1. 1 A Steganographic Scheme for Secure Communications Based on the Chaos and Eular Theorem Der-Chyuan Lou and Chia-Hung Sung IEEE Transactions on Multimedia, Vol. 6, No. 3, June 2004, pp. 501-509 National Defense University, Chung Cheng Institute of Technology Reporter: Jen-Bang Feng ( 馮振邦 )

2. 2 Outline Euler Theorem RSA Cryptosystem The Proposed Scheme 1.Choosing Positions 2.Embedding Method Experimental Results Comments

3. 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 Example: n=7, then Φ(7) =6 2 6 = 3 6 = 4 6 = 5 6 = 6 6 = 1 mod 7

4. 4 RSA Cryptosystem AliceBob e, n C e, n: Public Keysd: 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 = C d = M e×d = M a×n+1 = M mod n

5. 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. 6 1. Choosing Positions Stego-matrix OK (5,7)  (400, 68) (16, 20)  (90, 30)

7. 7 Stego matrix Sender Receiver N S =143, P S =Φ(143) =120N R =253, P R =Φ(253) =220 P 1 =5, P 2 =7, k=15, N pub =10 K PR =170 K PS =100 P S ’=LCM(120, N pub )=120 P R ’=LCM(220, N pub )=220 P SS =P S ’-K PS mod P s =20 mod 120 =20 P SR =P R ’-K PR mod P R =50 mod 220 =50 Keep in secret Public key Keep in secret Public

8. 8 For Ex., (13, 32) is going to be transformed and N S < N R. reblocking problem

9. 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. 10 2. Embedding Method ab cx e d f 10090 6050 2010 8070 4030 100 g u,x = (a+b+c)/3 g u,x = (d+e+f)/3 g m,x = (g u,x + g u,x )/2 b’ i = 0 if |g u,x – g l,x | ≤ 3T x’ = g m,x – T else if g u,x ≤ g l,x x’ = g u,x – T else x’ = g u,x + T b’ i = 1 if |g u,x – g l,x | ≤ 3T x’ = g m,x + T else if g u,x ≤ g l,x x’ = g l,x + T else x’ = g l,x – T Rules: For Ex.B = {0, 1} Cover image

11. 11 Ex. of Embedding 10090 6050 2010 8070 4030 100 b i = 0 if |g u,x – g l,x | ≤ 3T x’ = g m,x – T else if g u,x ≤ g l,x x’ = g u,x – T else x’ = g u,x + T b i = 1 if |g u,x – g l,x | ≤ 3T x’ = g m,x + T else if g u,x ≤ g l,x x’ = g l,x + T else x’ = g l,x – T B = {0, 1} Cover image 10090 6093 2010 80 40 10 b 1 = 0 g u,x = 83 g l,x = 20 x’ = g u,x + T = 93 T = 10 90 93 10 8070 330 100 b 2 = 1 g u,x = 88 g l,x = 13 x’ = g l,x – T = 3 10090 6093 2010 8070 330 100 Stego-image

12. 12 Ex. of Extraction 10090 6093 2010 8070 330 100 Stego-image when |g u,x – g l,x | ≤ 3T if x < g m,x b i = 0 else b i = 1 when |g u,x – g l,x | > 3T if |x – g u,x |< |x – g l,x | b i = 0 else b i = 1 B = {0, 1} g u,x = 83 g l,x = 7 b 0 = 0 g u,x = 87 g l,x = 13 b 1 = 1 T = 10

13. 13 Embedding Method Use data compression first May cause data error naturally  Use data redundancy Acceptable PSNR

14. 14 Experimental Results Original Cover Image Stego Image with PSNR = 32.58, L = 4096 bits

15. 15 Experimental Results

16. 16

17. 17 Comments A nearly public key system Still need secret information held by both sides..  Consider B i ’ = k 1 + k 2 xB i mod L Embedding method naturally cause data error.  LSB?  Redundancy is contradict to compression.