Abstract

The performance of fully phase- and amplitude-based encryption processors is analyzed. The effects of noise perturbations on the encrypted information are considered. A thresholding method of decryption that further reduces the mean-squared error (MSE) for the fully phase- and amplitude-based encryption processes is provided. The proposed thresholding scheme significantly improves the performance of fully phase- and amplitude-based encryption, as measured by the MSE metric. We obtain analytical MSE bounds when thresholding is used for both decryption methods, and we also present computer-simulation results. These results show that the fully phase-based method is more robust. We also give a formal proof of a conjecture about the decrypted distribution of distorted encrypted information. This allows the analytical bounds of the MSE to be extended to more general non-Gaussian, nonadditive, nonstationary distortions. Computer simulations support this extension.

© 2000 Optical Society of America

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References

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  1. Special Issue on Optical Security, Opt. Eng.35, 2451–2541 (1996).
  2. Special Issue on Optical Security, Opt. Eng.38, 1–119 (1999).
  3. B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
    [CrossRef]
  4. P. Réfrégier, B. Javidi, “Optical image encryption using input and Fourier plane random phase encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef]
  5. Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).
  6. B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
    [CrossRef]
  7. K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
    [CrossRef]
  8. H.-Y. Li, Y. Qiao, D. Psaltis, “Optical network for real-time face recognition,” Appl. Opt. 32, 5026–5035 (1993).
    [CrossRef] [PubMed]
  9. B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 1752–1756 (1995).
    [CrossRef]
  10. J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
    [CrossRef]
  11. C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, T. H. Chao, eds., Proc. SPIE3073, 373–382 (1997).
    [CrossRef]
  12. N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
    [CrossRef]
  13. M. Drake, M. Lid, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
    [CrossRef]
  14. F. Goudail, F. Bollaro, B. Javidi, P. Réfrégier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
    [CrossRef]
  15. J. Campello, J. T. Gill, M. Morf, M. J. Flynn, “Smart photonic networks and computer security for image data,” in Multimedia Networks: Security, Displays, Terminals, and Gateways, B. Derryberry, C. R. Holliday, L. S. Lome, V. Markandey, B. Vasudev, M. V. Bove, A. G. Tescher, eds., Proc. SPIE3228, 272–279 (1998).
  16. H. F. Heanue, M. C. Bashaw, L. Hesselink, “Encrypted holographic data storage based on orthogonal-phase-code multiplexing,” Appl. Opt. 34, 6012–6015 (1995).
    [CrossRef] [PubMed]
  17. O. Matoba, B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
    [CrossRef]
  18. B. Javidi, L. Bernard, N. Towghi, “Noise performance of double-phase encryption compared to xor encryption,” Opt. Eng. 38, 9–19 (1999).
    [CrossRef]
  19. L. O’Gorman, I. Rabinovich, “Secure identification documents via pattern recognition and public-key cryptography,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1097–1102 (1998).
    [CrossRef]
  20. N. Towghi, B. Javidi, Z. Luo, “Fully phase encrypted optical processor,” J. Opt. Soc. Am. A 16, 1915–1927 (1999).
    [CrossRef]
  21. V. Gnedenko, The Theory of Probability (Mir, Moscow, 1976).
  22. B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
    [CrossRef]
  23. I. I. Gikhman, A. V. Skorokhod, Introduction to the Theory of Random Process (Dover, New York, 1969).

1999 (4)

Special Issue on Optical Security, Opt. Eng.38, 1–119 (1999).

O. Matoba, B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
[CrossRef]

B. Javidi, L. Bernard, N. Towghi, “Noise performance of double-phase encryption compared to xor encryption,” Opt. Eng. 38, 9–19 (1999).
[CrossRef]

N. Towghi, B. Javidi, Z. Luo, “Fully phase encrypted optical processor,” J. Opt. Soc. Am. A 16, 1915–1927 (1999).
[CrossRef]

1998 (2)

L. O’Gorman, I. Rabinovich, “Secure identification documents via pattern recognition and public-key cryptography,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1097–1102 (1998).
[CrossRef]

F. Goudail, F. Bollaro, B. Javidi, P. Réfrégier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
[CrossRef]

1997 (1)

B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

1996 (4)

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

M. Drake, M. Lid, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Special Issue on Optical Security, Opt. Eng.35, 2451–2541 (1996).

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

1995 (4)

B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 1752–1756 (1995).
[CrossRef]

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

P. Réfrégier, B. Javidi, “Optical image encryption using input and Fourier plane random phase encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

H. F. Heanue, M. C. Bashaw, L. Hesselink, “Encrypted holographic data storage based on orthogonal-phase-code multiplexing,” Appl. Opt. 34, 6012–6015 (1995).
[CrossRef] [PubMed]

1994 (1)

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

1993 (1)

1991 (1)

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Bashaw, M. C.

Bernard, L.

B. Javidi, L. Bernard, N. Towghi, “Noise performance of double-phase encryption compared to xor encryption,” Opt. Eng. 38, 9–19 (1999).
[CrossRef]

Bollaro, F.

Campello, J.

J. Campello, J. T. Gill, M. Morf, M. J. Flynn, “Smart photonic networks and computer security for image data,” in Multimedia Networks: Security, Displays, Terminals, and Gateways, B. Derryberry, C. R. Holliday, L. S. Lome, V. Markandey, B. Vasudev, M. V. Bove, A. G. Tescher, eds., Proc. SPIE3228, 272–279 (1998).

Caulfield, J.

Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).

Drake, M.

M. Drake, M. Lid, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Fiddy, M. A.

M. Drake, M. Lid, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Fielding, K. H.

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Flynn, M. J.

J. Campello, J. T. Gill, M. Morf, M. J. Flynn, “Smart photonic networks and computer security for image data,” in Multimedia Networks: Security, Displays, Terminals, and Gateways, B. Derryberry, C. R. Holliday, L. S. Lome, V. Markandey, B. Vasudev, M. V. Bove, A. G. Tescher, eds., Proc. SPIE3228, 272–279 (1998).

Gikhman, I. I.

I. I. Gikhman, A. V. Skorokhod, Introduction to the Theory of Random Process (Dover, New York, 1969).

Gill, J. T.

J. Campello, J. T. Gill, M. Morf, M. J. Flynn, “Smart photonic networks and computer security for image data,” in Multimedia Networks: Security, Displays, Terminals, and Gateways, B. Derryberry, C. R. Holliday, L. S. Lome, V. Markandey, B. Vasudev, M. V. Bove, A. G. Tescher, eds., Proc. SPIE3228, 272–279 (1998).

Gnedenko, V.

V. Gnedenko, The Theory of Probability (Mir, Moscow, 1976).

Goudail, F.

Guibert, L.

B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

Heanue, H. F.

Hesselink, L.

Horner, J. L.

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Howlader, M.

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

Huang, Q.

Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).

Huignard, J.-P.

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Javidi, B.

N. Towghi, B. Javidi, Z. Luo, “Fully phase encrypted optical processor,” J. Opt. Soc. Am. A 16, 1915–1927 (1999).
[CrossRef]

O. Matoba, B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
[CrossRef]

B. Javidi, L. Bernard, N. Towghi, “Noise performance of double-phase encryption compared to xor encryption,” Opt. Eng. 38, 9–19 (1999).
[CrossRef]

F. Goudail, F. Bollaro, B. Javidi, P. Réfrégier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
[CrossRef]

B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

P. Réfrégier, B. Javidi, “Optical image encryption using input and Fourier plane random phase encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 1752–1756 (1995).
[CrossRef]

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

Li, H.-Y.

Li, J.

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 1752–1756 (1995).
[CrossRef]

Lid, M.

M. Drake, M. Lid, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Luo, Z.

Makekau, C. K.

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Matoba, O.

Morf, M.

J. Campello, J. T. Gill, M. Morf, M. J. Flynn, “Smart photonic networks and computer security for image data,” in Multimedia Networks: Security, Displays, Terminals, and Gateways, B. Derryberry, C. R. Holliday, L. S. Lome, V. Markandey, B. Vasudev, M. V. Bove, A. G. Tescher, eds., Proc. SPIE3228, 272–279 (1998).

O’Gorman, L.

L. O’Gorman, I. Rabinovich, “Secure identification documents via pattern recognition and public-key cryptography,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1097–1102 (1998).
[CrossRef]

Paek, E. G.

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, T. H. Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

Psaltis, D.

Qiao, Y.

Rabinovich, I.

L. O’Gorman, I. Rabinovich, “Secure identification documents via pattern recognition and public-key cryptography,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1097–1102 (1998).
[CrossRef]

Rajbenbach, H.

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Réfrégier, P.

Riza, N.

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

Rodolfo, J.

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Sergent, A.

B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

Skorokhod, A. V.

I. I. Gikhman, A. V. Skorokhod, Introduction to the Theory of Random Process (Dover, New York, 1969).

Tang, Q.

B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 1752–1756 (1995).
[CrossRef]

Towghi, N.

N. Towghi, B. Javidi, Z. Luo, “Fully phase encrypted optical processor,” J. Opt. Soc. Am. A 16, 1915–1927 (1999).
[CrossRef]

B. Javidi, L. Bernard, N. Towghi, “Noise performance of double-phase encryption compared to xor encryption,” Opt. Eng. 38, 9–19 (1999).
[CrossRef]

Watson, C. I.

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, T. H. Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

Wilson, C. L.

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, T. H. Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

Zhang, G.

B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

L. O’Gorman, I. Rabinovich, “Secure identification documents via pattern recognition and public-key cryptography,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1097–1102 (1998).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (10)

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

M. Drake, M. Lid, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Special Issue on Optical Security, Opt. Eng.35, 2451–2541 (1996).

Special Issue on Optical Security, Opt. Eng.38, 1–119 (1999).

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

B. Javidi, L. Bernard, N. Towghi, “Noise performance of double-phase encryption compared to xor encryption,” Opt. Eng. 38, 9–19 (1999).
[CrossRef]

Opt. Lett. (2)

Other (5)

J. Campello, J. T. Gill, M. Morf, M. J. Flynn, “Smart photonic networks and computer security for image data,” in Multimedia Networks: Security, Displays, Terminals, and Gateways, B. Derryberry, C. R. Holliday, L. S. Lome, V. Markandey, B. Vasudev, M. V. Bove, A. G. Tescher, eds., Proc. SPIE3228, 272–279 (1998).

Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, T. H. Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

I. I. Gikhman, A. V. Skorokhod, Introduction to the Theory of Random Process (Dover, New York, 1969).

V. Gnedenko, The Theory of Probability (Mir, Moscow, 1976).

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Figures (10)

Fig. 1
Fig. 1

Vector representation of decryption for (a) amplitude-based encryption, (b) full phase encryption.

Fig. 2
Fig. 2

Block diagram of the implementation of full phase encryption with double-random phase-encoding technique. FT, Fourier transform.

Fig. 3
Fig. 3

Fully phase- and amplitude-based encryption for gray-scale information in additive white noise with standard deviation of σ = 0.2: (a) original gray-scale image of Franklin D. Roosevelt; (b) Gaussian white noise, n(x, y), with σ = 0.2; (c) fully phase-encrypted information ψ p (x, y) plus white noise n(x, y); (d) amplitude-based encrypted information ψ A (x, y) plus white noise n(x, y); (e) recovered information from full phase encryption with intensity decryption (MSE = 0.0021); (f) recovered information from amplitude-based encryption with intensity decryption (MSE = 0.0401); (g) Recovered information from full phase encryption with thresholding method (MSE = 0.0013); (h) recovered information from amplitude-based encryption with thresholding method (MSE = 0.0243).

Fig. 4
Fig. 4

Theoretical upper-bound MSE and experimental MSE for fully phase- and amplitude-based encryption in the presence of white Gaussian noise. Error plots for amplitude-based decryption: inverted triangles, experimental MSE with intensity decryption; pluses, theoretical MSE experimental upper bound for a thresholding method; circles, experimental MSE for a thresholding method. Error plots for phase-based decryption: dashed curve, experimental MSE with intensity decryption; solid curve, theoretical MSE upper bound for a thresholding method; asterisks, experimental MSE with a thresholding method.

Fig. 5
Fig. 5

Theoretical upper-bound MSE and experimental MSE for fully phase- and amplitude-based encryption in the presence of colored non-Gaussian of 5 × 5 pixel bandwidth with the thresholding method. Decryption errors for fully phase- and amplitude-based decryption; pluses, theoretical MSE estimate for amplitude-based decryption; circles, experimental MSE for amplitude-based decryption; solid curve, theoretical estimate for fully phase-based decryption; asterisks, experimental MSE for fully phase-based decryption.

Fig. 6
Fig. 6

Fully phase- and amplitude-based encryption for binary information in additive colored noise with standard deviation of σ = 0.2: (a) original binary text, (b) colored noise n(x, y), (c) fully phase-encrypted information ψ p (x, y) plus colored noise n(x, y), (d) amplitude-based encrypted information ψ A (x, y) plus colored noise n(x, y), (e) recovered information from full phase encryption with intensity decryption (MSE = 0.0025), (f) recovered information from amplitude-based encryption with intensity decryption (MSE = 0.0432), (g) recovered information from full phase encryption with thresholding method (MSE = 0.0010), (h) recovered information from amplitude-based encryption with thresholding method (MSE = 0.0200).

Fig. 7
Fig. 7

Spatial distribution of one realization of an amplitude-based decryption of nonstationary, non-Gaussian perturbation of encrypted information. Decryption transforms the noise into a complex-valued additive white Gaussian distribution. (a) Distribution of the real part of the distorted portion, (b) distribution of the imaginary part of the distorted portion.

Fig. 8
Fig. 8

Spatial distribution of one realization of the distorted portion of decrypted information for fully phase-encrypted information distorted by multiplicative non-Gaussian noise of mean 1 and variance 0.25. (a) Distribution of the real part of the distorted portion, (b) distribution of the imaginary part of the distorted portion.

Fig. 9
Fig. 9

Performance of fully phase encryption when the encrypted information is distorted by multiplicative non-Gaussian noise of mean 1.0 and standard deviation of 0.25. (a) Original information. (b) Fully phase-encoded information. (c) Noisy encrypted information. (d) Decrypted information with thresholding method; actual MSE = 0.0014, predicted theoretical MSE = 0.0014.

Fig. 10
Fig. 10

Performance of amplitude-based encryption when the encrypted information is distorted by multiplicative non-Gaussian noise of mean 1.0 and standard deviation 0.25. (a) Original information. (b) Noisy encrypted information. (c) Decrypted information with only the real part; actual MSE = 0.0104, predicted theoretical MSE = 0.0109. (d) Decrypted information with thresholding; actual MSE = 0.0065, predicted theoretical MSE = 0.00643.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

ψax, y=fx, yexpi2πpx, y * hx, y,
ψax, y=fx, yexpi2πpx, y * hx, y+nx, y.
fax, y=fx, y+FT-1Nv, wexp-i2πbv, w×exp-i2πpx, y=fx, y+n0x, y,
fatx, y=0if Refax, y<0Refax, yif 0Refax, y11if 1<Refax, y.
ψpx, y=expiπfx, yexpi2πpx, y * hx, y.
ψpx, y=expiπfx, y+i2πpx, y * hx, y+nx, y.
Ax, yexpiπfpx, y=expiπfx, y+n0x, y,
n0x, y=FT-1Nv, wexp-i2πbv, w×exp-i2πpx, y.
n0x, y=n0Rx, y+in0Ix, y,
Ax, yexpiπfpx, y=cosπfx, y+n0Rx, y+isinπfx, y+n0Ix, y.
|fpx, y|=|ArgAx, yexpiπfpx, y/π|,
fptx, y=1if Ax, yexpiπfpx, y is in Quadrant III0if Ax, yexpiπfpx, y is in Quadrant IVArgAx, yexpiπfpx, y/πotherwise.
Errfrx, y=E1N×Mx=0N-1y=0M-1|fx, y-frx, y|2,
gψx, y=FT-1Kψu, vexp-i2πbu, v×exp-i2πpx, y,
n0x, y=gψx, y-fdx, y,
Varn01M×N2x=0N-1y=0M-1 E|gψx, y-fdx, y|2.
Err|fa|σ2.
E|f-fat|2=12πσ0-f1-f x2 exp-x22σ02dx+f2f1-fexp-x22σ02dx+1-f1-f2 exp-x22σ02dx,
Errfat18erfc122σ0+σ02 erf122σ0+σ02πexp-18σ02,
erfx=2π0xexp-t2dt,erfcx=2πxexp-t2dt.
Err|fp|σ02π2erf122σ0+6σ02π1-exp-18σ02+29erfc122σ0+12erf22σ0-erf122σ0+σ022π2erf32σ0-erf22σ0+14erfc32σ0.
Errfpt1π2NMx=0N-1y=0M-1 Efptx, y-122.
|fptx, y-fx, y|2tan-1n0Rx, y1+n0Ix, y2n0I>-1=π/22n0I-1.
π2 Errfpt12πσ02-1-×tan-1x1+y2exp-x22σ02×exp-y22σ02dxdy+12πσ02--1-π22×exp-x2σ02exp-y2σ02dxdy.
ϕ=ψ+ψ,
g=f+f,
gx=expi2πpx1Nu=0N fuexp2πibu+uxN,
Varg1N2u=0N E|fu|2,
limnu=0NτLN x2dFuxLN=0
Varg1N2u=0N |fu|2.
fatx, y=0if Refax, y<0Refax, yif 0Refax, y11if 1<Refax, y,
fax, y=fx, y+FT-1Nv, wexp-iπbv, w×exp-i2πpx, y=fx, y+n0x, y.
E|fx, y-fatx, y|2=σ02-f22erff22σ0+σ02-f22erf1-f22 σ0+σ02πf exp-f22σ02+1-fexp-1-f22σ02.
E|f-fat|=12πσ0-f1-f x2 exp-x22σ02dx+f2f1-fexp-x22σ02dx+1-f1-f2 exp-x22σ02dx.
Errfat18erfc122 σ0+σ02 erf122σ0+σ02πexp-18σ02.
12πσ02-1-tan-1x1+y2 exp-x22σ02×exp-y22σ02dxdy+12πσ02--1-π22×exp-x2σ02exp-y2σ02dxdy.
Px=12πσ0exp-x22σ02,
qx, y=tan-1x1+y2.
12πσ02-1-tan-1x1+y2×exp-x22σ02exp-y22σ02dxdy=2 -1-c1 qx, yPxPydxdy+2 -c01 qx, yPxPydxdy+2 0c1 qx, yPxPydxdy+2 c11 qx, yPxPydxdy+2 11 qx, yPxPydxdy+2 101 qx, yPxPydxdy+2 -1-c01 qx, yPxPydxdy+2 -c001 qx, yPxPydxdy+2 0c01 qx, yPxPydxdy+2 c101 qx, yPxPydxdy=I + II + III + IV + V=T1.
V0=σ0 exp-1/2σ02+πσ02erf1/σ022π,
t11=V0erfc/σ0221-c2,
t12=erf1/σ02erfc/σ022,
t1=erf1/σ02erf1-c/σ022+mint11, t12,
t2=erf1/σ02erf1-c/σ02-erfc/σ022,
t3=8V0erf1/σ02+erfc1/σ022+erf1/σ028.
T1t1+t2+t3.
12πσ02--1-π24exp-x22σ02exp-y22σ02dxdy,
t4=π2 erfc1/σ028.
Errfptt1+t2+t3+t4π2.

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