Geometrically Structured Codebooks

Security Analysis of Robust Data-Hiding with

Geometrically Structured Codebooks

E. Topak^(a), S. Voloshynovskiy^(a), O. Koval^(a), M. K. Mihcak^(b) and T. Pun^(a)

(a)

Stochastic Image Processing (SIP) Group, University of Geneva, Switzerland

&

(b)

Microsoft Research, Redmond, USA

Agenda

Problem formulation;

Channels with geometrical attacks;

Information theoretic (IT) framework for geometrically robust data-hiding;

Structured codebooks;

Analysis of security leaks and attacking strategies;

Conclusions;

Future research directions.

Problem formulation

Objectives:

To analyze the conditions of reliable communications in channels with geometrical transformations;

To study capacity achieving geometrically-robust data-hiding codes;

To investigate security leakages of structured codebooks and corresponding attacking strategies.

Problem formulation

Data-hiding problem: Given users each with its own key , communicate reliably message , , embedded in the host image through the channel

X ∈ X

^N .

∈ M

m ^{M =} { ^{1 , 2 , L , 2}

^NR

} ^{k ∈} { ^{1 , 2 , L K} }

K

)

| ( v y p

Encoder W _Decoder

M

X

Y V Mˆ

K K

)

| (v y p

→

= W ( M X , , K ) W

→

= W ( M , K ) W

If the host state is taken into account or not in watermark generation:

Random binning approach Random coding approach

+

Problem formulation

[ ]

∑

∈

=

≠

=

M

M m N

e

M m M m

P

_{( )}

1 Pr ˆ |

Performance criterion:

Random coding

Random binning

[

^I

(

^K

) (

^{I K}

) ]

R N1 U;V U;X

−

≤

(

^K

)

N I

R 1 W ;V

≤

Practical set-up ( )

N < ∞

Theoretical set-up ( )

N → ∞

)

0

(^N

≠

P

e

)

0

(^N

=

P

e

Conditions for reliable communications:

Problem formulation

Encoder W _Decoder

M

X

Y V Mˆ

K A K

) ,

| (v y a p

Problem: To analyze conditions of reliable communications in the case of geometrical attacks avoiding security leakages

+

Data-hiding in channels with geometrical attacks

Trade-offs:

Geometrical channels →→→→ Synchronization framework Syncronization framework →→→→ Security leakages

( )^J A∈

a A =

Channels with geometrical attacks

Average

probability of error:

( ) ( )

∑

( )

∈ ∈

=

J

N e N

G

e p P

P

a A

A a ^{( )} a

) (

( a

1

^, K ^, a

J

) ^,

=

a ^a

i

~ ^p

A

( ) ^a

Assumption: Applied transformation belongs to the set of typical geometrical transformations:

Theoretical set-up

( )

N → ∞

Pe^(N)

( )

^{a = 0}

( )

⁰

)

(^N a ≠

Pe

Practical set-up ( )

N < ∞

)

0

(^N

→

G

P

e

)

1

(^N

→

G

P

e

Geometrical attacks completely destroy reliable communications No impact on communications performance in price of increase

in decoding complexity A decoder without a synchronization framework has to perform an exhaustive decoding through all possible geometrical transformations!

Channels with geometrical attacks

Data-hider strategy: add synchronization part into the codebook.

( )^J

≤ A∈

A'

( ) ( )

∑

∈ ′

=

a A

A a ^{( )} a

)

( ~ _N

e N

G

e p P

P Average probability of error:

Constrained search space:

rate loss due to synchronization

( )^J A∈

a A =

A′

R ~

Geometrical synchronization based on structured codebooks:

Compensation of the estimate:

channel state compensation (CSC)

Estimation of the applied geometrical transformation

from the attacked data:

channel state estimation (CSE)

Channels with geometrical attacks

Structured codebooks

Redundant-based structured codebooks (codewords have special statistics

to aid CSE and CSC) Template-based structured codebooks

(a specially designed template is used to perform CSE and CSC)

Problem: How to combine these conflicting requirements?

Our objectives capacity achieving data-hiding

host interference problem to be solved based on random binning

dependent on host data

robustness to geometrical attacks

codewords with synchronization features to be generated according to statistics

that are independent from those of the host data

IT framework for geometrically robust data-hiding

Encoder

) , ,

( _{1 1}

1 M X K

W

)

|

(v v

p ′′ ′

Decoder M1

X

K1 V

ˆ 2

M

K2

M2

Encoder

K2

) ,

( _{2 2}

2 M K

W M

A

Y _CSC

K2

CSE

Aˆ ^Decoder

ˆ1

M

V′ Mˆ

Practical implementation principles:

CDMA/SDMA signalling Genie-aided decoding

(Multistage decoder)

K1

1 → W

2 → W

carries only information about . has synchronization features using:

redundant-based design, template-based design.

M1

( )

^⋅

TA

Y V ′′

Equivalent Channel

Proposed set-up

Attacking Channel

+

A code for MAC consists of:

(

^{2NR1,2NR2 , N}

)

{

^{1,2, ,2 1}

}

^{, 2}

{

^{1,2, ,2 2}

}

^;

1

NR

NR L

L =

= M

M

Encoding functions:

{ } ^{{ }}

{

^{1,2, ,2}

} ^{

^{1,2, ,}

^}

^;

:

; ,

, 2 , 1 2

, , 2 , 1 :

2 2

2

1 1

1

2 1

NR N

N N

NR

f f

W K

W X

K

→

×

→

×

×

L L

L L

Decoding function: ^{g : V N ×}

{

^{1,2,L, K1}

} {

^{× 1,2,L, K 2}

}

^→

{

^1,2,L,2NR1

} {

^{× 1,2,L,2NR2}

}

^⋅

( )

[ ( ) ( ) ( ) ]

( ) ≠ = = ⋅

=

∑

× + ∈

2 1 2 1 2 1

,

2 2

1 1

2 1 2

1 )

( Pr , , , | ,

2 1

M m M

m R R N N

e g K K m m M m M m

P V

Average probability of error for code:

(

^{2NR1,2NR2 , N}

)

Index sets:

IT framework for geometrically robust data-hiding

The achievable rates:

( ) ( )

[

2 1 1

]

1 1 ; | , ; |

K I

K N I

R ≤ U V W − U X

(

2 2

)

2 1 ; | ,

K N I

R ≤ W V U

( ) ( )

[

2 1 2 1

]

2

1 1 , ; | , ; |

K I

K K N I

R

R + ≤ W U V − U X

R

1

R

2

( ) ( )

[

; | 1 ; | 1

]

1 I K I K

N U V − U X

(

2; | , 2

)

1 I K

N W V U

(

2; | 2

)

1 I K

N W V

( ) ( )

[

; | 2, 1 ; | 1

]

1 I K I K

N U V W − U X

IT framework for geometrically robust data-hiding

The capacity region:

Structured codebooks

Template-based structured codebooks

+

W2 1 2 N

1 2 N

W1

1 2 N₁

W1 1 2 N2 W₂ N

N N₁+ ₂ =

CDMA signalling:

m1

2^NR1

1

1

K1 K₁

Codebook Codebook Codebook

SDMA signalling:

m1

1

2^N1^R

1

1 K1 K₁

Codebook Codebook Codebook

Structured codebooks

Redundant-based structured codebooks

CDMA signalling:

m1

2^NR1

1

1

K1 K₁

1 2 N

+

W1

W2 1 2 N

Codebook Codebook Codebook

m2

2^NR2

1

SDMA signalling:

m1

1

2^N1^R

1

1

K1 K₁

1 2 N1

W1 1 2 N2 W₂

Codebook Codebook Codebook

m2

2

2^N2^R

1

N N N₁+ ₂ =

Analysis of security leaks and attacking strategies

Attacker’s objective: To destroy reliable communications.

Assumptions based on Kerckhoff principle:

Attacker has access to:

encoding and decoding algorithms,

^codebooks.

Attacker does not know:

secret keys and ,

indexes and ,

the original host image . K1 K₂

M1 M₂

X Attacker’s approach: To exploit all available prior information and all security leakages.

Analysis of security leaks and attacking strategies

Codebook construction,

host and watermark

statistics Exhaustive search in

codebooks for the

communicated watermark in order to subtract it from

the stego data To destroy reliable

communications completely Key space

search attacks

- Signal desynchronization

To increase the decoding complexity on the

data-hider side Geometrical

attacks

Host and watermark

statistics Subtracting an estimate of the

watermark sequence from the stego data and adding noise to avoid the attack inversion To decrease the

rate of reliable communications Statistical signal

processing attacks

Required Priors Attacking Strategy

Goal Attack Type

Analysis of security leaks and attacking strategies

Key space search attacks

Attacks against template-based structured codebooks

Attacks against redundant-based structured codebooks

Security consideration:

Template is only key-dependent

and unique

for a particular key .

W2

k K

₂

=

Security consideration:

By observing stego data, the attacker could estimate

the statistics of

even when is not available.

W2

K

2

Analysis of security leaks and attacking strategies

, but is fixed and is the same for all users

, and there is no relationship between the

codebooks of and , and there is a one- to-one correspondence between

the codebooks of and for a given

Attack complexity Particular scenario

K K

K

₁

=

₂

=

W2

W1

K

[^{R R} ]

N + ′

+ 2 ¹ K 2

W2

W1

2

1

K

K ≠

[^{R R} ]

N + ′

+ ₁ 2 ¹

2 K

K

2

1

K

K ≠ K

₂ 1+ K ₁ 2^N[^R1+R′]

Attacks against template-based structured codebooks

Analysis of security leaks and attacking strategies

Attacks against redundant-based structured codebooks

The statistics of are different for all user codebooks and there is a one-to-one relationship between

the codebooks of and

The statistics of are the same for all codebooks

Attack complexity Particular scenario

W2

W2

W1

W2

K

₂

2

^NR2

+ K

₁

2

^N

[

^R1+R′

]

[

^{R R}

]

N

NR²

+ 2

¹+ ′ 2

2

K

Analysis of security leaks and attacking strategies

Random binning

Random coding

2

^H

(

^W|Y

) = 2

^H

( ) (

^{W −I W;Y}

) (

^U|Y

) 2 ( ) (

^U

[

^U;Y

) (

^U;X

) ]

2

^H

≤

^{H − I −I}

Assumption: Generate codebooks in the way that each one contains unique codewords and every possible codeword is included in a unique codebook.

Random binning

Random coding

2

^H

( )

^W

= K 2

^NR

( )

^N

[

^{R R}

]

H

= 2

+ ′

2

^U

K

Trial efforts without security leakage analysis:

Trial efforts with security leakage analysis:

Conclusions

The conditions of reliable communications based on structured

codebooks in channels with geometrical transformations are analyzed from an information-theoretic point of view;

The MAC framework is developed to design capacity achieving geometrically robust data-hiding;

The analysis of security leakages for each codebook structure is performed.

Future directions

Consideration of collusion attacks;

Emphasizing the impact of host data statistics on the security;

Extension of the proposed set-up to practical scenarios, with ;

Particular low-complexity search algorithms reducing the complexity of the attacker search based on the security leakages.

∞

<

N