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Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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Page 1: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 2: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 3: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 4: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 5: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

Neural Cryptography

A B

E

Wolfgang Kinzel, Minerva 2005 – p.5/19

Page 6: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 7: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 8: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 9: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 10: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 11: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 12: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

Nonlinear dynamics

Neural cryptography

A τ

x

B

Synchronization of chaotic systems

A Bf(t)

Wolfgang Kinzel, Minerva 2005 – p.12/19

Page 13: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

Lorenz equations

��� �� � � � � � � � � � � � � �� � � � � � � � � �

� � �� � � �� � � � � � � � � �

� �� � � � � � � �

�� �

� � � � � � � � � � �� � sign

� � � � � � �� � � � � � � � �� � � � � � � �� �

Exchange signal� � �

: nonlinear and time-delayedWolfgang Kinzel, Minerva 2005 – p.13/19

Page 14: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 15: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 16: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 17: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 18: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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

Page 19: Neural Cryptography Synchronization of chaotic systems ... · Generating secrets over public channels Neural Cryptography Synchronization of chaotic systems Würzburg: Richard Metzler,

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