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Generating secrets over public channels
Neural Cryptography
Synchronization of chaotic systems
Würzburg: Richard Metzler, Andreas Ruttor, W.K.
Bar Ilan: I. Kanter, M. Rosen-Zvi, E. Klein, R.Mislovaty, L. Shacham, Y. Perchenok
Wolfgang Kinzel, Minerva 2005 – p.1/19
Key exchange protocol
BA
E
Two partners A and B generate a common secret,without previous secret contact.An attacker E can only listen to the communication, butotherwise E knows everything what A knows about B.
Is this possible?
Wolfgang Kinzel, Minerva 2005 – p.2/19
Number Theory
Diffie and Hellmann 1976,Rivest Shamir Adelman (RSA), El Gamal
Large integers: ��� �� �� �. Public: �� �. Secret A: �, B:
�
A sends � � � � mod � and B calculates
� � � � mod �
B sends � � � � mod � and A calculates
� � � mod �
�
is the secret key, which is used to encrypt secretmessages
� � � � mod � is fast,
���� � � � � �� � � mod � is nonfeasible
Wolfgang Kinzel, Minerva 2005 – p.3/19
Physics
Quantum mechanics
Stochastic process:Synchronization of neural networks by mutuallearningEnsemble of random walks with stochastic repulsiveand attractive forces
Nonlinear dynamics:Synchronization of chaotic differential equations
Mutual interaction has an advantage over one-directional
information
Wolfgang Kinzel, Minerva 2005 – p.4/19
Neural Cryptography
A B
E
Wolfgang Kinzel, Minerva 2005 – p.5/19
Realization: Tree Parity Machine
τ
x
Π
w
σ
Discrete weights����� � � � � � � � � � � � � � � �� ���� � � �� � � � ��� � � � � � � � � �
�� � sign
��� �� �� � � � �� �� ��
Secret key: Common time dependent weights � � � � �
after synchronization
Wolfgang Kinzel, Minerva 2005 – p.6/19
Distribution of synchronization times
0 500 1000 1500 2000tsync
0
0.001
0.002
0.003
0.004
0.005
P(t sy
nc)
K=3 L=3 N=1000 (r)
Wolfgang Kinzel, Minerva 2005 – p.7/19
Learning rule
Each participant has three units
� � � � � � �
and oneoutput bit, which is sent to its partner,
� � � � �� � � �� � � �� � � � � �� � � �� � � ��
+++
+
A + +
−−
x
B
Rule for A: If � ��� � � �then do not move.
If � �� � � �then move unit
�
.
Wolfgang Kinzel, Minerva 2005 – p.8/19
Stochastic forces
� = probability that two corresponding hidden units aredifferent, synchronisation: � � �
attractive
repulsive
−
+ attractive
+
−
E
+++
A +
x
attractive
repulsive
+
−
B+
− repulsive
Prob(A/B repulsive) � ��
Prob(A/E repulsive) � �
Wolfgang Kinzel, Minerva 2005 – p.9/19
Scaling for large key size
Average synchronisation time between A and B:
��� � �� � � �
Probability that an attacker E synchronises with A and B:
��� � � �Security for
� �
Wolfgang Kinzel, Minerva 2005 – p.10/19
Ongoing research
Advanced algorithms versus advanced attacks
Combination with chaotic maps
Inputs from feeback
Inputs with restrictions (queries)
Attacking by ensembles
Attacking by genetic algorithms
See 12 common publications since 2002
Hardware: 20000 keys per second on an tiny chipVolkmer, Wallner, IEEE Trans. Comp. 2005
Wolfgang Kinzel, Minerva 2005 – p.11/19
Nonlinear dynamics
Neural cryptography
A τ
x
B
Synchronization of chaotic systems
A Bf(t)
Wolfgang Kinzel, Minerva 2005 – p.12/19
Lorenz equations
��� �� � � � � � � � � � � � � �� � � � � � � � � �
� � �� � � �� � � � � � � � � �
� �� � � � � � � �
�� �
� � � � � � � � � � �� � sign
� � � � � � �� � � � � � � � �� � � � � � � �� �
Exchange signal� � �
: nonlinear and time-delayedWolfgang Kinzel, Minerva 2005 – p.13/19
Synchronization
Lyapunov exponent
�
:
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
K
Parties
Attacker
Synchronization is possible for a limited range of coupling
strenght�
and delay times � �.Wolfgang Kinzel, Minerva 2005 – p.14/19
Attack 1
Synchronization of single Lorenz systems:Success probability
�
and synchronization time�
as afunction of the number � of recognized digits of � � � � .
-4
-3.5
-3
-2.5
-2
-1.5
-1
3 5 7 9 11 13 15
log
10(P
)
parties synchronize 14 digits
parties synchronize 30 digits
0
20
40
60
80
100
120
0 5 10 15 20 25
tim
e
Wolfgang Kinzel, Minerva 2005 – p.15/19
Attack 2
Calculating � � � � from
� � � � :Record
� � � � � � � � � � � � � � � � � � � � � � � � � and measurethe volume in � space which belongs to a tiny volume in
�
-space.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
x-<x>
P(x
-<x>
)
dynamic A
static A
BAK
W
Wolfgang Kinzel, Minerva 2005 – p.16/19
Improving security
Dynamic amplitude of the nonlinearity of
� � � � :
� � � �
�
�� � � � � � � � � � � ��� � ��
Ring of
�
Lorenz equations with directed internalcouplings .
-2.5
-2
-1.5
-1
-0.5
0
0 5 10 15 20 25 30 35N
log
10(P
)
0
20
40
60
80
100
120
140
0 20 40 60 80
N
Syn
ch
ron
izati
on
Tim
e
K
W
A B
Wolfgang Kinzel, Minerva 2005 – p.17/19
Summary
Secret keys can be generated over public channelsby physical mechanisms.
Neural Cryptography:Mutual learning of neural networksStochastic attractive and repulsive forcesScaling relation for the securityFast and simplemany keys per transactionrealization in small chips
Lorenz Synchronization:
Wolfgang Kinzel, Minerva 2005 – p.18/19
Lorenz Synchronization:Synchronization of chaotic differential equationswith nonlinear and time-delayed couplingsScaling relation for the securityAnalog signalsElectronic circuits, chaotic lasers
Wolfgang Kinzel, Minerva 2005 – p.19/19