Abstract

An affine mapping mathematical expression of the double-random-phase encryption technique has been deduced utilizing the matrix form of discrete fractional Fourier transforms. This expression clearly describes the encryption laws of the double-random-phase encoding techniques based on both the fractional Fourier transform and the ordinary Fourier transform. The encryption process may be regarded as a substantial optical realization of the affine cryptosystem. It has been illustrated that the encryption process converts the original image into a white Gaussian noise with a zero-mean value. Also, the decryption process converts the data deviations of the encrypted image into white Gaussian noises, regardless of the type of data deviations. These noises superimpose on the decrypted image and degrade the signal-to-noise ratio. Numerical simulations have been implemented for the different types of noises introduced into the encrypted image, such as the white noise with uniform distribution probability, the white noise with Gaussian distribution probability, colored noise, and the partial occlusion of the encrypted image.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. D. R. Stinson, Cryptography: Theory and Practice, 2nd ed. (CRC Press, 2002).
  14. L. Yuanlie, Applied Stochastic Process (Qinghua U. Press, 2003) (in Chinese).
  15. S. M. Ross, Stochastic Processes (Wiley, 1983).

2004

N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transforms in digital holography," Opt. Commun. 235, 253-259 (2004).
[CrossRef]

O. Matoba and B. Javidi, "Secure holographic memory by double-random polarization encryption," Appl. Opt. 43, 2915-2919 (2004).
[CrossRef] [PubMed]

2002

2001

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1505 (2001).
[CrossRef]

S. Liu, L. Yu, and B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering," Opt. Commun. 187, 57-63 (2001).
[CrossRef]

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (2001).
[CrossRef]

2000

C. Candan, M. A. Kutay, and H. M. Ozaktas, "The discrete fractional Fourier transformation," IEEE Trans. Signal Process. 48, 1329-1337 (2000).
[CrossRef]

1998

1996

R. Wang and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

1995

Bollaro, F.

Candan, C.

C. Candan, M. A. Kutay, and H. M. Ozaktas, "The discrete fractional Fourier transformation," IEEE Trans. Signal Process. 48, 1329-1337 (2000).
[CrossRef]

Chatwin, C.

R. Wang and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Goudail, F.

Javidi, B.

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transforms in digital holography," Opt. Commun. 235, 253-259 (2004).
[CrossRef]

Kishk, S.

Kutay, M. A.

C. Candan, M. A. Kutay, and H. M. Ozaktas, "The discrete fractional Fourier transformation," IEEE Trans. Signal Process. 48, 1329-1337 (2000).
[CrossRef]

Liu, S.

S. Liu, L. Yu, and B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering," Opt. Commun. 187, 57-63 (2001).
[CrossRef]

Matoba, O.

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transforms in digital holography," Opt. Commun. 235, 253-259 (2004).
[CrossRef]

Ozaktas, H. M.

C. Candan, M. A. Kutay, and H. M. Ozaktas, "The discrete fractional Fourier transformation," IEEE Trans. Signal Process. 48, 1329-1337 (2000).
[CrossRef]

Qiwen, R.

R. Qiwen, Wavelet Transform and Fractional Fourier Transform Theory and Applications (HaErbin U. Press, 2001) (in Chinese).

Refregier, P.

Ross, S. M.

S. M. Ross, Stochastic Processes (Wiley, 1983).

Shih, C. C.

C. C. Shih, "Fractionalization of Fourier transform," Opt. Commun. 48, 495-498 (1995).
[CrossRef]

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transforms in digital holography," Opt. Commun. 235, 253-259 (2004).
[CrossRef]

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (2001).
[CrossRef]

Stinson, D. R.

D. R. Stinson, Cryptography: Theory and Practice, 2nd ed. (CRC Press, 2002).

Sun, C.-C.

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1505 (2001).
[CrossRef]

Unnikrishnan, G.

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (2001).
[CrossRef]

Wang, B.

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1505 (2001).
[CrossRef]

Wang, R.

R. Wang and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Yu, L.

S. Liu, L. Yu, and B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering," Opt. Commun. 187, 57-63 (2001).
[CrossRef]

Yuanlie, L.

L. Yuanlie, Applied Stochastic Process (Qinghua U. Press, 2003) (in Chinese).

Zhu, B.

S. Liu, L. Yu, and B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering," Opt. Commun. 187, 57-63 (2001).
[CrossRef]

Appl. Opt.

IEEE Trans. Signal Process.

C. Candan, M. A. Kutay, and H. M. Ozaktas, "The discrete fractional Fourier transformation," IEEE Trans. Signal Process. 48, 1329-1337 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

C. C. Shih, "Fractionalization of Fourier transform," Opt. Commun. 48, 495-498 (1995).
[CrossRef]

S. Liu, L. Yu, and B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering," Opt. Commun. 187, 57-63 (2001).
[CrossRef]

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (2001).
[CrossRef]

N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transforms in digital holography," Opt. Commun. 235, 253-259 (2004).
[CrossRef]

Opt. Eng.

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1505 (2001).
[CrossRef]

R. Wang and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Opt. Lett.

Other

R. Qiwen, Wavelet Transform and Fractional Fourier Transform Theory and Applications (HaErbin U. Press, 2001) (in Chinese).

D. R. Stinson, Cryptography: Theory and Practice, 2nd ed. (CRC Press, 2002).

L. Yuanlie, Applied Stochastic Process (Qinghua U. Press, 2003) (in Chinese).

S. M. Ross, Stochastic Processes (Wiley, 1983).

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Figures (5)

Fig. 1
Fig. 1

Diagram of the implementation.

Fig. 2
Fig. 2

(a) Original image, (b) encrypted image (real part).

Fig. 3
Fig. 3

(a) Probability distribution of the pixel value of the encrypted image (real part), (b) autocorrelation function of the encrypted image.

Fig. 4
Fig. 4

(a) Colored noise, (b) one-fourth occlusion of the encrypted image (real part).

Fig. 5
Fig. 5

Probability distribution of the pixel value of the noise corresponding to data deviation: (a) colored noise, (b) one-fourth occlusion.

Equations (122)

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f ( x )
g ( x )
θ 0 ( x )
φ 0 ( u )
[ 0 , 1 ]
θ ( x )
φ ( u )
θ ( x ) = exp [ i 2 π θ 0 ( x ) ]
φ ( u ) = exp [ i 2 π φ 0 ( u ) ]
f ( x )
g ( x )
θ ( x )
φ ( u )
f ^ = [ f ( 0 ) , f ( 1 ) ,   …   ,   f ( N 1 ) ] T ,
g ^ = [ g ( 0 ) , g ( 1 ) ,   …   ,   g ( N - 1 ) ] T ,
θ ^ = [ θ ( 0 ) , θ ( 1 ) ,   …   ,   θ ( N - 1 ) ] T ,
φ ^ = [ φ ( 0 ) , φ ( 1 ) ,   …   ,   φ ( N 1 ) ] T .
F ˜ α = l = 0 3 A l ( α ) F ˜ l ,
A l ( α ) ( l = 0 , 1 , 2 , 3 )
A l ( α ) = 1 4 1 exp [ i 2 π ( α l ) ] 1 exp [ i 2 π ( α l ) 4 ] = cos [ ( α l ) π 4 ] cos [ 2 ( α l ) π 4 ] exp [ 3 i ( α l ) π 4 ] .
F ˜
F ˜ = 1 N [ W 0 × 0 W 0 × 1 W 0 × ( N 1 ) W 1 × 0 W 1 × 1 W 1 × ( N 1 ) W ( N 1 ) × 0 W ( N 1 ) × 1 W ( N 1 ) × ( N 1 ) ] ,
W = exp [ i ( 2 π N ) ]
F ˜
F ˜ m l = 1 N exp [ i ( 2 π ml / N ) ]
F α ( m , l )
F ˜ α
α 1 th
θ ( x ) f ( x )
X ( m ) = l = 0 N 1 F α 1 ( m , l ) θ ( l ) f ( l ) , ( m = 0 , 1 ,   …   ,   N 1 ) .
F ˜ α 1
F ˜ α 1
θ ( 0 )
θ ( 1 )
θ ( N 1 )
F ˜ θ α 1
F ˜ θ α 1
f ^
φ ( u )
X ( m ) φ ( m ) = φ ( m ) l = 0 N 1 F α 1 ( m , l ) θ ( l ) f ( l ) ,
( m = 0 , 1 ,   …   ,   N 1 ) .
F ˜ θ α 1
F ˜ θ α 1
φ ( 0 )
φ ( 1 )
φ ( N 1 )
F ˜ θ φ α 1
F ˜ θ φ α 1
f ^
a 2
g ^
g ( m ) = k = 0 N 1 l = 0 N 1 φ ( k ) F α 2 ( m , k ) F α 1 ( k , l ) θ ( l ) f ( l ) ,
( m = 0 , 1 ,   …   ,   N 1 ) .
H ˜
F ˜ α 2
F ˜ θ φ α 1
H ˜
f ^
g ^ = H ˜ f ^ ,
H ˜
N × N
F ˜
H ˜
H ˜ 1
f ^ = H ˜ 1 g ^
f ( x )
g ( x )
H ˜
f ^
g ^
g ^
f ^
α 1 = 1 , α 2 = 1
F ˜ α 1
F ˜ α 2
F ˜
F ˜ 1
g ( x )
E [ 1 N m = 0 N 1 g ( m ) ] = 1 N m = 0 N 1 E [ k = 0 N 1 l = 0 N 1 φ ( k ) F α 2 ( m , k ) × F α 1 ( k , l ) θ ( l ) f ( l ) ] ,
θ ( x )
φ ( u )
f ( x )
E [ 1 N m = 0 N 1 g ( m ) ] = 0 .
E [ 1 N m = 0 N s 1 g ( m ) g * ( m s ) ] = 1 N m = 0 N s 1 E { [ k 1 = 0 N 1 l 1 = 0 N 1 φ ( k 1 ) F α 2 ( m , k 1 ) F α 1 ( k 1 , l 1 ) × θ ( l 1 ) f ( l 1 ) ] [ k 2 = 0 N 1 l 2 = 0 N 1 φ * ( k 2 ) ( F α 2 ) * ( m s , k 2 ) × ( F α 1 ) * ( k 2 , l 2 ) θ * ( l 2 ) f ( l 2 ) ] } ,
s = 0 , 1 , . . . , N 1
R ( s ) = E [ g ( m ) g * ( m s ) ]
E [ 1 N m = 0 N s 1 g ( m ) g * ( m s ) ] = R ( s ) s N R ( s ) .
E [ 1 N m = 0 N 1 g ( m ) g * ( m s ) ] = { 1 N l = 0 N 1 | f ( l ) | 2 , s = 0 0 , s 0 .
A 0 δ ( τ )
A 0
σ 2 = A 0
θ ( x )
φ ( u )
f ( x )
σ 2 = 1 N l = 0 N 1 | f ( l ) | 2 .
θ ( x )
φ ( u )
φ ( u ) = 1
θ ( x ) = 1
F ˜ α 1
F ˜ α 2
F ˜ α 1
F ˜ α 2
f ( m ) = k = 0 N 1 l = 0 N 1 θ * ( k ) F α 1 ( m , k ) F α 2 ( k , l ) φ * ( l ) g ( l ) ,
( m = 0 , 1 ,   …   ,   N 1 ) .
g = g + Δ g
Δ g = ( Δ g 0 , Δ g 1 ,   …   ,   Δ g N 1 ) T
f = f + Δ f
Δ f = ( Δ f 0 , Δ f 1 ,   …   ,   Δ f N 1 ) T
Δ f m = k = 0 N 1 l = 0 N 1 θ * ( k ) F α 1 ( m , k ) F α 2 ( k , l ) φ * ( l ) Δ g l ,
( m = 0 , 1 ,   …   ,   N 1 ) .
Δ f
E ( 1 N m = 0 N 1 Δ f m ) = 0 ,
E ( 1 N m = 0 N 1 Δ f m Δ f m s * ) = { 1 N l = 0 N 1 Δ g l Δ g l * , s = 0 0 , s 0 .
σ 2 = 1 N l = 0 N 1 Δ g l Δ g l * .
256 × 256
( 0.9 , 0.9 )
( 1.0 , 1.0 )
σ 2 2
σ 2
σ 2 2
σ 2

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