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Framework for the analysis and designof encryption strategies
based on discrete-timechaotic dynamical systems
David Arroyo Guardeno
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From chaos to cryptography
1
Why?
2
How?
Criticalcontexts
3
Design Rules 3
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Perfect secrecy
Good mixingproperties. . .
Hopf: doughrolling andfolding. . .
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Sensitivity
Initial condition
Controlparameter
Diffusion
Mixing Ergodicity Confusion
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ENCRYPTION
T = R
Chaos incontinuous time
T = Z
Chaos indiscrete time
Chaos incontinuous time
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ENCRYPTION
T = R
Chaos incontinuous time
Synchronization
T = Z
Chaos indiscrete time
Chaos incontinuous time
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ENCRYPTION
T = R
Chaos incontinuous time
Synchronization
Security problems
T = Z
Chaos indiscrete time
Chaos incontinuous time
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ENCRYPTION
T = R
Chaos incontinuous time
Synchronization
Security problems
T = Z
Chaos indiscrete time
Chaos incontinuous time
DifferentialEquations
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ENCRYPTION
T = R
Chaos incontinuous time
Synchronization
Security problems
T = Z
Chaos indiscrete time
Chaos incontinuous time
DifferentialEquations
Dimension > 2
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ENCRYPTION
T = R
Chaos incontinuous time
Synchronization
Security problems
T = Z
Chaos indiscrete time
Chaos incontinuous time
DifferentialEquations
Dimension > 2
Efficiency problems
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ENCRYPTION
T = R
Chaos incontinuous time
Synchronization
Security problems
T = Z
Chaos indiscrete time
Chaos incontinuous time
DifferentialEquations
Dimension > 2
Efficiency problems
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How to design
secure digital
chaos-based cryptosystems
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Avoid critical contexts
Conventional cryptography
Standards
Commitments
Conventional attacks
Chaos theory
Loss of chaoticity
Reconstruction of the
underlying dynamics
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Avoid critical contexts
Conventional cryptography
Standards
Commitments
Conventional attacks
Chaos theory
Loss of chaoticity
Reconstruction of the
underlying dynamics
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1
Why?
2
How?
Criticalcontexts
Loss of chaoticity
3
Design Rules 3
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For xk+1 = f (λ ,xk) = fλ(xk)
it can not be assumed
chaos for all λ
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C. Chee and D.Xu,“Chaotic encryption using discrete-time synchronous chaos,” Physics
Letters A, 2006, 348, 284-292
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xk+1 =
[uk+1
vk+1
]=
[1−δ ·u2
k +vk
β ·vk
]
δ = ψ (pk) ·µ1 (vk)
β = µ2 (vk)
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−0.4 −0.2 0 0.2 0.4
1.2
1.4
1.6
1.8
2
β
δ
Periodic
Unbounded
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0 0.5 1 1.5 2 2.5 3
x 1014
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Plaintext block values
Asy
mpt
otic
val
ues
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David Arroyo et al.,“Cryptanalysis of a discrete-time syn-chronous chaotic encryption system,”
Physics Letter A, 2008, 372, 1034-1039
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1
Why?
2
How?
Criticalcontexts
Reconstruction of dynamics
3
Design Rules 3
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Estimation of λ and/or x0 after applyingconventional attacks
1 Access to chaotic orbits2 We can measure the entropy of the
underlying chaotic map3 Access to samples of chaotic orbits4 Access to coarse-grained versions of
chaotic orbits
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xi+1
xia bxc
xi+1 = f (xi)
Orbit : {x0,x1, . . .}f (a) = f (b), f (xc)≤ b
xc = Single turning point
f continuous in [a,b]
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xi+1
xi
Logistic map: xi+1 = λxi(1−xi)
λ
0 1xc
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xi+1
xi
Skew tent map: xi+1 =
{xi/λ 0 < xi < λ
(1−xi)/(1−λ ) λ ≥ xi < 1
λ
0 1
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Access to chaotic orbits
Ciphertext is a function of a chaotic orbit
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Access to chaotic orbits
Ciphertext is a function of a chaotic orbit
Only the chaotic orbit is secret
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Access to chaotic orbits
Ciphertext is a function of a chaotic orbit
Only the chaotic orbit is secret
Kerckhoff’s principle:we know the function and
xn+1 = f (λ ,xn),xn ∈ Rm
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Access to chaotic orbits
Ciphertext is a function of a chaotic orbit
Only the chaotic orbit is secret
Kerckhoff’s principle:we know the function and
xn+1 = f (λ ,xn),xn ∈ Rm
Estimation of λ from m +1 units of ciphertext
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B. Ling et al.,“Chaotic filter bank for computercryptography,” Chaos, Solitons
and Fractals, 2007, 34, 817-824
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Plaintext: {pn}
tn = K ∑∀j
pjh2n−j
t ′n = K ′∑∀j
pjh′2n−j
vn = tn + t ′n +sn
v ′n = t ′n−vn−s′n
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Plaintext: {pn}
tn = K ∑∀j
pjh2n−j
t ′n = K ′∑∀j
pjh′2n−j
vn = tn + t ′n +sn
v ′n = t ′n−vn−s′nLogistic map
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Plaintext: {pn}
tn = K ∑∀j
pjh2n−j
t ′n = K ′∑∀j
pjh′2n−j
vn = tn + t ′n +sn
v ′n = t ′n−vn−s′n
Ciphertext: {vn} ,{v ′n}, Key: λ ,λ ′,s0,s′0
Logistic map
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Known-plaintext attack: {pn}, {vn}, {v ′n}
sn = vn− tn− t ′ns′n = t ′n−vn−v ′n
λ =sn+1
sn(1−sn)
λ′ =
s′n+1s′n(1−s′n)
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David Arroyo et al., “Cryptanalysisof a computer cryptography schemebased on a filter bank,” Chaos, Soli-tons and Fractals, 2009, 41, 410-413
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1
Why?
2
How?
Criticalcontexts
Entropy of the underlying chaotic map
3
Design Rules 3
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Entropy
Orbit⇒ Probability distribution
Discretization ofthe phase space
Discretization in thefrequency domain
Relative energy ofresolution levels
Relative number ofvalues in subintervals
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n-gram conditional entropySplit the phase space into J disjoint intervals
Convert chaotic orbits into sequences of symbols
Group the symbols into words of length n
pr (n)i : probability of i-th word, 0≤ i ≤ Jn
Hn =−∑Jn
i=1 pr (n)i logpr (n)
i
hn = Hn+1−Hn, h0 = H1
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Conditional entropy of the logistic map
3.5 3.6 3.7 3.8 3.9 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ
h nn=4n=6n=8n=10n=12
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Conditional entropy of the skew tent map
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ
h n
n=4n=6n=8n=10n=12
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Multiresolution Entropy
1000 2000 3000 4000 5000 6000 7000 8000 90000
0.2
0.4
MR
ET
1
λ=3.5λ=3.8123λ variable
1000 2000 3000 4000 5000 6000 7000 8000 90000
0.2
0.4
MR
ET
2
λ=3.5λ=3.8123λ variable
1000 2000 3000 4000 5000 6000 7000 80000
0.2
0.4
Temporal variable
MR
ET
3
λ=3.5λ=3.8123λ variable
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High level of entropy
without leaking
the values of λ
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1
Why?
2
How?
Criticalcontexts
Samples of chaotic orbits
3
Design Rules 3
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Shape of histogramsof chaotic orbitsdepending on λ
Sampling on chaotic orbits
Estimation of λ
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A.N. Pisarchik et al. “Encryp-tion and decryption of images
with chaotic map lattices,” Chaos,2006, 16, Art. No. 033118
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Logistic map, xmin = λ 2
4 (1− λ
4 ), xmax = λ
4 , plaintext {pi}Ji=1
r = 1,{
y0i
}= {pi}
x0 =
{y r−1
J if i = 1y r
i i .o.c
Iterate n times the logistic map from x0 to get xn
y ri = xn +y r−1
i and subtract xmax −xmin until y ri ∈ [xmin,xmax ]
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x0 =
{y r−1
J if i = 1y r
i i .o.c
Iterate n times the logistic map from x0 to get xn
y ri = xn +y r−1
i and subtract xmax −xmin until y ri ∈ [xmin,xmax ]
r = r +1
r < R
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
λ/4λ2(1−λ/4)
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Ciphertext-only attack
xmax = max{
yRi
}
λ ≈ λ = 4 · xmax
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David Arroyo et al., “On the securityof a new image encryption scheme
based on chaotic map lattices,”Chaos, 2008, 18, Art. No. 033112
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1
Why?
2
How?
Criticalcontexts
Coarse-grained versions of chaotic orbits
3
Design Rules 3
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Assign a partition to the phase space
1 Stream cipher2 Searching based chaotic ciphers
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Stream cipherxi+1
xia bxcxL
i xRi
xi+1
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Stream cipherxi+1
xia bxcx0
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Stream cipherxi+1
xia bxcx0
L
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Stream cipherxi+1
xia bxc
xi+1 = xi
x0 x1
L R
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Stream cipherxi+1
xia bxc
xi+1 = xi
x0 x1x2
L R R
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Stream cipherxi+1
xia bxc
xi+1 = xi
x0 x1x2
0 1 1 ... Binary sequence
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A.P. Kurian and S. Puthusserypady,“Self-synchronizing chaoticstream ciphers,” Signal Pro-
cessing, 2008, 88, 2442-2452
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Logistic map
Skew tent map
≥ xc
⊕Shuffler
Plaintext
CiphertextBks
Binit
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Logistic map
Skew tent map
≥ xc
⊕Shuffler
0
Bsh = π(Binit||Bks) =Bsh(λ, x0)
Bks Bks
Binit
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Chosen-plaintext attack
Bsh(λ ,x0)⇒ Pr1 ={
pr (1)j
}2N
j=1
Bks(λ i ,xk)⇒ Pr(i ,k) ={
pr (i ,k)j
}2N
j=1
Wootters’ distance
DW (Pr1,Pr(i ,k)) = cos−1
(2N
∑j=1
√pr (1)
j ·pr (i ,k)j
)
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0 0.2 0.4 0.6 0.8 10
0.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
λ
Woo
tters
’ dis
tanc
e
x0
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3.83.85
3.93.95
0.2
0.4
0.6
0.8
1
1.1
1.2
1.3
1.4
1.5
λ
x0
Woo
tters
’ dis
tanc
e
![Page 66: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/66.jpg)
David Arroyo et al.,“Cryptanalysis of a family of self-
synchronizing chaotic streamciphers”, Submitted to Signal
Processing on 17 March, 2009
![Page 67: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/67.jpg)
1
Why?
2
How?
Criticalcontexts
Coarse-grained versions of chaotic orbits
3
Design Rules 3
![Page 68: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/68.jpg)
Searching based chaotic ciphersP
hase
spac
eP
laintextalphabet
a1
a2
ak
a|A|
Partition
![Page 69: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/69.jpg)
Searching based chaotic ciphersP
hase
spac
eP
laintextalphabet
ak
f Mλ (x0 )
M=ciphertext
![Page 70: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/70.jpg)
f (0)(x)
xa bxc
0 1
![Page 71: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/71.jpg)
f (x)
x
xc
a bxc
00 01 11 10
![Page 72: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/72.jpg)
f (2)(x)
x
xc
a bxc
000 001 011 010110 111 101 100
![Page 73: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/73.jpg)
X. Wang et al.,“A new chaotic cryptography based
on ergodicity,” International Journal ofModern Physics B, 2008, 22, 901-908
![Page 74: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/74.jpg)
Logistic map: x0 and λ secret key
pi is a word with w bits
Ciphertext: number ofiterations to find pi in the
binary sequence generatedfrom the logistic map
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Symbolic dynamics of unimodal maps
Chosen-ciphertext attack
![Page 76: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/76.jpg)
Gray Ordering NumberGM(λ ,x) = g0g1 · · ·gM−1, gi ∈ {0,1}gi = 0⇔ f (i)
λ(x) < xc
gi = 1⇔ f (i)λ
(x)≥ xc
g0 b0
b1g1
b2
bM−1
g2
gM−1
GON(GM(λ ,x)) = 2−1 ·b1 +2−2 ·b2 + . . .+2−(n−1) ·bn−1
![Page 77: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/77.jpg)
GON for the logistic map
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
GO
N(P
f λn(x
))
λ=3.4
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GON for the logistic map
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
GO
N(P
f λn(x
))
λ=3.6
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GON for the logistic map
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
GO
N(P
n f λ(x))
λ=3.8
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GON for the logistic map
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
GO
N(P
f λn(x
))
λ=4
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GON for the logistic map and x0 = fλ(xc)
3 3.2 3.4 3.6 3.8 40.65
0.7
0.75
0.8
0.85
0.9
0.95
1G
ON
(Pf λn(f
λ(xc))
)
λ
![Page 82: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/82.jpg)
GON for the logistic map and x0 = fλ(xc)
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Binary sequence of length N
Sliding window of length M and compute GON
Estimation of λ through a binary search from the maximum GON
GONM(λ , λ
4 ) = GONmax
Estimation of x0 using the estimation of λ and the binary sequence
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Chosen-ciphertext attack
Ask for the decryption of w · i
0 returns the first w bits,w the following w bits, . . .
GM(x0,λ )⇒ λ ,x0
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Parameter estimation error
0 2 4 6 8 10
x 105
10−12
10−10
10−8
10−6
10−4
c es
timat
ion
erro
r (L
ogar
ithm
ic s
cale
)
M
![Page 86: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/86.jpg)
Error in the estimation of the initialcondition
10 20 30 40 50 6010
−20
10−15
10−10
10−5
100
x 0 est
imat
ion
erro
r (L
ogar
ithm
ic s
cale
)
N
![Page 87: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/87.jpg)
David Arroyo et al.,“Cryptanalysis of a new chaotic
cryptosystem based on ergodicity,”International Journal of ModernPhysics B, 2009, 23, 651-659
![Page 88: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/88.jpg)
1
Why?
2
How?
Criticalcontexts
Searching based chaotic ciphers: unimodal maps
3
Design Rules 3
![Page 89: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/89.jpg)
Previous attack only works if
GONM(λ , fλ(xc))
depends on
on the control parameter
![Page 90: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/90.jpg)
Is the cryptosystem secure
if the logistic map
is replaced by
the skew tent map?
![Page 91: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/91.jpg)
David Arroyo et al., “Estimationof the control parameter from
symbolic sequences: Unimodalmaps with variable critical point,”
Chaos, 2009, 19, Art. No. 023125
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λ can be estimatedfrom the PDF oforder patterns
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xi+i = f (xi)
[x0,x1,x2, . . . ,xL−1]
π(x0) = [π0,π1, . . . ,πL−1]
πi permutation |πi 7→ i
f π0(x0) < f π1(x0) < · · ·< f πL−1(x0)
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xi+1
xi
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
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xi+1
xi
[0.31225,
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 96: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/96.jpg)
xi+1
xi
[0.31225,
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 97: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/97.jpg)
xi+1
xi
[0.31225,0.6245
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 98: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/98.jpg)
xi+1
xi
[0.31225,0.6245
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 99: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/99.jpg)
xi+1
xi
[0.31225,0.6245,0.751,
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 100: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/100.jpg)
xi+1
xi
[0.31225,0.6245,0.751,
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 101: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/101.jpg)
xi+1
xi
[0.31225,0.6245,0.751,0.498]
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 102: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/102.jpg)
xi+1
xi
[0.31225,0.6245,0.751,0.498]⇒ π(0.31225) = [0,3,1,2]
f : [0,1]→ [0,1],xi+1 = f (xi) =
{2xi , 0 < xi < 0.52(1−xi), 0.5≥ xi < 1
0 1
![Page 103: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/103.jpg)
The intersections between
f 0(x), f 1(x), . . . , f L−1(x)
determine intervals
with initial conditions
leading to the same order pattern
![Page 104: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/104.jpg)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f0(x)f1(x)
f2(x)f3(x)
![Page 105: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/105.jpg)
Order patterns
can be used to assign a partition
to the definition domain
![Page 106: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/106.jpg)
fλ : I→ I, I ⊂ R, λ ∈ J ⊂ R
Pπ = {x ∈ I : x generates the order pattern π}
Pπ depends on λ through fλ
![Page 107: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/107.jpg)
xi+1
xi
Skew tent map: xi+1 =
{xi/λ , 0 < xi < λ
(1−xi)/(1−λ ), λ ≥ xi < 1
λ
0 1
![Page 108: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/108.jpg)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
f λ (k) (x
)
fλ(1)(x)
fλ(2)(x)
fλ(3)(x)
fλ(0)(x)
[0,1,2,3]
�
[0,1,3,2]
�[0,3,1,2]
�[3,0,1,2]
�
[0,3,1,2]
�
[0,2,1,3]
�[2,3,0,1]
�
[2,0,3,1]
�
[2,0,1,3]
�[3,1,0,2]
�[1,3,2,0]
�
[1,2,3,0]
�
[1,2,0,3]
�[1,2,3,0]�
![Page 109: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/109.jpg)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
f λ (k) (x
)
fλ(3)(x)
fλ(2)(x)
fλ(1)(x)
fλ(0)(x)
[0,1,2,3]
�
[0,1,3,2]
�
[0,3,1,2]
�[3,0,1,2]
�[0,3,1,2]
�
[0,2,1,3]
�
[2,0,3,1]
�[2,3,0,1]
�
[2,0,3,1]
�
[2,0,1,3]
�
[3,1,0,2]
�[1,3,2,0]
�[1,2,3,0]�[1,2,0,3]
[1,2,3,0]
�
![Page 110: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/110.jpg)
Order pattern [0,1, . . . ,L−1]
determined by the
leftmost intersection
of the iterates f L−2λ
and f L−1λ
![Page 111: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/111.jpg)
fλ ergodic with invariant measure µ
Ofλ (x) = {f n(x) : n ∈N∪{0}}
Ofλ (x) visits Pπ withrelative frequency µ(Pπ)
![Page 112: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/112.jpg)
Orbit of length M
Sliding window of width L
M−L+1 order L-patterns
Compute the relative fre-quency of each order pattern
![Page 113: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/113.jpg)
For some fλ(x)
1-to-1 relation between
the relative frequency
of some order pattern
and the control parameter λ
![Page 114: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/114.jpg)
Skew tent map
f nλ(x) =
{x/λ n, if 0≤ x ≤ λ n
(λ n−1−x)/λ n−1(1−λ ), if λ n ≤ x ≤ λ n−1
P[0,1,...,L−1] = (0,φL(λ )), with
φL(λ ) =λ L−2
2−λ
![Page 115: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/115.jpg)
L = 4⇒ φ4 = λ 2
2−λ
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
Ord
er p
atte
rn fr
eque
ncy
![Page 116: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/116.jpg)
Skew tent map
Unimodal map
x1 < x2⇒G(x1)≤G(x2)
Order patterns from “coarse-grained” orbits
![Page 117: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/117.jpg)
Error in the estimation of λ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
−4
10−3
10−2
λ
Mea
n er
ror
valu
e (L
ogar
ithm
ic s
cale
)
![Page 118: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/118.jpg)
Finite precision arithmetics
Digital degradation of dynamics
Non-perfect recovery of λ
![Page 119: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/119.jpg)
1
Why?
2
How?
Criticalcontexts
3
Design Rules 3
![Page 120: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/120.jpg)
Digital chaos-based cryptosystem
Encryption architecture
Stream cipher
Linear complexity
Correlation attacks
. . .
Block cipher
Differential attack
Linear attacks
. . .
Chaotic map
Loss of chaoticity
Bijections in entropy measures
Leaking of the underlying order
Defective probability distribution
![Page 121: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems](https://reader034.fdocuments.us/reader034/viewer/2022042814/5550b655b4c905fa618b4b13/html5/thumbnails/121.jpg)
Design rules I
1 Assure the chaotic behavior of theunderlying dynamical systems
2 Guarantee avalanche effect3 High level of entropy without leaking of
the values of control parameters4 Definition of the ciphertext avoiding the
reconstruction of the underlying chaoticdynamics
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Design rules II
5 Chaotic maps with flat histograms andwidth of the phase space independent ofthe control parameters
6 Selection of chaotic maps with highsensitivity to control parameter mismatch
7 The number of iterations of chaotic mapscan not be part of the key
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0 50 1000
50
100
150Control parameter a=3.8204607418
Tim
e in
sec
onds
n × j
0 50 1000
50
100
150Control parameter a=3.8294707872
Tim
e in
sec
onds
n × j
0 50 1000
50
100
150Control parameter a=3.8743936381
Tim
e in
sec
onds
n × j0 50 100
0
50
100
150Control parameter a=3.9771765651
Tim
e in
sec
onds
n × j
j=1
j=2
j=3
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David Arroyo et al.,“On the security of a new image
encryption scheme based onchaotic map lattices,” Chaos,2008, 18, Art. No. 033112
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SCI
Chaos-basedcryptography 5
Unimodalmaps 7
CONFERENCES
International 8
National 8
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Future work
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Problems detected in unimodal maps
Multimodal maps
Discrete chaos
Other sources of chaos
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Chaotic map
Encryptionarchitecture
Practicalimplementation
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Design ofchaos-based cryptosystems
needs of cryptography+
analysis of chaotic dynamics
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Framework for the analysis and designof encryption strategies
based on discrete-timechaotic dynamical systems
http://hdl.handle.net/10261/15668