Hugo Solis-Sanchez Elena Gabriela Barrantes Universidad de

Hugo Solis-Sanchez Elena Gabriela Barrantes Universidad de

Using Coupled Maps from
the Nature for PKC
Hugo Solis-Sanchez
Elena Gabriela Barrantes
Universidad de Costa Rica
Coupled Maps
Coupled Maps
Populations
Chemical Reactions
Bio-Systems
Oscillations
VOLUME 90, N UMBER 4
PHYSICA L R EVIEW LET T ERS
week ending
31 JANUARY 2003
Using Distributed Nonlinear Dynamics for Public Key Encryption
Roy Tenny,1,2 Lev S. Tsimring,1 Larry Larson,2 and Henry D. I. Abarbanel1,3
1
Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402
Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, California 92093-0354
3
• The equations
wereofused
in simulationUniversity
of DDE. of California,
Department of Physics and Marine Physical Laboratory
(Scripps below
Institution
Oceanography),
San Diego, La Jolla, California 92093-0402
• The
simulation
system
we present
(Received 15 May
2002;
published
28 January
2003)intent to demonstrate encoding decoding and give a genera
can be implemented. In the future we plan to optimize and increase the dimension of the D
We introduce a new method for asymmetric
key/private
key)and
encryption
exploiting properties
order(public
to enhance
efficiency
security.
of nonlinear dynamical systems. A high-dimensional dissipative nonlinear dynamical system is
Chaos-Based Communication
and Encryption
213
distributed between transmitter and receiver, so we call the method distributed dynamics encryption
(DDE). The transmitter dynamics is public, and the receiver
dynamics: is hidden. A message is encoded
Transmitter
by modulation of parameters of the transmitter, and this results in a shift of the overall
system
attractor.
Ø
Ø
Ø
Ø
Ø
Ø
An unauthorized receiver does not know the hidden dynamics
in the receiver
decode
theØ
temp
= 2.4and
· t0cannot
[n] ° 1.3
· Øsr [n]
message. We present an example of DDE using a coupled map
lattice.
t0 [n +
1]
= °0.6 · temp ° 0.07 · t21 [n] ° 0.5 · s2r [n] + 1
t1 [n + 1]
= °0.1 · temp ° 0.8 · t21 [n] ° 0.1 · t22 [n] + 1
DOI: 10.1103/PhysRevLett.90.047903
PACS
numbers:
05.45.Ra
2
2
2
2
t2 [n + 1]
= 05.45.Vx,
°0.07 · t05.45.Gg,
1 [n] ° 0.6 · t2 [n] ° 0.07 · t3 [n] + 0.5 · sr [n] + 1
t3 [n + 1]
= °0.1 · t22 [n] ° 0.8 · t23 [n] ° 0.1 · t24 [n] + 1
t4 [n + 1]
= °0.07 · t23 [n] ° 0.6 · t24 [n] ° 0.07 · t25 [n] ° 0.5 · x2 [n] + 1
plex timplementations
wherein
inherent
During the last decade there has been a great interest in
= °0.1
· t24 [n]the
° 0.8
· t25 [n] °security
0.1 · t26 [n]can
+1
5 [n + 1]
be strengthened
as=one
wishes.
developing secure communication schemes utilizing
t6 [n + 1]
°0.07
· t25 [n] ° 0.6 · t26 [n] ° 0.07 · t27 [n] + 0.5 · x2 [n] + 1
The
distributed
dynamics
encryption
chaos. Most of the proposed schemes are based on chaos
t7 [nbasic
+ 1] idea =of°0.1
· t26 [n] ° 0.8
· t27 [n] ° 0.1
· t28 [n] + 1
2
2
(DDE)
dynamical
D2 !
synchronization [1], controlling chaos [2], and chaotic
t8 [nis+to1]split a=
°0.07 · t27 system
[n] ° 0.6of· tdimension
8 [n] ° 0.07 · t9T[n] ° 0.5 · x [n] + 1
2 $
shift keying [3]. Also, chaos based block ciphers were
DR into
DT· t28transmitter
variables
t9 [n two
+ 1] parts=with
°0.1
[n] ° 0.8 · t29 [n]
° 0.1 · tt"n#
+1
1 0 [n]
2
2
2
2
t10.[n
=
°0.07
·
t
[n]
°
0.6
·
t
0
[n]
°
0.07
·
t
studied in [4]. In conventional cryptography, all encryp%t1 "n#;
. .+
; t1]
"n#&,
and
D
receiver
variables
r"n#
$
1
DT
R 9
11 [n] + 0.5 · sr [n] + 1
2
t11 [n
°0.07
· t210 [n]
° 0.8 · tthe
+ 1 signal
tion schemes are divided into symmetric and asymmetric
%r1 "n#;
. . .+; r1]DR "n#&.=The
receiver
receives
scalar
11 [n]
2
7 Security of
methods [5]. Symmetric methods require sharing of the
st "n# from the transmitter, and the transmitter receives
Receiver
:
same key by both transmitter for message encryption and
the scalar
signal
sr "n# from the receiver. At each discrete
receiver for message decryption. Asymmetric methods
time n $ 1; 2; . . . , these satisfy
r0 [n + 1]
= °0.8 · r02 [n] ° 0.1 · r12 [n] + 1
have one ‘‘public’’ key known to all users for encoding
r1 [n + 1]
= 0.01 · s2t [n] ° 0.07 · r02 [n] ° 0.6 · r12 [n] + 1
messages, but another ‘‘private’’ key for use by trusted
t"n ! 1# $ FT !t"n#;
sr "n#; m"n#";
(1)
receivers for decoding the message. Decoding the mesCoupling signals
: 1# $ FR !r"n#; st "n#";
r"n
!
sage using the public key which was used to encrypt the
g. 7.15. Public and private keys
of DDE.
Private
key: FRunfeasible.
(•),GR (•),r(n).
Ø
P11 Ø
message
is made
computationally
Communi-Public
st [n]
= °0.2 · i=0 Øti [n]Ø + 1 + m · A
cationsonly
strategies
that use asymmetric
public[2].
key)
"n# $ GT !t"n#"
and sr "n#+$
G !r"n#" are signals
where sst[n]
y: FT (•),GT (•),sr (n),st (n). Known
to transmitter:
t(n),m.(orFrom
= °0.8 · r02 [n]
1; R
r
methods for encryption have much greater inherent setransmitted from the transmitter to the receiver and back,
curity than symmetric
eliminateisthemodurespectively. Here FT "'# is a DT dimensional vector field,
Fig.methods
7.16.since
A they
message
problem of key management, which itself can pose the
FR "'# is DR dimensional, GR "'# and GT "'# are scalars,
lated
by
altering
the
parameters
of m"n# is the message. Of these quantities FT "'# and
most serious security risk. However, in the application of
and
thedynamics
transmitter
which
results inGaT "'# are public, while m"n#; FR "'#, and GR "'# are priideas from nonlinear
to secure
communica-
Our Prototype
Encrypt
Decrypt
Attack
Coupled Map
PHYSICA
ELSEVIER
Physica A 245 (1997) 446-452
Bifurcation scenarios and quasiperiodicity
in coupled maps
K.E. Kiirten, G. NicolislPhysica A 245 (1997) 446-452
44
Karl E. K/irten a,,, Gr6goire Nicolis b
Institutfiir Experimentalphysik,
Universitdt
0.49
I
i
I Wien,I AustriaI
I
b Center .[or Nonlinear Phenomena and Complex Systems, UniversitO Libre de Bruxelles, Belgium
a
0.48
Received 8 April 1997
0.47
Abstract
0 -~"
i
r
l
l
l
l
l
l
0.9."
0.91
~
0.46
The appearance of coherent structures in a model of two coupled logistic maps is studied. Ana- 0.~
0.45 attractors and transitions from periodic to quasiperiodic
lytical results for the existence of multiple
motion are presented. The theoretical prediction for the Hopf radius agrees well with computer 3.8g
0.44 region.
experiment throughout the quasiperiodic
Keywords: Coupled map lattices; Bifurcations;
Quasiperiodicity; Chaos
0.43
1. Introduction
3.88
0.42
).87
0.41
0.86
I
f
I
I
I
I
0.4
0.85
I
In recent years, coupled maps have
extensively
as a privileged
0.84been
0.84studied
0.85 0.85
0.86 0.86
0.87 0.87 model
0.36
for understanding the onset o f complex behaviour in spatially extended systems. On
I
I
0.4
0.44
I
I
0.46
I
I
0.52
I
0.56
Results
100000000#
Log10(Number-of-States)-
10000000#
1000000#
100000#
10000#
0#
2#
4#
6#
8#
Dimensions-
10#
12#
14#
Figure$1.$Data$Transmi7on$using$tradi7onal$map$
1"
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Bit$
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Sequence$Number$
Figure$1.$Data$Transmi7on$using$tradi7onal$map$
1"
0.9"
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Bit$
0.6"
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0"n=50"
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1" 2" 3" 4" 5" 6" 7" 8" 9" 10"11"12"13"14"15"16"17"18"19"20"21"22"23"24"25"26"27"28"29"30"31"32"33"34"35"36"37"38"39"40"41"42"43"44"45"46"47"48"49"50"
Sequence$Number$
Figure$2.$Data$transmi6on$using$quasi$periodic$map$
1"
0.9"
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Bit$
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0"n=50"
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1" 2" 3" 4" 5" 6" 7" 8" 9" 10"11"12"13"14"15"16"17"18"19"20"21"22"23"24"25"26"27"28"29"30"31"32"33"34"35"36"37"38"39"40"41"42"43"44"45"46"47"48"49"50"
Sequence$Number$
Figure$2.$Data$transmi6on$using$quasi$periodic$map$
1"
0.9"
0.8"
0.7"
Bit$
0.6"
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0"n=50"
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1" 2" 3" 4" 5" 6" 7" 8" 9" 10"11"12"13"14"15"16"17"18"19"20"21"22"23"24"25"26"27"28"29"30"31"32"33"34"35"36"37"38"39"40"41"42"43"44"45"46"47"48"49"50"
Sequence$Number$
Future Work