
32
17
5673
10
74
3596
24
54118
85
91
37
99
108
104
13
19
44
8884
61
109
40
102
20
29
121
42
6466
43
107
100
106 22
28120
51
69
8
122
58
7
110
45
2126
86
65
68
21
25
62
83
103
125
117 76
11
38
3
10181
90
36
23
98
12
87
30
92
57
0 5
59
18
70
34
123
128
112
27
47
49
14
114
79
5316
5294
75
105
41
116
71
77
c)Fig. 1. The state-mapping network
::::SMNs
:of
::the
:::::digital Logistic map with
µ = 121/25:,:::::::::implemented
::::under
::::::different
::::::::fixed-point
:::::::precisions: a) 5-bit
precision::::n = 5; b) 6-bit precision
::::n = 6; c) 7-bit precision
::::n = 7.
:::(The
:::first
:::::2n + 1
::::nodes
:::are
::::::plotted.)
from::::based
:::on
:Property 2.
As (Nµ/2nµ+2) ∈ [0, 1], one further
:::::::::furthermore
:has∣∣F ∗n+1(2i+ 1)− 2 · F ∗n(i)
∣∣≤∣∣R (4− (1 + 4i)/2n−1
)∣∣+{2 if rn ∈ [0.25, 0.75);
1 otherwise,
≤
{6 if rn ∈ [0.25, 0.75);
5 otherwise.
SMN F ∗nµ has dominate influence on F ∗n . From Property 1and Corollary 4, one can see that F ∗n can be considered asthe result obtained by incrementally expanding F ∗j within afixed small scope from j = nµ to n. Furthermore, the errorin each expansion step fall in a small known scope. So, theinitial structure of F ∗nµ has critical influence on that of F ∗n .
From the proof of Corollary:::::From
:::the
:::::proof
:::of
:::::::Property 4,
one can see that the upper bound of |F ∗n+1(2i+1)−2 ·F ∗n(i)|depends on the value
:::::values
:of i and Nµ. Figure 2a) depicts
the values of |F ∗n+1(2i+1)−2 ·F ∗n(i)| and F ∗n+1(2i)−2F ∗n(i)for every node shown in Fig. 1a). The corresponding datum
:::data
:for the nodes shown in Fig. 1b) is
:::are plotted in Fig. 2b),
to further demonstrate the difference:::::::::differences
:between F ∗n
and F ∗n+1.The
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 5
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
-1
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
-1
0
1
2
3
4
5
6
a)
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
-1
0
1
2
3
4
5
6
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
-1
0
1
2
3
4
5
6
b)Fig. 2. Distribution of the difference
:::::::differences
:between F ∗
n and F ∗n+1: a)
n = 5; b) n = 6.
Fact 1.:::::Round
:::::::function
::::R(·)
:::::::satisfies
::::that
:::::::::::R(x) = R(y)
::if
:::and
:::only
::if:{
1/2 ≤ y − bxc < 3/2 if x− bxc ≥ 1/2;
−1/2 ≤ y − bxc < 1/2 if x− bxc < 1/2,:::::::::::::::::::::::::::::::::::::
:::::where
::::::::x, y ∈ R.
:
Property 5.:::The
::::::::in-degree
::of:::::
node:::::::::F ∗n(2n−1)
::in
:::the
:::::SMN
::of
::the
::::::::Logistic
::::map
:::::::::::implemented
::by
:::::n-bit
:::::finite
::::::::precision
::is
2 ·⌊√
(d− R(d) + 1/2) · 2nµ+n/Nµ⌋
+ 1,
:::::::::::::::::::::::::::::::::::
(9)
:::::where
:::::::::::::::::::::::::::::::d = Nµ/2
nµ−n+2 − bNµ/2nµ−n+2c.:
Proof:::::::::Assuming
::::that
:::::::::::::::::Fn(i) = Fn(i+ kt),
:::::::namely
R(fn(i) · 2n) = R(fn(i+ kt) · 2n),:::::::::::::::::::::::::::::
(10)
:::one
:::can
::::::ensure
:::the
::::::degree
::of
::::the
::::node
:::::Fn(i)
::::::among
:::the
:::::SMN
::be
::::Nkt ,::::::
where::::::kt ∈ Z
::::and
::::Nkt ::
is:::the
:::::::number
::of
:::all
:::::::possible
:::::values
::of
:::kt::::::::
satisfying::::
Eq.::::(10).
:
::::::Putting
:::Eq.
:::(8)
::::with
:::::::::i = 2n−1
:::into
::::Eq.
::::(10),
::::one
::::gets
:
R(Nµ/2nµ−n+2) = R(Nµ/2
nµ−n · (1/4− k2t /22n)).:::::::::::::::::::::::::::::::::::::::::::
(11)
::::::::Referring
::to
::::Fact
::1,
::::one
:::has
:
R(d)− 1/2 ≤ d−Nµk2t /2nµ+n < R(d) + 1/2::::::::::::::::::::::::::::::::::::::
(12)
::::from
::::Eq.
:::::(11),
::::::where
:::::one
:::of
:::the
:::::two
:::::cases
:::in
:::::Fact
::1
:is:::::::::
selected,::::::::::
depending::::
on::::
the::::::
value:::
of:::::::::::::
R(d) ∈ {0, 1}.::::Since
::::::::::::::d− R(d) < 1/2
::::for
:::any
:::d,
:::the
::::right
::::part
:::of
:::the
:::::above
::::::::inequality
::is
::::::always
:::::::satisfied.
:::::::Solving
:::the
:::left
::::part
::of
::::::::inequality
::::(12),
::::one
::::has
::::::::::::::::::::::::::::::::::|kt| ≤
√(d− R(d) + 1/2) · 2nµ+n/Nµ.
::::So,
:::::::::::::::::::::::::::::::::::::::::Nkt = 2 · b
√(d− R(d) + 1/2) · 2nµ+n/Nµc+ 1,
:::::::::which
::::::::completes
:::the
:::::proof
:::of
:::the
::::::::property.
::::Since
::::::::::::::::::::::f ′(x) = µ · (1− 2x) > 0
::::for
:::::::::::x ∈ [0, 1/2]
::::and
:::the
::::value
:::of
::::the
::::::::Logistic
:::::map
:::::::::::::monotonously
::::::::increases
:::::from
:::zero
:::to
:::the
:::::::::maximum
::::::value
:::::::::::::f(1/2) = µ/4,
:::::which
::::::::becomes
::::::::::::::R(Nµ/2
nµ−n+2):::
in::::
the::::::n-bit
::::::finite
:::::::::arithmetic
::::::::domain.
:::The
:::::point
::::1/2
:::in
:::the
:::::::::::::::infinite-precision
:::::::domain
::::::::::corresponds
::to
:::the
::::::node
:::::::::i = 2n−1
:::in
:::the
:::::::SMN.
::::::When
:::::::::::n 6= nµ + 1,
::::::::::::::::::::::::::::::::Nµ/2
nµ−n+2 − bNµ/2nµ−n+2c < 1/2::::::::
always::::::
holds,::::
so
::::::::::::::::::::::::::::::R(Nµ/2
nµ−n+2) = bNµ/2nµ−n+2c.:::::Since
:::::::::::::::f ′′(x) ≡ −2µ < 0
::for
::::::::::::x ∈ [0, 1/2],
::::::f ′(x)
:::::::::::::monotonously
:::::::::decreases
::::::from
::µ
::to
:::::zero,
::::the
:::::node
::::::::::F ∗n(2n−1)
:::::owns
::::the
:::::::::maximal
::::::degree
:::::::::::::::::::::2 ·⌊√
2nµ+n−1/Nµ
⌋+ 1
:::in
:::::the
:::::::::associate
::::::SMN
::::::when
::::::::::n > nµ + 1.
:
::::Since
::::::f ′(x)
:::::::::::::monotonously
::::::::decreases
:::::from
:::µ
::to
:::::zero
::in
::the
:::::::studied
::::::::interval,
::::the
:::::::::in-degree
::::(the
::::::::number
:::of
:::::edges
::::::directed
:::::into
::a
:::::node
::in
::a::::::::
directed::::::::network)
::::::::::::corresponding
::to
:y:::::::::::::
monotonously::::::::increases
::as
::y::::::::increases
:::::from
::::zero
::to
:::the
::::::::maximum
:::::value
:::::::::::::f(1/2) = µ/4.
::::But,
::::due
::to::::
the::::::::::quantization
::in
:::the
:::::n-bit
:::::::::arithmetic
:::::::domain,
::::not
:::::every
::::::::possible
:::::value
::in
::the
:::::::::codomain
::::can
:::be
::::::::accessed
::by
::::the
::::::digital
:::::::version
::of
:::the
:::::::Logistic
::::map
::::(see
::::Fig.
:::3).
:::::::::::Fortunately,
:::the
:::::::::::quantization
:::can
::::only
::::::change
:::the
::::::::::::monotonicity
:::::when
::::the
::::::degree
::is
::::::::relatively
:::::small,
::as
::::::::::::demonstrated
::in
::::Fig.
:::3.
:::As
::to
:::the
:::::node
:::::::::F ∗n(2n−1),
::its
::::::::in-degree
::::(the
::::::::number
::of
::::::edges
::::::linking
:::to
:::the
::::::node)
::in
::the
::::::::associate
:::::SMN
::::has
::::been
::::::exactly
:::::::derived
::in
::::::::Property
::5.
::To
:::::obtain
:::the
::::::overall
::::::::::relationship
::::::among
:::the
::::most
::::::::important
:::::nodes
::in
:::the
::::SMN
:::in
:::::terms
::of
:::::::degrees,
:::::::assume
:::that
:::the
:::::::::::monotonicity
:is:::::::retained
::in:::the
:::::right
:::part
:::of
::the
:::::::interval
:::::::::::::::(0, F ∗n(2n−1)/2n)
::in
::the
:::::::::following
:::::::analysis.
::::::::In-degree
::::::::::distribution
:::::p(k),
:::the
::::::fraction
::of
:::::nodes
:::in
:::the
:::::::network
:::::with
::::::::in-degree
:::k,
::is
::a
::::::::::fundamental
:::::::::::characteristic
::of
:a::::::::
directed:::::::network.
:::To
:::::derive
::::the
:::::::::distribution
::of
:::the
:::::SMN
:::of
::::the
:::::::Logistic
:::::map,
::::first
::::::::compute
:::its
::::::variant
::in
:::::::Theorem
:::3,
:::::where
:cumulative in-degree distribution of the
state-mapping network::::P (k)
::::::means
:::the
:::::::fraction
:::of
:::::nodes
::in
::the
::::::::network
::::with
:::::::::in-degrees
:::::larger
::::than
:::k.
0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128
0
1
2
4
6
9
::a)
:
872 880 888 896 904 912 920 928 936 944 952 960 968 976
0
2
4
6
8
12
18
23
::b)
:
-103
-102
-101
-100 0 10
0
0
100
101
:c):
Fig. 3.::The
:::::degree
:::of
:::::F ∗n(i)::
in:::
the::::
SMN:::
of:::the
::::::Logistic
::::map
:::with
:::::::::µ = 121/25:
::a)
:::::n = 6;
::b)
::::::n = 10;
:c)::::::n = 13.
Theorem 1.:::The
:::::::::cumulative
::::::::in-degree
::::::::::distribution
::of
:::the
::::SMN
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 6
F ∗n satisfies
P (k) =
(2
µk− k
2n+1
)2
.
:::::::::approaches
:
P (k) =4
µ2k2:::::::::::
::as
::n
::::::::increases.
:
Proof. In a computing domain of fixed-point arithmetic pre-cision n with
:a:specified quantization scheme, the definition
domain and value domain::::::domain
::::and
::::::::codomain
:of any 1-D
map are both divided into intervals of fixed length ∆ = 1/2n.As demonstrated in
::As
::::::::::::demonstrated
:::by
:Fig. 4, as
::::::assume
::a:::::
point:::x0::::
and
:::::::::y0 = f(x0)
::::are
::::both
:::::::::multiples
::of
:::∆.
:::As
:for the interval to
which y0 = f(x0) belong::y0 :::::::
belongs, the number of inter-vals owning
::::::having
:pre-image of y0 in the neighbourhood
:::::::::::neighborhood
:of x0 is
ck:
=
⌈|x− f−1(f(x)−∆)|
∆
|x0 − f−1(f(x0)−∆)|∆
:::::::::::::::::::
⌉,
where dxe:::d·e
:gives the smallest integral value not less
than argument. As::the
:::::::::argument.
:::::Since
:::::::moving
::::::::location
::of
::::::::coordinate
::::::origin
::::does
:::not
::::::::influence
:::the
:::::value
::of
:::the
::::::::calculated
::::::degree,
:::the
:::::above
::::::::equation
::is
:::::::::applicable
::to
::::any
::::other
:::::point
::in
::the
:::::map.
:
0 0( )y f x
0 -y
0x
0-x
Fig. 4.::::::::::Demonstration
::of
::::::counting
::the
::::::number
::of
:::::::preimages
::of
:a:::map
::in
::the
::::digital
::::::domain.
::::Since
::::the Logistic map (7) owns
:::has
::a symmetric property,
i.e.
f(x) = f(1− x), (13)
the in-degree of the node corresponding to y = f(x) is doubleto that existing
::of
::::that
:in the left half part of the definition
domain.Demonstration of counting the number of preimages of map
in digital domain.In
:::::::domain.
::In
::::the interval [0, 1/2], the inverse function of
map (7) is
f−1(y) = (1−√
1− 4y/µ)/2.
Furthermore, f ′(x) = µ · (1− 2x) > 0 for x ∈ [0, 1/2), andf(x) monotonously increases with respect to x. So, one hasthe in-degree of the node to which y belongs
:,::as
:
k = 2 ·⌈f−1(y)− f−1(y − 1/2n)
2−n
⌉= 2 ·
⌈√1− 4(y − 1/2n)/µ−
√1− 4y/µ
21−n
⌉.
=:
2 ·√
1− 4(y − 1/2n)/µ−√
1− 4y/µ
21−n+ ε,
::::::::::::::::::::::::::::::::::::
From the above equation, one can get
y =µ
4− 1
µ · (k + ε)2− µ · (k + ε)2
22n+4+ 2−n−1,
where ε is the change caused by the quantization function,and 0 ≤ ε < 0.5.
:::::::::0 ≤ ε < 2.
::::::::Squaring
::::both
:::::sides
::of
:::the
:::::above
:::::::equation
:::::twice,
::::one
::::gets
:
y =µ
4− 1
µ · (k − ε)2− µ · (k − ε)2
22n+4+ 2−n−1.
::::::::::::::::::::::::::::::::::::::
(14)
As the relative influence of ε is very small, such similar casesare neglected in the following discussion.
As f ′′(x) ≡ −2µ::::Since
:::::::::::::::::f ′′(x) ≡ −2µ < 0
::for x ∈
[0, 1/2]and the in-degree corresponding to y monotonouslydecreases as it increase from zero to the maximum valuef(1/2) = µ/4. So, the rank of
::::::(order)
::of
:::the
:::::::interval
:::the
:state
y is:::::::belongs,
::::::among
::all
::::::::intervals,
::is:
r =
⌈µ/4− y
1/2n
⌉,.
and the number of nodes
N =
⌈µ/4
1/2n
⌉.
According to the definition of::the
:cumulative in-degree distri-
bution, one can get
P (k) = 1− 4y
µ.
:::has
P (k)::::
=:r/N:::
≈:
1− 4y
µ,
::::::
:::::where
:::::::::::N = d µ/41/2n e:::
is:::the
::::::::number
::of
::::::nodes
:::in
:::the
::::::SMN.
Incorporating Eq. (14) (14) into the above equation, one has
::::::obtains
P (k) =
(2
µk− k
2n+1
)2
.
Obviously, P (k) monotonously increases with respect to n.So,
::by
:increasing the value of n, the cumulative in-degree
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 7
distribution of::the
:SMN for the Logistic map approach its
limitlimn→∞
P (k) =4
µ2k2
more close. To verify this point, we draw::::tends
:::to
::its
:::::limit:
::::::::::::::::limn→∞
P (k) = 4µ2k2 .
::To
::::::verify
:::::::::Theorem
:::1,
:::::draw
::::the
::cumulative in-degree
distribution::::::::::distributions of SMN F ∗5 ∼ F ∗20 :
as:in Fig. 5, where
N is fixed as::to
::be
:220to demonstrate
:,::to
::::::clearly
::::::::::demonstrate
::the
:evolution of the studied distributionsclearly
::::::::::distributions.
The corresponding in-degree distributions are shown in Fig. 6,which agrees with the proof of
:::::agree
::::with Corollary 1.
100
101
102
103
10−6
10−5
10−4
10−3
10−2
10−1
100
k + 1
ratio
P (k) = 4µ2k2
Fig. 5. Cumulative in-degree distribution::::::::distributions
:of SMN F ∗
5 ∼ F ∗20,
where µ = 12125
.
Corollary 1. The in-degree distribution of the state-mappingnetwork
::::SMN
:F ∗n satisfies
p(k).==
:
16(k + 1)
µ2k2(k + 2)2.
Proof: Due to the symmetrical nature::::::::symmetry
:::::::property
of the Logistic map,::the
::in-degree of
::its
:SMN is always
evenexcept that:,::::::except
:::the
::::one corresponding to the critical
point f(1/2). According to
::::::::According
:::to
::::the
:definition of the cumulative in-degree
distribution, in-degree distribution p(k) can be calculated as
::by
:
p(k) = P (k)− P (k + 2).
So, one has
p(k) =
(2
µk− k
2n+1
)2
−(
2
µ(k + 2)− k + 2
2n+1
)2
= (k + 1)
(16
µ2k2(k + 2)2− 1
22n
).
Obviously, p(k) monotonously approach::::tends
::to::
its limitvalue
:::
limn→∞
p(k) = 16(k+1)µ2k2(k+2)2 with increase of n.
:.:
C. Digital::::SMN
::of::::
the::::::digital Tent map
In the::::same
::digital domain discussed in the above
sub-section:::last
::::::::::subsection, the Tent map f(x) = µ ·
(1− 2 |x− 1/2|) become:::::::becomes
fn(i) = ((Nµ/2nµ) · (1− 2 |(i/2n)− 1/2|)) . (15)
100
101
102
103
10−6
10−5
10−4
10−3
10−2
10−1
100
k + 1
ratio
p(k) = 16(k+1)µ2k2(k+2)2
Fig. 6. In-degree distribution::::::::distributions
:of SMN F ∗
5 ∼ F ∗20, where µ =
12125
To facilitate description, we::the
:::::::::following
:::::::::discussion, draw the
SMN of:::the
:Tent map, F ?n , with µ = 31/25 under domain
:in
::the
::::::::domains
:of fixed-point 5-bit and 6-bitin Fig
:,::::::::::respectively,
::as
::::::shown
::in
::::Figs. 7a) and b), respectively.
31
16
22
2
28
1
4
8
30
239
29
17
15 6
20
5
7
10
25
1914
27
18
26
3
0
13 24
12
2111
32
a)
12
40
24
55
47
17
62328
0
52
60
23
38
57
58
14
34
61
26
6
30
50
15
46
49
13
51
20
48
44
3556
16
39
25
29
31
18
4
63
12
3
33
9
7
275
59
11
53
22
28
4342
3754
10
45
19
36
64
4121
b)
Fig. 7. The SMN of::the Tent map with µ = 31/25: a) 5-bit precision; b)
6-bit precision.
Corollary 2. The nodes of odd label::::with
:::odd
:::::label
:::::::numbers
in the state network of F ?n+1 and that in the state network ofF ?n satisfy
∣∣F ?n+1(2i+ 1)− 2 · F ?n(i)∣∣ ≤ {4 if rn ∈ [0.25, 0.75);
3 otherwise,
where i ∈ {0, · · · , 2n − 1}.
Proof: The proof is very similar to that ofCorollary
:::::::Property 4. As
:::::Since
:
(Nµ/2
nµ−1)∈ [0, 2],
one has∣∣F ?n+1(2i+ 1)− 2 · F ?n(i)∣∣
≤∣∣R (Nµ/2nµ−1)∣∣+{
2 if rn ∈ [0.25, 0.75);
1 otherwise,
≤
{4 if rn ∈ [0.25, 0.75);
3 otherwise.
From the proof of Corollary 2, one can see that the upperbound of |F ?n+1(2i + 1) − 2 · F ?n(i)| just depends on the
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 8
value of:::::::depends
::::only
::on
:Nµ. Figure 8a) depicts the values of
|F ?n+1(2i + 1) − 2 · F ?n(i)| and F ?n+1(2i) − 2F ?n(i) for everynode shown in Fig. 7, which agrees with Property
::::::::Properties 1
and Corollary 4. The corresponding datum under:::data
::::with
n = 6 is::are
:shown in Fig. 8b)to ,
::::::which further demonstrate
the difference::::::::differences
:between F ?n and F ?n+1. Now, we
::one
can see that:::the
::::SMN
:::of
:::the Tent map also follow Property ??
as Logistic map in the studied:::::::::::incrementally
:::::::expand,
::::just
::as
::the
::::::::Logistic
::::map,
:::in
:::the
:::::same digital domain.
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
−1
0
1
2
3
4
5
6
i
Fn+1(2i)− 2Fn(i)
upper bound
|Fn+1(2i + 1)− 2Fn(i)|
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
−1
0
1
2
3
4
5
6
i
a)
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
−1
0
1
2
3
4
5
6
i
Fn+1(2i)− 2Fn(i)
upper bound
|Fn+1(2i + 1)− 2Fn(i)|
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
−1
0
1
2
3
4
5
6
i
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
−1
0
1
2
3
4
5
6
i
b)Fig. 8. Distribution
::::::::Distributions
:of the difference
:::::::differences between F ?
nand F ?
n+1: a) n = 5; b) n = 6.
1
0
31
13
2
3
17
19
26
65
29
27
12
11
22
21
20 10
24
32
48
16
15
30
9
18
14
23
25
7
28
a)
522651
45
381924
1227
2010
4054 37
44
5
5829
35
6
3
13
33 624
31
2
63
47
156
14
23
2550
57
41
46
28
367
4243
2153
2261 11
5518
4915
9
16
60
34
17
30
8
64 3259
39
48
0
b)
Fig. 9. SMN of::the Tent map of µ = 1 with
::::µ = 1
:::after
:round quantization:
a) 5-bit precision; b) 6-bit precision.
Property 6. The degree of SMN of tent::the
:::::SMN
::of
:::the
::::Tent
map has only three possible values:
k =
1 the node denoting maximual value;2 other nodes owing pre-images;0 other nodes.
Proof. When x = 1/2, the maximum value f(x) = µ isobtained. The node corresponding to x = 1/2 only pointsto one node corresponding to y = µ. So, the in-degree of thenode to which y = µ belongs is k = 1.
Due to the symmetrical nature of the tent map, we onlyconsider
::::::::symmetry
:::::::property
:::of
:::the
:::::Tent
:::::map,
::::only
:the left
half part of the definition domain . In::::::domain
::is::::::::::
considered.
::In
:::the
:interval [0, 1/2], the inverse function of the tent
:::Tent
map y = f(x) is
f−1(y) = y/ (2µ) .
As:::::Since
:f ′(x) = 2µ > 0 for x ∈ [0, 1/2], and f(x)
monotonously increases with respect to x. Just like:::::::Similarly
::to the proof of Theorem 1, one has the in-degree of the nodeto which y belongs,
::as
k = 2 ·⌈f−1(y)− f−1(y − (1/2n))
2−n
⌉= 2 ·
⌈y/(2µ)− (y − 1/2n)/(2µ)
2−n
⌉= 2 · d1/(2µ)e= 2
when y 6= µ.
From Property 6, the links among the nodes of SMN of
:::::edges
::in
:::the
:::::SMN
::of
::::the Tent map are not accumulated with
increase of::as the implementation precision (See
::::::::increases
:::(see
Fig. 10), which is different from that of::the
:Logistic map.
Based on the above discussions, we:::one
:can conclude that
f ′′(x) > 0 in the whole definition domain is only a sufficientbut not necessary condition for that
:a:::::::::necessary
::::::::condition,
:::for
:::::which
:the associate SMN of the corresponding map follows a
power-law distribution.
1 2 310
−5
10−4
10−3
10−2
10−1
100
rati
o
k + 1
a)
1 2 310
−6
10−5
10−4
10−3
10−2
10−1
100
rati
o
k + 1
b)
Fig. 10. Statistics of SMN F ?5 ∼ F ?
20with ,:µ = 31/25: a) accumulative
in-degree distribution; b) in-degree distribution.
III. ANALYSIS OF STATE-MAPPING NETWORK:::::SMNS
:OF
DIGITAL CHAOTIC MAPS IN FLOATING-POINT ARITHMETICDOMAIN
A. Influence caused by floating-point arithmetic
To obtain a:
trade-off between range and:a:::::wide
:::::range
:::and
:a::::high
:precision, digital computers adopt two kinds of binary
representation formats to represent real numbers: fixed-pointformat , and floating-point format. The former is more suitablefor integers or real numbers with a fixed precision, and thelatter approximate
:::for
:::::::::::::approximating
:real numbers with a
higher and variable precision. Before standardization by IEEEand ANSI in 1985, different machine may use
:::::::machines
::::used
dramatically different representation forms for the floating-point arithmetic[58].
Following the floating-point standard:::::::standards, a sequence
of n bits, {b(i)}n−1i=0 , is divided into three parts: a sign, asigned exponent, and a significand. The represented number
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 9
is interpreted as the signed product of the significand and tworaised
:::the
:::::::number
:2:
to the power of its exponent:
v =
0 if e = 0, os = 0;
(−1)s · (m∑i=1
bl+i · 2−i) · 22−2l−1
if e = 0, os 6= 0;
(−1)s · ∞ if e = 2l − 1, os = 0;
“not a number” if e = 2l − 1, os 6= 0;
(−1)s ·(
1 +m∑i=1
bl+i · 2−i)· 2e−os otherwise,
where s = b0, e =l−1∑i=0
b1+i · 2i, os = 2l−1 − 1.
As for::to
:::the
:single-precision floating-point format (bina-
ry32), e.g. “float” in C language:”::in
::::::::::C-language and “single”
:”:in Matlab, (l,m) = (8, 23), whereas (l,m) = (11, 52)
in double-precision floating-point forma:::::format
:(binary64).
In IEEE 754-2008, half-precision floating-point format (bina-ry16) is designed for storage with
::::::::storaging
::::with
::a:
higherprecision, not for performing arithmetic computations, where(l,m) = (5, 10). Meanwhile, due to
:::the intermediate scale
of the data generated by binary16, its simulation is widelyadopted for experiments [59].
In:::the
:floating-point arithmetic domain, equality (13) does
not exist:::hold
:in general. Some concrete intermediate datum
during calculating::::data
::::from
::::::::::calculating
:::the
:Logistic map are
shown in Table I. Precisely::::::::::Specifically,
µ · x · (1− x)?= µ · (1− x) · (1− (1− x)),
:::::which
:is caused by the difference between fl(x) and fl(1 −
fl(1 − fl(x))), where fl(x) denotes the normalized floatingpoint number closest to x in the given floating-point domain. Ifa number fall in
:::falls
::::into
:::the
:interval [0.5, 1], its complement
in terms of subtraction from 1 in the:::that
:domain is fixed, so
fl(1−fl(1−fl(x))) ≡
{1− fl(1− fl(x)) if x ≤ 0.5;
fl(x) if x > 0.5.(16)
For any x ∈ R ∩ [0, 1], there exists a unique integer e suchthat x = (
∑∞i=0 xi · 2−i) · 2e, ::::::::::::::::::::
x = (∑∞i=0 xi · 2−i) · 2e,:where
x0 = 1. In the floating-point domain with parameter (l,m), itbecomes
fl(x) =
{(∑mi=1 xi · 2−i) · 22−2
l−1
if x ∈ (0, 22−2l−1
);
(1 +∑mi=1 xi · 2−i) · 2e if x ∈ [2e, 2e+1),
where e ∈ {2− 2l−1, · · · ,−2}. Then, one has
1− fl(x) =∑2l−1−2i=1 2−i + (
∑mi=1 xi · 2−i + 2−m) · 22−2l−1
if x ∈ (0, 22−2l−1
);∑−(e+1)i=1 2−i + (
∑mi=1 xi · 2−i + 2−m) · 2e
if x ∈ [2e, 2e+1),
(17)
where xi = 1 − xi. Observing the:::first
:::::item
:::in
:::the
:right-
hand side of Eq. (17), one can see that the exponent for::the
representation of (1− fl(x)) in the:::that
:domain is minus one,
i.e.
fl(1− fl(x)) =
(1 +
m∑i=1
xi · 2−i)· 2−1,
::::::namely,
:
fl(1− fl(x)) =
(1 +
m∑i=1
xi · 2−i)· 2−1,
::::::::::::::::::::::::::::::::::
where xi ∈ {0, 1}. In addition, 2l−1 − 2 ≥ m + 1 is anecessary condition for assuring the
:::::::ensuring
:a:sufficient scope
of represented numbers by the floating-point format. So, thetwo cases in Eq. (17) need to be further divided into threecases:
fl(1− fl(x)) =(∑m+1i=1 2−i) if x ∈ (0, 22−2
l−1
);
(∑m+1i=1 2−i) if x ∈ [2e1 , 2e1+1);
(∑−e2−1i=1 2−i) + (
∑m+1+e2i=1 xi · 2−i) · 2e2
if x ∈ [2e2 , 2e2+1),
(18)
where e1 ∈ {2−2l−1, · · · ,−m−2}, e2 ∈ {−m−1, · · · ,−2}.Referring to the first case in Eq. (16) and subtracting
Eq. (18) from Eq. (17), one has::::::obtains
fl(1− fl(1− fl(x)))− fl(x)
= (1− fl(x))− fl(1− fl(x))
=
(∑2l−1−2i=m+2 2−i) + (
∑mi=1 xi · 2−i + 2−m) · 22−2l−1
if x ∈ (0, 22−2l−1
);
(∑−(e1+1)i=m+2 2−i) + (
∑mi=1 xi · 2−i + 2−m) · 2e1
if x ∈ [2e1 , 2e1+1);
(∑mi=m+2+e2
xi · 2−i + 2−m) · 2e2if x ∈ [2e2 , 2e2+1).
(19)
From Eq. (19), one can see:it:::::::
follows:
that the differencebetween 1 − (1 − x) and x decreases monotonically inevery selected interval, which is verified by the differencesshown in Fig. 11. As shown in the inset in Fig. 11,the two segments corresponding the first two cases inEq. (19), i.e. intervals [2−10+2−24 , 22−2
4
] = [2−24, 2−14]and {[2e1 , 2e1+1]}−10−2e1=2−24 = {[2e1 , 2e1+1]}−12e1=−14
::::::::::::::::::::::::::::::::::::::{[2e1 , 2e1+1]}−10−2e1=2−24 = {[2e1 , 2e1+1]}−12e1=−14,
::::can be
connected smoothly. The corresponding difference causingfor
:::::::yielding
:the final result of
::the
:Logistic map is shown in
Fig. 12.Assume
:::that
:::::the initial condition
:is
x(0) = (0.b1b2 · · · bj · · · bL−1bL)2 6= 0, wherebL = 1 (the least significant 1-bit) and1− x(0) = (0.b′1b
′2b′3 · · · b′j · · · b′L−1bL)2:::::::::::::::::::::::::::::::::
1− x(0) = (0.b′1b′2b′3 · · · b′j · · · b′L−1bL)2.
Then, the iteration of the Tent map becomes
x(1) =
2x(0) = x(0)� 1 = (0.b2 · · · bj · · · bL−1bL)2,
if 0 ≤ x(0) < 0.5,
2(1− x(0)) = (b′1.b′2b′3 · · · b′j · · · b′L−1bL)2,
if 0.5 ≤ x(0) ≤ 1,
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 10
TABLE ITHE INTERMEDIATE
::::::::::INTERMEDIATE VALUES OF CALCULATING
::THE
:LOGISTIC MAP IN BINARY16.
x 1− x 1− (1− x) f(x) f(1− x)0.0099945068359375 0.98974609375 0.01025390625 0.037384033203125 0.038360595703125
0.04998779296875 0.94970703125 0.05029296875 0.179443359375 0.18054199218750.0899658203125 0.90966796875 0.09033203125 0.309326171875 0.3105468750.0999755859375 0.89990234375 0.10009765625 0.340087890625 0.3405761718750.199951171875 0.7998046875 0.2001953125 0.6044921875 0.604980468750.289794921875 0.7099609375 0.2900390625 0.77783203125 0.77832031250.389892578125 0.60986328125 0.39013671875 0.89892578125 0.89941406250.489990234375 0.509765625 0.490234375 0.9443359375 0.94482421875
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5x 10
−4
x
dif
fere
nce
2−11
2−10
2−9
2−8
2−7
2−6
2−5
2−4
2−3
2−2
2−1
20
0
5.5x 10
−4
2−24
2−14
2−11
Fig. 11. Subtracting x from 1− (1− x) under:::with various value
:::::values of
x in Binary16.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10−3
x
dif
fere
nce
2−11
2−10
2−9
2−8
2−7
2−6
2−5
2−4
2−3
2−2
2−1
20
0
2x 10
−3
2−24
2−14
2−11
Fig. 12. Subtracting f(x) from f(1−x) under:::with various value
::::values of
x in Binary16.
where � denotes the left bit-shifting operation. Note thatb1 = 0 when 0 ≤ x(0) < 0.5. Apparently, after
::::After L − 1
iterations,::one
:::::::obtains x(L−1) ≡ (0.bL)2 = (0.1)2. Then,
::So
x(L) ≡ 1 , and x(L+ 1) ≡ 0. That is, the number of requirediterations to converge to
:::::::::converging
::to
:zero is Nr = L + 1.
Note that Nr = 0 when x(0) = 0. To visualize the operationsof
:::the digital Tent map with a typical example, we present here
the evolution process of the number:is::::::::presented
:corresponding
to the node labelled with “13”::::::labeled
::::with
:::::“13” in Fig. 13a):
(0.01101)2 → (0.1101)2 → (0.011)2 → (0.11)2 → (0.1)2 →(1)2 → (0)2.
From the above analysis, it is clear that no any quantizationerror is introduced in
:::into
:the digital chaotic iterations, which
is because the chaotic iterations can be exactly carried out
4
32
2
30
0
12
26
6
288
2024
165
147
18
10
9
22113
13
15
1
a)
8
24
64
48
16
320
11
21
27
25
5
3656
23
18
50
9
28
46
14 7
58
38 52
26
12
6
29
19
3
62
60
4
30
34
2
31
15
13
17
1
44
2040
54
22
42
10
b)Fig. 13. SMN of
::the
:Tent map of
:::with
:µ = 1 under
:in:::the binary floating-
point domain: a) 7-bit binary floating-point domain with (l = 3, m = 3); b)8-bit binary floating-point domain with (l = 3, m = 4).
with the digital operation�. The value of L can be estimatedaccording to two different conditions of x(0) 6= 0:• x(0) is a normalized number:
x(0) = (1.bm−1 · · · b0)× 2−e = (0.
e−1︷ ︸︸ ︷0 · · · 0 1bm−1 · · · b0)2.
Assuming:::::::::::::::::::::::::::::::::::::::::::::::x(0) = (1.bm−1 · · · b0)× 2−e = (0.
e−1︷ ︸︸ ︷0 · · · 0 1bm−1 · · · b0)2.
::::::::Assuming
::::that
:the least 1-bit of x(0) is bi = 1, one can
immediately get x(0) = (0.
e−1︷ ︸︸ ︷0 · · · 0 1
m−i︷ ︸︸ ︷bm−1 · · · bi
i︷ ︸︸ ︷0 · · · 0)2
and deduce L = (e − 1) + 1 + (m − i) = e + (m − i).Considering e ∈ [1, 2l−1 − 2] and i ∈ [0,m − 1],
:::one
::::has
L ∈ [2, 2l−1 − 2 +m].• x(0) is a non-zero denormalized number:
x(0) = (0.bm−1 · · · b0)× 22−2l−1
= (0.
2l−1−2︷ ︸︸ ︷0 · · · 0 bm−1 · · · b0)2.
Assuming:::::::::::::::::::::::::::::::::::::::::::::::::x(0) = (0.bm−1 · · · b0)× 22−2
l−1
= (0.
2l−1−2︷ ︸︸ ︷0 · · · 0 bm−1 · · · b0)2.
::::::::Assuming
::::that
:the least 1-bit of x(0) is bi = 1, one can
immediately get x(0) = (0.
2l−1−2︷ ︸︸ ︷0 · · · 0
m−i︷ ︸︸ ︷bm−1 · · · bi
i︷ ︸︸ ︷0 · · · 0)2 and
deduce L = 2l−1−2+(m−i) = 2l−1−2+m−i. Consideringi ∈ [0,m− 1],
::one
::::has L ∈ [2l−1 − 1, 2l−1 − 2 +m].
To sum up:::::::::::Summarizing, in both conditionsL ≤ 2l−1 − 2 +m,
:::::::::::::::L ≤ 2l−1 − 2 +m.
Firstly, let us::::Next,
:consider the mathematical expectation
of i ∈ {0, · · · ,m − 1}. Without loss of generality, for adenormalized number or a normalized number with a fixedexponent e, assume
:::that
:the mantissa fraction (bm−1 · · · b0)2
distributes uniformly over the discrete set {0, · · · , 2m − 1}.Then, the probability that
::of
:(bi = 1, bi−1 = · · · = b0 = 0)
is 2m−1−i
2m = 12i+1 , and the probability that
:of
:(bm−1 = · · · =
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 11
b0 = 0) is 12m . Then
::::Thus, the mathematical expectation of i
is
E(i) ≈m−1∑i=0
i · 1
2i+1+m · 1
2m
=1
2·m−1∑i=1
i
2i+
m
2m
= 1− 1
2m≈1..
Then, let us consider::::Next,
::the mathematical expectation
of e ∈ {1, · · · , 2l−1 − 2}:is::::::::
analyzed. From the uniformdistribution of x(0) in the interval [0,1], one has
::it
::::::follows
:::that
:the probability of the exponent e is about Prob[2−e ≤
x < 2−(e−1)] = 2−e. Thus, the mathematical expectation of eis
E(e) ≈2l−1−2∑e=1
e
2e= 2− 2l−1
22l−1−2 ≈ 2.
From the above deductions, we can immediately deduce::one
:::can
::::::deduce
::::that
:
E(L)
= Prob[normalized numbers] · (E(e) + (m− E(i))) +
Prob[denormalized numbers] ·(2l−1 − 2 +m− E(i)
)=
2l−1 − 2
2l−1 − 1· (E(e) + (m− E(i))) +
1
2l−1 − 1·(2l−1 − 2 +m− E(i)
)≈ 2l−1 − 2
2l−1 − 1· (2 + (m− 1))+
1
2l−1 − 1· (2l−1 − 2 +m− 1)
=(2l−1 − 2)(m+ 2) +m− 1
2l−1 − 1. (20)
To verify the mathematical expectation of Nr, some experi-ments were made for test
::::::::performed
:::for
::::::testing
:on 10,000 ini-
tial conditions, pseudo-randomly generated with the standardRand function of Microsoft Visual Studio 2010. The occur-rence frequency of different values is shown in Fig. 14, wherethe average values of Nr for the three arithmetical
::::::::arithmetic
domains are about 11.95, 24.97, and 54.01, respectively. Everydistribution
:::All
:::::::::::distributions
::::well
:agree with the theoretical
expectationswell.
B. The relation::::::::::Relationship
:between SMN
:::the
:::::SMNs obtained
in the two arithmetic domains
In the floating-point arithmetic domain with::the
:exponent
width of l bits and:::the mantissa width of m bits, the min-
imum fixed interval is 2(1−(2l−1−1)) · 2−m = 22−2
l−1−m.Under
:a::
fixed-point computing environment of precisionn,
::the
:fixed interval is 2−n. If 22−2
l−1−m = 2−n, namely
:::::::::::::::22−2
l−1−m = 2−n,::::
i.e.:n = m + 2l−1 − 2, the two SMN
:::::SMNs
:obtained by implementing one map in the two domains
satisfy strong correlationshown in::::have
:a:::::strong
::::::::::correlation,
::as
:::::::::::characterized
::by
:Theorem 2.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Nr
freq
uen
cy
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
binary16
binary32
binary64
Fig. 14. The occurrence frequency of different values of Nr in the::::within
:a
total::of 10,000 values.
Theorem 2. Given binary floating-point format parameter
:::::::::parameters
:l and m, the node with label
:“i
:”:
in Fl,m andthat with label
:“i
::” in Fn satisfy
Fn(i)− Fl,m(i)
≤
{1 if Fn(i) ∈ [0, 2m);
2n−m−1−j if Fn(i) ∈ [2n−j−1, 2n−j),(21)
where j ∈ {2l−1 − 3, 2l−1 − 4, · · · , 1, 0}, and
n = m+ 2l−1 − 2. (22)
Proof. Utilizing the property of:::the
:integer quantization func-
tion|x− R(y)| = |R(x− y)|, x ∈ Z,
one can get:::gets
:
|Fl,m(i)− Fn(i)|= |fl,m(i) · 2n − R (fn(i) · 2n)| .= |R ((fl,m(i)− fn(i)) · 2n)| ,
where Fl,m(i) = fl,m(i) · 2n.As for node labelled
:::the
::::node
:::::::labeled with “i”
:” in Fn, the
corresponding value i/2n can be both accurately representedin
::::both
:the floating-point domain with (l,m) and
:::the n-bit
fixed-point domain. Furthermore, the intermediate processesof calculating f(i/2n) are the same in the two arithmeticdomains. So, the difference between fl,m(i) and fn(i) is onlycaused
::::::caused
::::only
:by the final quantization step.
When Fn(i) ∈ [2n−1−j , 2n−j), one has
fl,m(i) =2−j−1 ·
(1 +
m∑i=1
ai · 2−i),
fn(i) =2−j−1 ·
(1 +
n∑i=1
ci · 2−i),
where j ∈ {0, · · · , 2l−1 − 3}. Obviously, ai = ci for i = 1 ∼m. So,
:::one
::::has
fn(i)− fl,m(i) = 2−j−1 ·
(n∑
i=m+1
ci · 2−i)
< 2−m−j−1.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 12
Then::::Thus, one can conclude that
Fn(i)− Fl,m(i)
= R((fn(i)− fl,m(i)) · 2m+1+j · 2n−m−1−j
)≤ R
(2n−m−1−j
)= 2n−m−1−j .
When Fn(i) ∈ [0, 2m), fl,m(i) is a subnormal number, andfl,m(i) and fn(i) can be expressed as
fl,m(i) = 22−2l−1
·
(m∑i=1
ai · 2−i),
fn(i) = 22−2l−1
·
(n∑i=1
ci · 2−i).
Similar:::::::Similarly
:to the above cases, one has
fn(i)− fl,m(i) = 22−2l−1
·
(n∑
i=m+1
ci · 2−i)
< 2−m+2−2l−1
= 2−n.
So, one gets
|Fl,m(i)− Fn(i)| = R (|(fl,m(i)− fn(i)) · 2n|)≤ R (1) = 1.
:::gets
:
|Fl,m(i)− Fn(i)|::::::::::::::
= R (|(fl,m(i)− fn(i)) · 2n|)::::::::::::::::::::::::
≤ R (1) = 1.::::::::::
To demonstrate:::::::illustrate Theorem 2, we draw three con-
nected components of SMN of::the
::::::SMN
::of
:::the
:Logistic map
with µ = 121/25, F12, in Fig. 15a). Relative relations of thenodes in F4,6 are shown in Fig. 15b). The corresponding partsof F4,6 are shown in Fig. 15b).
Due to the limitation of presentation space:::::space
::::::::limitation,
only the involved nodes and their neighbours, not::::::::neighbors,
:::but
:::not
::::the
:original connected components, are shown in
Fig. 15b). Concretely:::::::Moreover, differences between some
mapping::::::::mappings
:in F12 and that in F4,6 are listed in Table II,
which validates Theorem 2 also::as
::::well.
From Theorem 2, we:::one can see that SMN of
::the
:::::SMN
::of
::the
:Logistic map implemented by
::the
:floating-point arithmetic
(m, l) can be considered:::::::regarded as a rewired version of
::the
sub-network of its SMN implemented by:in
:the corresponding
fixed-point precision. Precisely::::More
::::::::precisely, SMN Fl,m can
be generated from SMN Fn as follows: all nodes linking toFn(i) are redirected to the node labelled
:::::nodes
::::::labeled
:2n−j−
(k+1)·2n−m−1−j when 2n−j−(k+1)·2n−m−1−j < Fn(i) <2n−j − k · 2n−m−1−j ; all nodes linking to Fn(i), except thenode labelled
::::::labeled
:2n, are redirected to the nodes with label
2n−j − k · 2n−m−1−j::::::::::::::::::2n−j − k · 2n−m−1−j or 2n−j − (k +
1) · 2n−m−1−j when Fn(i) = 2n−j − k · 2n−m−1−j , wherej ∈ {2l−1 − 4, · · · , 1, 0}, and k = 0 ∼ 2m − 1. When 0 <
1200
68
252
348
95
57212
1296380
444
3552
14882624 119
58
768
86
23
122
216
4401472
436 4064
120
32
85
121
1088316
312
3008
254
896
3200
888
214
760
2336
3328
a)
2901
436
3748
1473
3200
896
3660
1204
2647
348
1497
3338
2336
444
3795
3975
1459
103
768
1195
2896
3328
3021
2816
3008
3208
1088 1200
2360
2637
1304
3552
3993
1784
380
2341
2800
3884
1296
758
121
4073
3350
318
760
1280
2624
3566
1472
86
40641485
3976 32
120
4090
44023
6
3582
1488 441
2630
3655
888
3846
250212
3784
216
312
316
1103
1090
3780
767
252
894
254
4038
3882
901
38423844
214
58
122
448437
3974
57
119
68
3977
4078
15 774253213 18
4081
351
95
4001
315
4011
85
b)Fig. 15. Some connected components of
::the
:SMN of
::the
:Logistic map in
floating-point domain and relative relations of their nodes in the correspondingfixed-point domain: a) 11-bit floating-point domain with (l = 4, m = 6); b)12-bit finite-precision domain, where µ = 121/25.
TABLE IIDIFFERENCES BETWEEN SMN
::::SMNS IMPLEMENTED IN TWO
ARITHMETIC DOMAIN::::::
DOMAINS WITH n = 12, l = 4, m = 6.
i Fn(i) Fl,m(i)∣∣Fn(i)− Fl,m(i)
∣∣ 2n−m−1−j
6 23 22 1 118 68 67 1 133 124 123 1 140 150 148 2 253 198 196 2 267 249 248 2 282 304 300 4 4
112 412 408 4 4130 476 472 4 4156 567 560 7 8238 848 840 8 8284 999 992 7 8316 1103 1088 15 16576 1872 1856 16 16648 2063 2048 15 16768 2360 2336 24 321280 3328 3296 32 322080 3871 3840 31 32
Fn(i) < 2m+1, all nodes linking to Fn(i) are redirected to thenodes with label Fn(i) or Fn(i)− 1.
The process can be verified by comparing Fig. 16a) andFig. 16b) with Fig. 17a) and Fig. 17b), respectively. Table IIIpresents concrete difference
::the
::::::::::differences
:between the two
SMN in:::::SMNs
::in
:::the
:two arithmetic domains with n = 6 ,
and (l,m) = (3, 4).
16
17
54
5
0
48
5046
19
6 13
29
15
52
8
369
2044
1
3056
3
60
34
23
2
27
11 10
2818
42
4
31
14
1264
2140
587
32
226225
2624
38
a)
47
64
2048
0
16
17
53
44
46
3414
42
18
31
50
9
740
5158
23
1357
41
24
32 10
4362
54
255 33
30
216
36
28
3
11
6135
29
4938
19
22 5212
6056
1542637
45
63
27 1
8
25 5
39 59
b)Fig. 16. SMN of
::the
:Logistic map with µ = 62/24: a) 8-bit floating-
point domain with (l = 3, m = 4); b) 6-bit fixed-point precision and roundquantization.
Referring to Theorem::::::::Theorems 1 and Theorem 2, one can
conclude that the cumulative in-degree distribution and::the
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 13
27 40 230 1264 4222
54
413
24
10
7
32
29
6018
26
62
3
50
14
21
48
3856
15
288
19
34
44
46
36
17
52
31
6 11
20
58
25
9
16
1
2
5
30
a)
7
61
42
33
3
45
13815 28
4
56
63 3662
32
59 5
17
9
260
1
55
39
25
52 22
23
47
12
18 1950494824 37
30 2610
5134
543864
41
40
1158
57
2131
53
43 6
44 1446
16 3529200 27
b)Fig. 17. SMN of
::the Tent map with µ = 15/24: a) 8-bit binary floating-
point domain with (l = 3, m = 4); b) 6-bit fixed-point precision and roundquantization.
in-degree distribution of SMN of:::the
:::::SMN
::of::::
the:Logistic
map implemented in:::the
:floating-point arithmetic domain
approximate that implemented in the corresponding fixed-pointarithmetic domain. This is verified by comparing Fig. 5 andFig. 18 with Fig. 6 and Fig. 19, respectively.
100
101
102
10−5
10−4
10−3
10−2
10−1
100
k + 1
ra
tio
m = 6m = 7m = 8m = 9m = 10m = 11m = 12m = 13m = 14
P (k) = 4µ2k2
Fig. 18. Cumulative In-degree distribution::::::in-degree
::::::::distributions
:of
::the SMN
of::the
:Logistic map in various floating-point domains, where l = 4, m =
6 ∼ 14.
100
101
102
10−4
10−3
10−2
10−1
100
k + 1
ra
tio
m = 6m = 7m = 8m = 9m = 10m = 11m = 12m = 13m = 14
p(k) = 16(k+1µ2k2(k+2)2
Fig. 19. In-degree distribution::::::::distributions of
::the
:SMN of
::the
:Logistic map
in various floating-point domains, where l = 4, m = 6 ∼ 14.
IV. TESTING:::THE
:RANDOMNESS OF VARIOUS
PSEUDO-RANDOM NUMBER GENERATORS::::::PRNGS
:BASED
ON CHAOS:::::::::ITERATING
::A
::::::::CHAOTIC
::::MAP
:VIA SMN
In this section , we demonstrate::::This
::::::section
:::::::::::demonstrates
that SMN can::be
::::used
:::to
:::::::classify
:::the
:::::::::structures
:::of
::::::various
::::::PRNGs
:::::based
:::on
:::::::iterating
::a::::::chaotic
:::::map.
:::::::::::Furthermore,
::it:::can
work as a coarse visual tool on randomness test of PRNGbased on iterating chaotic map, which is complementary
::for
::::::::evaluating
:::::their
::::::::::randomness
::::::levels,
:::as
::a:::::::::::complement
:to all
kinds of test indexes enclosed in the famous:::::::standard
:test
suites, e.g. NIST SP 800-22 [60] and TestU01 [61]. To achievethis , we scrutinize SMN of enhanced Logstic mapsby variousmethods:
:::For
::::this
::::::::purpose,
::::the
:::::::various
::::::::methods
:::for
:::::::::::chaos-based
:::::PRNG
::::are
:::::::::classified
::::into
::::the
::::::::following
::::six
:::::::::categories
:::by
:::::::::scrutinizing
::::the
:::::SMNs
:::of
::::::::enhanced
:::::::Logistic
::::::maps:
• Selecting state and control parameter:::::::::parameters
Observing::By
:::::::::observing
:Fig. 1, one can see that the nodes
labelled::::::labeled “0” and “2n”
:“::::and
:::::““2n”
:are pathological
seedsand ,::::::::
therefore:::::
they should be excluded for::::from
::::any
PRNG. More similar nodes can be found in::::from
:Fig. 1c),
which is very different to be found via:::are
::::very
::::::difficult
::to:::be
::::::::identified
::by
::a randomness test suite. In [15], it was claimed
that the discrete Tent map can achieve a satisfactory tradeoffamong period length, statistical
:::::::trade-off
::::::among
:::the
::::::period
::::::lengths,
::::the
::::::::statistical
:characteristics of the generated bit
sequences, and::the
:complexity of hardware implementations
in:::the
:fixed-point domain. As
::the
:control parameter is the
sole factor for SMN of:in
::::the
:::::SMN
::of
:::the
:Tent map under
::the
:given implementation environment, reference [19] only
select desired SMN via::::now
::::one
:::can
::::see
::::that
:::in
:::::[19]
:it
::::only
::::::selects
:::the
:::::::desired
::::::SMN
:::by
:choosing some control
parameters.• Increasing
:::the arithmetic precision
As discussed in [12], increasing::the
:arithmetic precision
can substantially enlarge the average length of the orbitof
::an
:SMN and enhance its complexity. However, from
Property ??, we can see that it:::such
::::::::::enhancing
:::::::method
cannot change its overall structure, which:as
:can be observed
in::::from
:Fig. 1. Figure 2 in [9] also confirms that increasing
::the
:precision does not always enlarge the average path period
of::an
:SMN. In addition, exhaustively searching smaller-scale
data incurred by:a lower arithmetic precision, e.g. binary16,
may also find::::::discover
:the rules found from the big data
generated by:a:higher-precision as
:in:
[9].• Perturbing states
Essentially, perturbing states::the
:::::state
:is to jump walk
from a path of::::from
::a:::::walk
::::path
:::in
:::an
:SMN to anoth-
er (namelyrewire:,::to
::::::rewire
::::the
:linking edges of SMN)
[50], [19], [20], [21], [51]::an
:::::::SMN)
::::::::::::::::::[50], [19], [20], [21].
To show:::the effect of this class
:::kind
:of methods, we perturb
SMN:::the
:::::SMN
::is
:::::::::perturbed
::as
:shown in Fig. 1b)
:, by the
method given in [50]and present the result .::::
The::::::
result::is
::::::::presented in Fig. 20a), where the perturbance
::::::::::perturbation
is performed by bit-wise XOR of the three least significantbits of the mapping value and the perturbing bit sequence(100)2. From Fig. 20a), we
:::one
:can see that the orbit starting
from some states, especially that in connected components ofsmall size, keep unchanged. The
::::sizes,
:::::::remains
::::::::::unchanged.
:A:
cycle may become even shorter after perturbing states .Essentially
::the
:::::states
:::are
:::::::::perturbed.
::::::::Generally, the enhancing
methods based on feed-back control:::::::proposed
:in [56], [16]
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 14
TABLE IIIREAL DIFFERENCE
:::::::::DIFFERENCES
:BETWEEN SMN IMPLEMENTED IN
:::THE TWO ARITHMETIC DOMAINS WITH n = 6, l = 3, m = 4.
i Fn(i) fn(i)− fl,m(i) 2−(m+1+j) i Fn(i) fn(i)− fl,m(i) 2−(m+1+j) i Fn(i) fn(i)− fl,m(i) 2−(m+1+j)
0 0 0 0.015625 16 30 0 0.015625 32 60 0 0.031251 2 0.013671875 0.015625 17 32 0.013671875 0.03125 34 56 0.00390625 0.031252 4 0.01171875 0.015625 18 34 0.02734375 0.03125 36 53 0.0078125 0.031253 6 0.009765625 0.015625 19 36 0.025390625 0.03125 38 49 0.01171875 0.031254 8 0.0078125 0.015625 20 38 0.0234375 0.03125 40 45 0.015625 0.031255 9 0.005859375 0.015625 21 39 0.021484375 0.03125 42 41 0.01953125 0.031256 11 0.00390625 0.015625 22 41 0.01953125 0.03125 44 38 0.0234375 0.031257 13 0.001953125 0.015625 23 43 0.017578125 0.03125 46 34 0.02734375 0.031258 15 0 0.015625 24 45 0.015625 0.03125 48 30 0 0.0156259 17 0.013671875 0.015625 25 47 0.013671875 0.03125 50 26 0.00390625 0.015625
10 19 0.01171875 0.015625 26 49 0.01171875 0.03125 52 23 0.0078125 0.01562511 21 0.009765625 0.015625 27 51 0.009765625 0.03125 54 19 0.01171875 0.01562512 23 0.0078125 0.015625 28 53 0.0078125 0.03125 56 15 0 0.01562513 24 0.005859375 0.015625 29 54 0.005859375 0.03125 58 11 0.00390625 0.01562514 26 0.00390625 0.015625 30 56 0.00390625 0.03125 60 8 0.0078125 0.01562515 28 0.001953125 0.015625 31 58 0.001953125 0.03125 62 4 0.01171875 0.015625
40 48
59
24
57
12
37
47
33
14 2930
3160
28
020 5446 1018
44
19
5
52
17
27
50
5626
39
1323
8
51
762
45
1642
955
41 4
1 63 22
25
6
21
15
4964
58
38
43
11
342
32
61
35 36
353
:a):
interacts with
interacts with
interacts with
interacts with
interacts with
interacts with
interacts with
interacts with
interacts with
interacts with
45
50
48
19
23
40
5
16
51
47
57
20
17
37
29
61
4412
59
43
39
1325
55
5227
229
833 36
14
35
34 26
1160
42 5641
0
18
64
30
15
4
5838
21
2
32
7
49
62
24
31
53
28
46
3
1
63
10
54
6
::b)
interacts with
interacts with
interacts with
interacts with
interacts with
interacts with
59
40
57
60
544
35
63
14 1
5
8
333031
36 47
39
49
64
46
11
16
52
38
23
44
19
2013
12
62
2
710 24
326
327
41
15
53
3418
61
48
50
21
22
559
26
4228
56
17
29
43
0
25
51
37
58
45
:c):
28 31
4618
2959
4
60
13
42
57
5
51
7
48
47
0
3
43
38
12
64
52
63 61
559 21
35
56
3630
2234
18
33
17
24
6
240
50
41
45
14
58
1520 44
6227
37 16
54
10 49
2311
26
19
39
53
32
25
::d)
Fig. 20.:::::Results
::of
:::::::enhancing
::::SMN
:::::shown
::in
:::Fig.
:::1b)
:::with
:::::various
:::::::methods:
::a)
:::::::perturbing
::::states
:::with
:::the
:::::method
::in:::[50]
:;::b)
:::::::perturbing
:::::control
:::::::parameter
:µ::::from
:::::121/25
::to::::::62/24;
::c)
::::::switching
::::with
::::SMN
::of
:::the
:::Tent
:::map
::::with
:::::::parameter
::::::::µ = 31/25
::::::::alternately;
::d)
:::::::cascading
:::with
::::SMN
::of:::
the:::Tent
::::map
:::with
::::::parameter
:::::::::µ = 31/25.
can be considered as perturbing nodes of:::the
::::::::::::corresponding
SMN adaptively.• Perturbing
:::the control parameters
Perturbing:::the
:control parameters is to walk from a path
of::an
:SMN corresponding to one control parameter to that
corresponding to another one by:::with
:a:
timely jump. So, thisclass
:::kind
:of methods is actually to cascade multiple SMN
:::::SMNs
:generated by the same chaotic map [52, Sec. 4]. To
visualize this strategy, we cascade SMN::::SMN
::::::::cascades
:::are
shown in Fig. 1b):,:::::along
:with that in Fig. 16b)and depict
the result:,:::and
::::the
:::::results
::::are
::::::::presented
:in Fig. 20b).
• Switching:::::among
:multiple chaotic maps
In each iteration, one:::the chaotic map is switched from some
candidates to:::one
::::::::candidate
:::to
:::::::another,
:::so
:::as
::to
:generate
the next state [54], [55]. From the viewpoint of SMN, theobtain
:::::::obtained orbit is to walk every SMN
::on
:::::every
:::::SMN
::for
:one step and
::::then jump to another SMN,
:depending on
the switching operation. As one state may be operated bydifferent chaotic maps, the out-degree
:::::::::out-degrees
:of some
nodes of the final SMN is may larger than one and:::may
:::be
:::::larger
::::than
:1:::but
:bounded by the number of available chaotic
maps. A demo on switching Logistic and Tent maps:::::::between
::the
::::::::Logistic
::::map
:::and
::::the
::::Tent
::::map
:is shown in Fig. 20c).
• Cascading::::::among multiple chaotic maps
This class::::kind
:of methods is to cascade some walks on
multiple SMN:::::SMNs
:into one walk [53]. The method given
::::used in [19] is an extreme case, where
:::the
:outputs of one
chaotic map are used to select:::the
:start of the path in
:::the
SMN of another chaotic map. Although two chaotic mapswere used, the final
:::::finally obtained SMN is not updated. A
demo::of
:::the
::::::SMN, on cascading two chaotic maps , SMN
shown in Fig. 1b) and Fig. 7 ,::::::::::respectively,
::is
:depicted in
Fig. 20d). The isolated::but
:connected components in the
original SMN can be connected in:::::::together
::::into
:the final
cascaded SMN once they own one pair of nodes connectedin any SMN.
::As
:::::above
:::::::::enhancing
::::::::methods
:::can
::be
:::::::::considered
:::to
::::make
:::the
::::::::dynamical
:::::::::properties
::of
:::an
:::::::existing
::::::chaotic
::::map
:::::::become
::::more
:::::::complex,
::::::SMN
:::can
:::::also
::be
:::::used
:::to
:::::::evaluate
:::its
:::::::::dynamical
:::::::::complexity
::as
::::did
::in
::::[62].
:
a) b) c) d) Results of enhancing SMN shown in Fig. 1b)with various methods: a) perturbing states with the methodin [50]; b) perturbing control parameter µ from 121/25 to62/24; c) switching with SMN of Tent map with parameterµ = 31/25 alternately; d) cascading with SMN of Tent mapwith parameter µ = 31/25.
V. CONCLUSION:::::::::::::CONCLUSIONS
This paper studied:::has
::::::studied
::::the
:dynamical properties
of digital chaotic maps with the methodology of complexnetworks. Some subtle properties of
:::the state-mapping network
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 15
of::::::::networks
::of
:::the Logistic map and Tent map were disclosed,
which offers a panorama of:::the
::::Tent
::::map
::::have
:::::been
:::::::revealed,
::::::offering
::a:::::::::panorama
::::with
:both microscopic and macroscopic
structure::::::::structures of the network. It demonstrated that
::has
::::been
:::::::::::demonstrated
::::that
:::the
:state-mapping network of digital
maps in:a::::::digital
::::map
:::in
::a:small-precision digital domain
can work as an efficient tool to show randomness of itsorbits. The analysis method can be
::for
:::::::::classifying
:::its
:::::::structure
:::and
::::::::coarsely
::::::verify
:::its
:::::::::::randomness.
:::::This
:::::::analysis
::::can
:::be
:::::further
:extended to higher-dimensional chaotic systems. Much
more properties on the state-mapping networks of variouschaotic systems deserve further exploration.
:::::::::::Achievements
:::::::::::::notwithstanding,
::::::more
:::::::::properties
:::::and
:::::::::::applications
:::of
:::the
:::::::::::state-mapping
:::::::network
::::::::::framework
::of
::::::various
::::::chaotic
:::::maps
:::call
::for
::::::further
::::::::::exploration
:::in
:::the
::::near
::::::future.
REFERENCES
[1] C. Li, D. Lin, B. Feng, J. Lu, and F. Hao, “Cryptanalysis of achaotic image encryption algorithm based on information entropy,” IEEEAccess, vol. 6, 2018.
[2] D. F. Voss, “Chaos on computers,” Science, vol. 246, no. 4934, pp.1172–1172, 1989.
[3] L. Kocarev, J. Szczepanski, J. M. Amigo, and I. Tomovski, “Discretechaos–I: Theory,” IEEE Transactions on Circuits and Systems–I: Regu-lar Papers, vol. 53, no. 6, pp. 1300–1309, Jun 2006.
[4] C. Li, D. Lin, J. Lu, and F. Hao, “Cryptanalyzing an image encryp-tion algorithm based on autoblocking and electrocardiography,” IEEEMultiMedia, vol. 25, no. 4, 2018.
[5] J. Oteo and J. Ros, “Double precision errors in the logistic map:Statistical study and dynamical interpretation,” Physical Review E,vol. 76, no. 3, p. 036214, 2007.
[6] Z. Galias, “The dangers of rounding errors for simulations and analysisof nonlinear circuits and systems–and how to avoid them,” IEEE Circuitsand Systems Magazine, vol. 13, no. 3, pp. 35–52, Aug 2013.
[7] S. C. Phatak and S. S. Rao, “Logistic map: A possible random-numbergenerator,” Physical Review E, vol. 51, no. 4, pp. 3670–3678, 1995.
[8] P.-M. Binder and N. H. Okamoto, “Unstable periodic orbits and dis-cretization cycles,” Physical Review E, vol. 68, no. 4, p. 046206, 2003.
[9] K. Persohn and R. Povinelli, “Analyzing logistic map pseudorandomnumber generators for periodicity induced by finite precision floating-point representation,” Chaos Solitons & Fractals, vol. 45, no. 3, pp.238–245, 2012.
[10] C. Grebogi, E. Ott, and J. A. Yorke, “Roundoff-induced periodicityand the correlation dimension of chaotic attractors,” Physical ReviewA, vol. 38, no. 7, p. 3688, 1988.
[11] S. Li, G. Chen, and X. Mou, “On the dynamical degradation of digitalpiecewise linear chaotic maps,” International Journal of Bifurcation andChaos, vol. 15, no. 10, pp. 3119–3151, 2005.
[12] T. Lin and L. O. Chua, “On chaos of digital filters in the real world,”IEEE Transactions on Circuits and Systems, vol. 38, no. 5, pp. 557–558,May 1991.
[13] L. Kocarev and G. Jakimoski, “Pseudorandom bits generated by chaoticmaps,” IEEE Transactions on Circuits and Systems I: FundamentalTheory and Applications, vol. 50, no. 1, pp. 123–126, Jan 2003.
[14] S. Callegari, R. Rovatti, and G. Setti, “Embeddable ADC-based truerandom number generator for cryptographic applications exploitingnonlinear signal processing and chaos,” IEEE Transactions on SignalProcessing, vol. 53, no. 2, pp. 793–805, Feb 2005.
[15] T. Addabbo, M. Alioto, A. Fort, S. Rocchi, and V. Vignoli, “Thedigital Tent map: performance analysis and optimized design as alow-complexity source of pseudorandom bits,” IEEE Transactions onInstrumentation and Measurement, vol. 55, no. 5, pp. 1451–1458, Oct2006.
[16] Y. Deng, H. Hu, W. Xiong, N. N. Xiong, and L. Liu, “Analysis anddesign of digital chaotic systems with desirable performance via feed-back control,” IEEE Transactions on Systems, Man, and Cybernetics:Systems, vol. 45, no. 8, pp. 1187–1200, Aug 2015.
[17] N. Masuda and K. Aihara, “Cryptosystems with discretized chaoticmaps,” IEEE Transactions on Circuits and Systems I: FundamentalTheory and Applications, vol. 49, no. 1, pp. 28–40, Jan 2002.
[18] S. M. Ulam and J. von Neumann, “On combination of stochastic anddeterministic processes,” Bulletin of the American Mathematical Society,vol. 53, no. 11, p. 1120, 1947.
[19] G. Heidari-Bateni and C. D. McGillem, “A chaotic direct-sequencespread-spectrum communication system,” IEEE Transactions on Com-munications, vol. 42, no. 234, pp. 1524–1527, Feb 1994.
[20] S.-L. Chen, T. Hwang, and W.-W. Lin, “Randomness enhancement usingdigitalized modified logistic map,” IEEE Transactions on Circuits andSystems II: Express Briefs, vol. 57, no. 12, pp. 996–1000, Dec 2010.
[21] C.-Y. Li, Y.-H. Chen, T.-Y. Chang, L.-Y. Deng, and K. To, “Period ex-tension and randomness enhancement using high-throughput reseeding-mixing PRNG,” IEEE Transactions on Very Large Scale Integration(VLSI) Systems, vol. 20, no. 2, pp. 385–389, Feb 2012.
[22] M. Jessa, “The period of sequences generated by Tent-like maps,”IEEE Transactions on Circuits and Systems I: Fundamental Theory andApplications, vol. 49, no. 1, pp. 84–89, Jan 2002.
[23] ——, “Designing security for number sequences generated by means ofthe sawtooth chaotic map,” IEEE Transactions on Circuits and Systems–I: Regular Papers, vol. 53, no. 5, pp. 1140–1150, May 2006.
[24] T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “Aclass of maximum-period nonlinear congruential generators derived fromthe Renyi chaotic map,” IEEE Transactions on Circuits and Systems I:Regular Papers, vol. 54, no. 4, pp. 816–828, Apr 2007.
[25] F. Chen, K.-W. Wong, X. Liao, and T. Xiang, “Period distribution of thegeneralized discrete Arnold Cat map for N = 2e,” IEEE Transactionson Information Theory, vol. 59, no. 5, pp. 3249–3255, May 2013.
[26] G. Alvarez and S. Li, “Some basic cryptographic requirements for chaos-based cryptosystems,” International Journal of Bifurcation and Chaos,vol. 16, no. 8, pp. 2129–2151, 2006.
[27] R. Sedgewick, T. G. Szymanski, and A. C. Yao, “The complexity offinding cycles in periodic functions,” SIAM Journal on Computing,vol. 11, no. 2, pp. 376–390, 1982.
[28] P. Flajolet and A. M. Odlyzko, “Random mapping statistics,” in Ad-vances in Cryptology – Crypto’89, ser. Lecture Notes in ComputerScience, vol. 434, 1990, pp. 329–354.
[29] I. Ozturk and R. Kilic, “Cycle lengths and correlation properties of finiteprecision chaotic maps,” International Journal of Bifurcation and Chaos,vol. 24, no. 09, p. 1450107, 2014.
[30] B. Yang and X. Liao, “Period analysis of the logistic map for the finitefield,” Science China Information Sciences, vol. 60, no. 2, p. 022302,2017.
[31] T. Miyazaki, S. Araki, and S. Uehara, “Some properties of Logistic mapsover integers,” IEICE Transactions on Fundamentals of Electronics,Communications and Computer Sciences, vol. 93, no. 11, pp. 2258–2265, 2010.
[32] T. Miyazaki, S. Araki, Y. Nogami, and S. Uehara, “Rounding logisticmaps over integers and the properties of the generated sequences,”IEICE Transactions on Fundamentals of Electronics, Communicationsand Computer Sciences, vol. 94, no. 9, pp. 1817–1825, 2011.
[33] S. Araki, H. Muraoka, T. Miyazaki, S. Uehara, and K. Kakizaki, “Adesign guide of renewal of a parameter of the logistic map over integerson pseudorandom number generator,” in International Symposium onInformation Theory and Its Applications. IEEE, 2016, pp. 781–785.
[34] J. Zhang and M. Small, “Complex network from pseudoperiodic timeseries: topology versus dynamics,” Physical Review Letters, vol. 96,no. 23, p. 238701, 2006.
[35] P. M. Binder and R. V. Jensen, “Simulating chaotic behavior with finite-state machines,” Physical Review A, vol. 34, no. 5, pp. 4460–4463, 1986.
[36] P. Binder, “Limit-cycles in a quadratic discrete iteration,” Physica D,vol. 57, no. 1-2, pp. 31–38, 1992.
[37] N. Marwan, J. F. Donges, Y. Zou, R. V. Donner, and J. Kurths, “Complexnetwork approach for recurrence analysis of time series,” Physics LettersA, vol. 373, no. 46, pp. 4246–4254, 2009.
[38] R. V. Donner, J. Heitzig, J. F. Donges, Y. Zou, N. Marwan, and J. Kurths,“The geometry of chaotic dynamics–a complex network perspective,”European Physical Journal B, vol. 84, no. 4, pp. 653–672, 2011.
[39] J. F. Donges, J. Heitzig, R. V. Donner, and J. Kurths, “Analyticalframework for recurrence network analysis of time series,” PhysicalReview E, vol. 85, no. 4, p. 046105, 2012.
[40] A. Shreim, P. Grassberger, W. Nadler, B. Samuelsson, J. E. S. Socolar,and M. Paczuski, “Network analysis of the state space of discretedynamical systems,” Physical Review Letters, vol. 98, no. 19, p. 198701,2007.
[41] C. Xu, C. Li, J. Lu, and S. Shu, “On the network analysis of thestate space of discrete dynamical systems,” International Journal ofBifurcation and Chaos, vol. 27, no. 4, p. art. no. 1750062, 2017.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 16
[42] B. Luque, L. Lacasa, F. J. Ballesteros, and A. Robledo, “Feigenbaumgraphs: a complex network perspective of chaos,” PLoS One, vol. 6,no. 9, p. art. no. e22411, 2011.
[43] T. Iba, “Scale-free networks hidden in chaotic dynamical systems,”arXiv:nlin/1007.4137, 2010.
[44] E. Borges, D. Cajueiro, and R. Andrade, “Mapping dynamical systemsonto complex networks,” The European Physical Journal B, vol. 58,no. 4, pp. 469–474, 2007.
[45] F. Kyriakopoulos and S. Thurner, “Directed network representation ofdiscrete dynamical maps,” in International Conference on Computation-al Science, ser. Lecture Notes in Computer Science, vol. 4488. Springer,2007, pp. 625–632.
[46] T. Iba, “Hidden order in chaos: The network-analysis approach todynamical systems,” in Proceedings of the Eighth International Confer-ence on Complex Systems, 2011, pp. 769–783, http://necsi.edu/events/iccs2011/papers/70.pdf.
[47] X. Xu, J. Zhang, and M. Small, “Superfamily phenomena and motifsof networks induced from time series,” Proceedings of the NationalAcademy of Sciences, vol. 105, no. 50, pp. 19 601–19 605, 2008.
[48] M. A. Dastgheib and M. Farhang, “A digital pseudo-random numbergenerator based on sawtooth chaotic map with a guaranteed enhancedperiod,” Nonlinear Dynamics, vol. 89, no. 4, pp. 2957–2966, 2017.
[49] C. Li, T. Xie, Q. Liu, and G. Cheng, “Cryptanalyzing image encryptionusing chaotic Logistic map,” Nonlinear Dynamics, vol. 78, no. 2, pp.1545–1551, 2014.
[50] T. Sang, R. Wang, and Y. Yan, “Perturbance-based algorithm to expandcycle length of chaotic key stream,” Electronics Letters, vol. 34, no. 9,pp. 873–874, 1998.
[51] L. Liu, J. Lin, S. Miao, and B. Liu, “A double perturbation method forreducing dynamical degradation of the digital baker map,” InternationalJournal of Bifurcation and Chaos, vol. 27, no. 7, p. art. no. 1750103,2017.
[52] J. Cernak, “Digital generators of chaos,” Physics letters A, vol. 214, no.3-4, pp. 151–160, 1996.
[53] Z. Hua and Y. Zhou, “One-dimensional nonlinear model for producingchaos,” IEEE Transactions on Circuits and Systems I: Regular Papers,vol. 65, no. 1, pp. 235–246, 2018.
[54] N. Nagaraj, M. C. Shastry, and P. G. Vaidya, “Increasing average periodlengths by switching of robust chaos maps in finite precision,” TheEuropean Physical Journal Special Topics, vol. 165, no. 1, pp. 73–83,2008.
[55] Y. Wu, Y. Zhou, and L. Bao, “Discrete wheel-switching chaotic systemand applications,” IEEE Transactions on Circuits and Systems I-RegularPapers, vol. 61, no. 12, pp. 3469–3477, Dec 2014.
[56] T. Addabbo, M. Alioto, A. Fort, S. Rocchi, and V. Vignoli, “A feedbackstrategy to improve the entropy of a chaos-based random bit generator,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 53,no. 2, pp. 326–337, 2006.
[57] S. Wang, W. Liu, H. Lu, J. Kuang, and G. Hu, “Periodicity ofchaotic trajectories in realizations of finite computer precisions and itsimplication in chaos communications,” International journal of modernphysics B, vol. 18, no. 17, pp. 2617–2622, 2004.
[58] D. Zuras and et al., “IEEE standard for floating-point arithmetic,” IEEEStd 754-2008, pp. 1–70, 2008.
[59] C. Rau, “Half-precision floating point library,” http://half.sourceforge.net/index.html, 2017.
[60] A. Rukhin and et al., “A statistical test suite for random and pseudoran-dom number generators for cryptographic applications,” NIST SpecialPublication 800-22rev1a, 2010, available online at http://csrc.nist.gov/groups/ST/toolkit/rng/documentation software.html.
[61] P. L. Ecuyer and R. Simard, “TestU01: A C library for empirical testingof random number generators,” ACM Transactions on MathematicalSoftware, vol. 33, no. 4, p. art. no. 22, 2007.
[62] Q. Wang, S. Yu, C. Li, and et al., “Theoretical design and FPGA-basedimplementation of higher-dimensional digital chaotic systems,” IEEETransactions on Circuits and Systems I: Regular Papers, vol. 63, no. 3,pp. 401–412, Mar 2016.
Chengqing Li (M’07-SM::::07–SM’13) received his
M.Sc.degree in Applied Mathematics:::::degree
::in
:::::applied
::::::::mathematics
:from Zhejiang University, Chi-
na in 2005 and his Ph.D.degree in ElectronicEngineering
:::::degree
::in:::::::electronic
::::::::engineering
:from
City University of Hong Kong in 2008. Thereafter,he had been working
:::::worked
:as a Postdoctoral
Fellow at The Hong Kong Polytechnic Universitytill September 2010. Then, he worked at the Collegeof Information Engineering, Xiangtan University,Chinaas an Associate Professor and Full Professor
recently. From April 2013 to July 2014, he worked at::the
:University of
Konstanz, Germany, under the support of the Alexander von HumboldtFoundation. He is serving as an associate editor for International Journal ofBifurcation and Chaos. Dr.
:::Since
:::::April
::::2018,
::he
:::has
:::been
::::::working
::::with
::the
::::School
::of:::::::
Computer::::::
Science:::and
:::::::Electronic
:::::::::Engineering,
:::::Hunan
:::::::University,
::::China
::as
:a:::full
:::::::professor.
:::Prof. Li focuses on security analysis of multimedia encryption schemes and
privacy protection schemes::::::dynamics
::::::analysis
::of
::::digital
::::::chaotic
:::::systems
:::and
:::their
::::::::applications
::in
::::::::multimedia
:::::security. He has published about fifty papers
on the:::focal
:subject in the past 13 years, receiving more than 1800
:::2500
citations with h-index 23.::28.
:
Bingbing Feng is pursuing his:::::obtained
:::his
::::B.Sc.
::::degree
:::and
:M.Sc.degree
:::::degree
:::both
:in computer
science at::::from the College of Information Engi-
neering, Xiangtan University, where he received hisB.Sc. degree in the same subject in 2015.
::::China
::in
:::2015
:::and
::::2018,
:::::::::respectively.
His research interests include complex networksand nonlinear dynamics.
Shujun Li (M’08-SM::::08–SM’12) received the B.E.
degree in information science and engineering, andthe Ph.D.
:degree in information and communication
engineering from Xi:’an Jiaotong University, Chi-
na, in 1997 and 2003, respectively. From 2003 to2005
:::2007, he was a Post-Doctoral Research Assis-
tant with The::the
:City University of Hong Kong.
From 2005 to 2007, he was ,:::
and::::
then:
a Post-Doctoral Fellow with the Hong Kong PolytechnicUniversity. From 2007 to 2008, he was a HumboldtResearch Fellow with the FernUniversita
:::::::conducting
:::::visiting
::::::research
::at
:::the
::::::::::FernUniversit‘at, Hagen, Germany
:,::as
:a:::::::
Humboldt
::::::Research
:::::Fellow. From 2008 to 2011, he was a Zukunftskolleg Fellow with
the Universita:‘at Konstanz, Germany. He is currently a Reader with
:In::::
2011,
::he
::::joined
:::::::University
::of
:::::Surrey,
:::UK,
:::::initially
::as
:a:::::Senior
::::::Lecturer
::and
::::then
::was
::::::promoted
::to
:::::Reader
::in
::::2017.
::::Since
:::::::November
::::2017,
::he:::
has::::been
:a::::::Professor
:of:::::
Cyber::::::Security
:at:
the University of Surrey, U.K., and a Deputy Directorywith the Surrey Centre for Cyber Security . His
::::Kent,
:::UK,
:::and
::::::directing
::the
:::Kent
:::::::::::Interdisciplinary
:::::::Research
:::::Centre
::in
:::::Cyber
::::::Security
:::::::(KirCCS),
:a:::
UK
::::::::government
:::::::recognized
::::::::Academic
:::::Centre
::of
::::::::Excellence
::in
::::Cyber
::::::Security
::::::Research
:::::::::(ACE-CSR).
:::Prof.
:::Li’s
:current research interests include
:::::mainly
::::focus
::on
:::::::interplays
:::::between
:::::several
:::::::::::interdisciplinary
::::::research
::::areas
:::::::including cyber security and
privacy, digital and multimedia forensics, human-centered::::::::cybercrime,
::::human
:::::factors,
::::::::multimedia computing, and visual quality assessment.
::::digital
::::chaos.
::He
::is
:a:::::Fellow
::of:::
the::::BCS
:–:::
The:::::::
Chartered::::::
Institute:::
for::IT
:::and
:::the
:::Vice
::::::President
:::for
::::::Internal
:::::::::::Communications
::&:::::
Public:::::::
Relations:::
of::the
:::::ABCP
::::::::(Association
::of
:::::British
::::::Chinese
::::::::Professors).
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 17
Jurgen Kurths received the Ph.D. degree from theGDR Academy of Sciences, Berlin, Germany, in1983. He was a Full Professor at the University ofPotsdam from 1994 to 2008. He has been a Professorof nonlinear dynamics at Humboldt University ofBerlin, Berlin, and the Chair of the research domainTransdisciplinary Concepts of the Potsdam Institutefor Climate Impact Research, Potsdam, Germany,since 2008. He has authored or coauthored morethan 500 papers that are cited more than 18,000times (H-index
:::::h-index: 57). His primary research
interests include synchronization, complex networks, and time-series analysisand their applications. He received an honorary doctorate in 2008 from N.I. Lobachevsky State University of Nizhny Novgorod and one in 2012 fromSaratov State University. He
::He became a member of the Academia Europaea
::::::Academy
::of
:::::Europe
:in 2010 and of the Macedonian Academy of Sciences
and Arts in 2012. He is the Editor-in-chief of Chaos.:::::CHAOS.
:
:::Prof.
:::::Kurths’
::::::primary
::::::research
::::::interests
:::::include
:::::::::::synchronization,
::::::complex
::::::networks,
:::and
::::::::time-series
:::::analysis
:::and
::::their
::::::::applications.
:
Guanrong Chen (M’89–SM’92–F’97::::::97–LF’19) re-
ceived the:::
MSc.:::::degree
::in
:::::::computer
:::::science
::::from
::Sun
::::::Yat-sen
:::::::University,
:::::::::Guangzhou,
::::China
::in::::1981
::and
:::the
:Ph.D.
:degree in applied mathematics from
Texas A&M University, Texas, TX, USA , in 1987.He has been a Chair Professor and the Director ofthe Centre for Chaos and Complex Networks, CityUniversity of Hong Kong, Hong Kong , since 2000,prior to that , he was a tenured Full Professor withthe University of Houston, Houston, TX, USA. Dr.
:::Prof. Chen was a recipient of the 2011 Euler Gold
Medal, Russia, and the Highly Cited Researcher Award in Engineering as wellas in Mathematics
::in
::::::::Engineering
:::::named by Thomson Reuters, and conferred
Honorary Doctorate by the Saint Petersburg State University, Russia , in 2011and by the University of Le Havre, Normandie, France , in 2014. He is amember of the Academia
::::::Academy
:of Europe and a fellow of The World
Academy of Sciences.