Reversible data hiding for encrypted
signals by homomorphic encryption
and signal energy transfer
Source: J. Vis. Commun. Image R. 41 (2016) 58–64
Authors: Xiaotian Wu , Bing Chen , Jian Weng
Speaker: Ran Tang
Date:
06/08/2017
1
Outline
• Related Works
– Paillier homomorphic encryption
– Signal energy transfer
• Proposed Method
• Experiment Results
• Conclusions
2
Related Works-Paillier homomorphic encryption
p, q: two large primes,
N =p*q;
receiver 𝓡 computer:
𝝀 = lcm(p−1,q−1);
selects
𝐠 ∈ ℤ∗𝑁2 𝑤𝑖𝑡ℎ gcd(L(𝑔 𝜆 𝑚𝑜𝑑 𝑁 2 ),N)=1
where 𝐿 𝑥 =
Public key:(N, g);
Private key:(p, q, 𝝀).
𝑥−1
∗
,
ℤ
𝑁2
𝑁
= 1,2, … , 𝑁 2 − 1 .
Plaintext: m ∈ ℤ𝑁 , 𝑟 ∈ ℤ∗𝑁 𝑖𝑠 𝑟𝑎𝑛𝑑𝑜𝑚𝑙𝑦 𝑠𝑒𝑙𝑒𝑐𝑡𝑑;
Ciphertext: c = 𝑔𝑚 𝑟 𝑁 𝑚𝑜𝑑 𝑁 2 .
Decryption: m =
𝐿(𝑐 𝝀 𝑚𝑜𝑑 𝑁2 )
.
𝐿(𝑔𝝀 𝑚𝑜𝑑 𝑁2 )
𝐸(𝑚1 + 𝑚2 ) = E(𝑚1 )𝐸 𝑚2 ;
𝐸(𝑚)𝑎 = 𝐸 𝑎𝑚 .
3
Related Works-Signal energy transfer
A signal can be represented by the sum of other signals.
𝑃𝑖 = 𝑡𝑖 + 𝑎0𝑖 + 𝑎1𝑖 ;
𝒙𝒊 = 𝑃𝑖 /2 .
𝒓𝒊 𝑖𝑠 𝑟𝑎𝑛𝑑𝑜𝑚𝑙𝑦 𝑠𝑒𝑙𝑒𝑐𝑡𝑑 𝑓𝑟𝑜𝑚 0,1,2, … , 𝑥𝑖 .
𝒂𝟎𝒊 , 𝒂𝟏𝒊 = 𝑟𝑖 ;
𝒕𝒊 = 𝑃𝑖 − 2𝑟𝑖 .
𝐸(𝑚1 + 𝑚2 ) = E(𝑚1 )𝐸 𝑚2 ;
𝐸(𝑚)𝑎 = 𝐸 𝑎𝑚 .
4
Proposed Method(1/2)
5
Proposed Method(2/2)
Original pixel:𝑃𝑖 = 97; 𝑠𝑒𝑐𝑟𝑒𝑡 𝑏𝑖𝑡: 𝑏 = 1;
Image encryption phase:
𝑥𝑖 = 𝑃𝑖 /2 = 97/2 = 48;
suppose that 𝑟𝑖 = 6 is randomly selectd from 0,1,2, … , 48 ;
𝑎0𝑖 = 𝑎1𝑖 = 𝑟𝑖 = 6; 𝑡𝑖 = 𝑃𝑖 − 𝑎0𝑖 − 𝑎1𝑖 = 97 − 12 = 85.
When 𝑷𝒊 is encrypted we get: 𝑬(𝒑𝒊 ) = E 85 E 6 E 6 .
data embedding phase:
𝐸
𝐸
𝐸
=
𝐸
′
E 𝑎0𝑖
=
′
E 𝑎1𝑖
𝑎0𝑖 − 𝑎1𝑖 − 1
𝑎0𝑖 + 𝑎1𝑖 + 1
𝑎0𝑖 + 𝑎1𝑖 + 1
𝑎0𝑖 − 𝑎1𝑖 − 1
𝑖𝑓 𝑏 = 0;
𝑖𝑓 𝑏 = 1;
𝑖𝑓 𝑏 = 0;
𝑖𝑓 𝑏 = 1;
′
′
∵ 𝑏 = 1; ∴ E 𝑎0𝑖
= 𝐸 13 ; E 𝑎1𝑖
= 𝐸 −1 .
′
marked encrypted pixel:𝑬(𝑷𝒊 ) = 𝐸 85 𝐸 13 𝐸 −1 ;
data extraction and image recovery phase:
directly decrypted: 𝐏𝐢′ = 85 + 13 + −1 = 97;
′
′
0,
𝑖𝑓𝑎0𝑖
< 𝑎1𝑖
;
b=
′
′
1,
𝑖𝑓𝑎0𝑖
> 𝑎1𝑖
.
′
′
∵ 𝑎0𝑖
> 𝑎1𝑖
; ∴ 𝑏 = 1.
6
Experiment Results(1/5)
7
Experiment Results(2/5)
8
Experiment Results(3/5)
9
Experiment Results(4/5)
10
Experiment Results(5/5)
-references
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[16] X. Zhang, Separable reversible data hiding in encrypted image, IEEE Trans. Inform.
Forensics Secur. 7(2) (2012) 826–832.
[20] W. Zhang, K. Ma, N. Yu, Reversibility improved data hiding in encrypted images,
Sign. Process. 94 (2014) 118–127.
[23] Y.-C. Chen, C.-W. Shiu, G. Horng, Encrypted signal-based reversible data hiding with
public key cryptosystem, J. Vis. Commun. Image Represent. 25(5) (2014) 1164–1170.
[24] C.W. Shiu, Y.C. Chen, W. Hong, Encrypted image-based reversible data hiding with
public key cryptography from difference expansion, Sign. Process. Image Commun.
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[25] X. Zhang, J. Wang, Z. Wang, H. Cheng, Lossless and reversible data hiding in
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11
Conclusions
Each unit in the original image is divided
into three parts, and they are encrypted by
Paillier homomorphic encryption.
The proposed method have optimal
visual quality and improved embedding rate.
Moreover, the proposed method can be
extended to deal with encoded multimedia.
12
Q&A
13