1

A Steganographic Scheme for Secure Communications Based on t

he Chaos and Eular Theorem

Der-Chyuan Lou and Chia-Hung SungIEEE Transactions on Multimedia, Vol. 6, No.

3, June 2004, pp. 501-509National Defense University , Chung Cheng In

stitute of Technology Reporter: Jen-Bang Feng (馮振邦 )

2

Outline

Euler Theorem RSA Cryptosystem The Proposed Scheme

1. Choosing Positions

2. Embedding Method

Experimental Results Comments

3

Euler Theorem

The function Φ(n) satisfies:

for all a < n and gcd(a, n) = 1.

Φ(n) < n and gcd(Φ(n), n) = 1

na n mod 1

Example: n=7, then Φ(7) =6

26 = 36 = 46 = 56 = 66 = 1 mod 7

4

RSA Cryptosystem

(mod )eC M n (mod )dM C n

Alice Bobe, n

C

e, n: Public Keys d: Private Key

n = p×q, two large primes p and q

GCD(e, Φ(n))=1, e×d=1 mod Φ(n)

Φ(n) = (p-1)×(q-1)

M = Cd = Me×d = Ma×n+1 = M mod n

5

The Proposed Scheme

A data hiding scheme

1. Choose the hiding positions by Chaos and Euler

2. Embed the encrypted secret

OK

Cover Image

Secret Message

Stego Image

OK

Secret Message

encrypt

transmit decrypt

6

1. Choosing Positions

Stego-matrix

nkQP

PkQS

kkkQ

mod 0

0

1

11

1

2

1

nIkQIkQkQP

PkQ

kQP

PkQkQ

P

PkQS

n

n

mod 0

0

...0

0

0

0

11

2

1

1

2

11

2

1

OK

(5,7)(400, 68)

(16, 20)(90, 30)

7

Stego matrix

Sender Receiver

NS=143, PS=Φ(143) =120 NR=253, PR=Φ(253) =220

P1=5, P2=7, k=15, Npub=10

KPR=170KPS=100

PS’=LCM(120, Npub)=120 PR’=LCM(220, Npub)=220

PSS=PS’-KPS mod Ps

=20 mod 120 =20

PSR=PR’-KPR mod PR

=50 mod 220 =50

Keep in secret

1

2

1

1

11

0

0

1

11

kkP

P

kkS

elsejiBSS

NNifjiBSSjiB

PR

R

SS

S

SS

S

PR

R

KN

PN

RSPN

KN

,,

,,',''

elsejiBSS

NNifjiBSSjiB

PS

S

SR

R

SR

R

PS

S

KN

PN

RSPN

KN

,,

,,, ***

Public key Public key

Keep in secretKeep in secret

Public

8

For Ex., (13, 32) is going to be transformed and NS < NR.

elsejiBSS

NNifjiBSSjiB

PS

S

SR

R

SR

R

PS

S

KN

PN

RSPN

KN

,','

,',', ***

elsejiBSS

NNifjiBSSjiB

PR

R

SS

S

SS

S

PR

R

KN

PN

RSPN

KN

,,

,,',''

RSKK

RSKK

RSPKPK

RSPK

NNS

NNSS

NNSS

NNSSj

i

PSPR

PSPR

SPSSPR

SSPR

modmod32

13

modmod32

13

modmod32

13

modmod32

13

'

'

mod'

RSKKKK

RSKK

RSPKPK

SRPK

NNSS

NNj

iS

NNj

iSS

NNj

iSS

j

i

PSPRPRPS

PRPS

RPRRPS

SRPS

modmod32

13

modmod'

'

modmod'

'

modmod'

'

mod'

*

*

reblocking problem

9

Choosing Positions

Use Chaos and Euler theorem

Encrypt the data by a mapping function from small (32x32) to large (512x512)

Redundancy by large to large (512x512)

10

2. Embedding Method

a b

c x

e

d

f

100 90

60 50

20 10

80 70

40 30

10 0

gu,x = (a+b+c)/3

gu,x = (d+e+f)/3

gm,x = (gu,x+ gu,x)/2

b’i = 0

if |gu,x – gl,x| ≤ 3T

x’ = gm,x – T

else

if gu,x ≤ gl,x

x’ = gu,x – T

else

x’ = gu,x + T

b’i = 1

if |gu,x – gl,x| ≤ 3T

x’ = gm,x + T

else

if gu,x ≤ gl,x

x’ = gl,x + T

else

x’ = gl,x – T

Rules:

For Ex. B = {0, 1}

Coverimage

11

Ex. of Embedding

100 90

60 50

20 10

80 70

40 30

10 0

bi = 0

if |gu,x – gl,x| ≤ 3T

x’ = gm,x – T

else

if gu,x ≤ gl,x

x’ = gu,x – T

else

x’ = gu,x + T

bi = 1

if |gu,x – gl,x| ≤ 3T

x’ = gm,x + T

else

if gu,x ≤ gl,x

x’ = gl,x + T

else

x’ = gl,x – T

B = {0, 1}

Coverimage

100 90

60 93

20 10

80

40

10

b1 = 0

gu,x = 83

gl,x = 20

x’ = gu,x + T

= 93

T = 10

90

93

10

80 70

3 30

10 0

b2 = 1

gu,x = 88

gl,x = 13

x’ = gl,x – T

= 3

100 90

60 93

20 10

80 70

3 30

10 0

Stego-image

12

Ex. of Extraction

100 90

60 93

20 10

80 70

3 30

10 0

Stego-imagewhen |gu,x – gl,x| ≤ 3T

if x < gm,x

bi = 0

else

bi = 1

when |gu,x – gl,x| > 3T

if |x – gu,x|< |x – gl,x|

bi = 0

else

bi = 1B = {0, 1}

gu,x = 83

gl,x = 7

b0 = 0

gu,x = 87

gl,x = 13

b1 = 1

T = 10

13

Embedding Method

Use data compression first

May cause data error naturally Use data redundancy

Acceptable PSNR

14

Experimental Results

Original Cover Image Stego Image with PSNR = 32.58, L = 4096 bits

15

Experimental Results

16

17

Comments

A nearly public key systemStill need secret information held by both

sides. Consider Bi’ = k1 + k2xBi mod L

Embedding method naturally cause data error. LSB? Redundancy is contradict to compression.