Abstract

The amplitude-encoding case of the double random phase encoding technique is examined by defining a cost function as a metric to compare an attempted decryption against the corresponding original input image. For the case when a cipher–text pair has been obtained and the correct decryption key is unknown, an iterative attack technique can be employed to ascertain the key. During such an attack the noise in the output field for an attempted decryption can be used as a measure of a possible decryption key’s correctness. For relatively small systems, i.e., systems involving fewer than 5×5  pixels, the output decryption of every possible key can be examined to evaluate the distribution of the keys in key space in relation to their relative performance when carrying out decryption. However, in order to do this for large systems, checking every single key is currently impractical. One metric used to quantify the correctness of a decryption key is the normalized root mean squared (NRMS) error. The NRMS is a measure of the cumulative intensity difference between the input and decrypted images. We identify a core term in the NRMS, which we refer to as the difference parameter, d. Expressions for the expected value (or mean) and variance of d are derived in terms of the mean and variance of the output field noise, which is shown to be circular Gaussian. These expressions assume a large sample set (number of pixels and keys). We show that as we increase the number of samples used, the decryption error obeys the statistically predicted characteristic values. Finally, we corroborate previously reported simulations in the literature by using the statistically derived expressions.

© 2009 Optical Society of America

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References

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  1. H. O. Yardley, The American Black Chamber (Naval Institute Press, 1931).
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    [CrossRef]
  7. B. Javidi, Optical and Digital Techniques for Information Security (Springer Verlag, 2005).
    [CrossRef]
  8. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39, 6595-6601 (2000).
    [CrossRef]
  9. L. E. M. Brackenbury and K. M. Bell, “Optical encryption of digital data,” Appl. Opt. 39, 5374-5379 (2000).
    [CrossRef]
  10. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887-889 (2000).
    [CrossRef]
  11. B. M. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269-271 (2003).
    [CrossRef] [PubMed]
  12. T. J. Naughton and B. Javidi, “Compression of encrypted three-dimensional objects using digital holography,” Opt. Eng. 43, 2233-2238 (2004).
    [CrossRef]
  13. B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
    [CrossRef]
  14. B. Javidi, A. Sergent, G. S. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992-998 (1997).
    [CrossRef]
  15. F. Goudail, F. Bollaro, B. Javidi, and P. Réfrégier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629-2638 (1998).
    [CrossRef]
  16. B. Javidi, N. Towghi, N. Maghzi, and S. C. Verrall, “Error-reduction techniques and error analysis for fully phase- and amplitude-based encryption,” Appl. Opt. 39, 4117-4130 (2000).
    [CrossRef]
  17. B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).
    [CrossRef]
  18. B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
    [CrossRef]
  19. U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, “A known-plaintext heuristic attack on the Fourier plane encryption algorithm,” Opt. Express 14, 3181-3186 (2006).
    [CrossRef] [PubMed]
  20. T. J. Naughton, B. Hennelly, and T. Dowling, “Introducing secure modes of operation for optical encryption,” J. Opt. Soc. Am. A 25, 2608-2617 (2008).
    [CrossRef]
  21. D. S. Monaghan, G. Situ, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Role of phase key in the double random phase encoding technique: an error analysis,” Appl. Opt. 47, 3808-3816 (2008).
    [CrossRef] [PubMed]
  22. D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. 46, 6641-6647 (2007).
    [CrossRef] [PubMed]
  23. D. S. Monaghan, G. Situ, G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Analysis of phase encoding for optical encryption,” Opt. Commun. 282, 482-492 (2008).
    [CrossRef]
  24. G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46, 5257-5262 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  27. D. S. Monaghan, “Practical implementations and theoretical analysis of optical encryption,” Ph.D. dissertation (University College Dublin, 2009).
  28. Lena Test Image, http://sipi.usc.edu/database/.
  29. Matlab 7.0.1, http://www.mathworks.com/.

2008 (3)

2007 (3)

2006 (1)

2005 (1)

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

2004 (2)

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

T. J. Naughton and B. Javidi, “Compression of encrypted three-dimensional objects using digital holography,” Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

2003 (2)

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269-271 (2003).
[CrossRef] [PubMed]

2000 (4)

1998 (1)

1997 (1)

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

1995 (1)

1994 (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 224-230 (1994).
[CrossRef]

1976 (1)

W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

Bell, K. M.

Bollaro, F.

Brackenbury, L. E. M.

Castro, A.

Deavours, C. A.

C. A. Deavours, Cryptology Yesterday, Today and Tomorrow (Artech House, 1987).

Diffie, W.

W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

Dowling, T.

Frauel, Y.

Gaines, G. F.

G. F. Gaines, Cryptanalysis: A Study of Ciphers and Their Solution (Dover, 1939).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gopinathan, U.

Goudail, F.

Guibert, L.

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

Hellman, M. E.

W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

Hennelly, B.

Hennelly, B. M.

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269-271 (2003).
[CrossRef] [PubMed]

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 224-230 (1994).
[CrossRef]

Javidi, B.

Joseph, J.

Maghzi, N.

Monaghan, D. S.

Naughton, T. J.

Réfrégier, P.

Sergent, A.

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

Sheridan, J. T.

D. S. Monaghan, G. Situ, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Role of phase key in the double random phase encoding technique: an error analysis,” Appl. Opt. 47, 3808-3816 (2008).
[CrossRef] [PubMed]

D. S. Monaghan, G. Situ, G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Analysis of phase encoding for optical encryption,” Opt. Commun. 282, 482-492 (2008).
[CrossRef]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46, 5257-5262 (2007).
[CrossRef] [PubMed]

D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. 46, 6641-6647 (2007).
[CrossRef] [PubMed]

U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, “A known-plaintext heuristic attack on the Fourier plane encryption algorithm,” Opt. Express 14, 3181-3186 (2006).
[CrossRef] [PubMed]

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269-271 (2003).
[CrossRef] [PubMed]

Singh, K.

Situ, G.

Tajahuerce, E.

Towghi, N.

Unnikrishnan, G.

D. S. Monaghan, G. Situ, G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Analysis of phase encoding for optical encryption,” Opt. Commun. 282, 482-492 (2008).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887-889 (2000).
[CrossRef]

Verrall, S. C.

Yardley, H. O.

H. O. Yardley, The American Black Chamber (Naval Institute Press, 1931).

Zhang, G. S.

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

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

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

Opt. Commun. (2)

D. S. Monaghan, G. Situ, G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Analysis of phase encoding for optical encryption,” Opt. Commun. 282, 482-492 (2008).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

Opt. Eng. (4)

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

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 224-230 (1994).
[CrossRef]

T. J. Naughton and B. Javidi, “Compression of encrypted three-dimensional objects using digital holography,” Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Optik (Stuttgart) (1)

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).
[CrossRef]

Other (8)

B. Javidi, Optical and Digital Techniques for Information Security (Springer Verlag, 2005).
[CrossRef]

C. A. Deavours, Cryptology Yesterday, Today and Tomorrow (Artech House, 1987).

H. O. Yardley, The American Black Chamber (Naval Institute Press, 1931).

G. F. Gaines, Cryptanalysis: A Study of Ciphers and Their Solution (Dover, 1939).

J. W. Goodman, Statistical Optics (Wiley, 2000).

D. S. Monaghan, “Practical implementations and theoretical analysis of optical encryption,” Ph.D. dissertation (University College Dublin, 2009).

Lena Test Image, http://sipi.usc.edu/database/.

Matlab 7.0.1, http://www.mathworks.com/.

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Figures (7)

Fig. 1
Fig. 1

Re [ A d ( p ) ] λ f ( p ) and Im [ A d ( p ) ] averaged over 1000 runs, for randomly chosen keys, for the 256 × 256 pixel Lena test image. The results are Gaussian and support the conjecture made in Eq. (15). As the total NRMS error value for an attempted decryption decreases, the area under the corresponding Gaussian noise graph decreases.

Fig. 2
Fig. 2

NRMS, Eq. (11), error values for each of the 1000 runs, with different keys used to generate Fig. 1. The average NRMS value here is calculated to be 1.0001, which is very close to 1.

Fig. 3
Fig. 3

The blue curve (hidden blue triangles), plotted using Eq. (11), represents the direct NRMS calculated from the standard error metric. The red curve (red overlapping circles) is plotted by using Eq. (34), derived in Section 4, and is a close approximation to the numerical NRMS values (small black triangle). The dashed box relates to Fig. 4.

Fig. 4
Fig. 4

(a) Enlarged region indicated by the dashed box in Fig. 3. The NRMS error is plotted against the percentage of correct pixel values retained in R 2 , d , during decryption. If 20% of R 2 , d is correct, then it implies that 80% of R 2 , d is made up of pseudorandomly generated, incorrect, quantization levels. (a) Two curves for 1 run of the simulation, large variations in the actual NRMS (blue curve) can be seen. (b) Curves plotted when averaged over 500 runs.

Fig. 5
Fig. 5

(i) ( E R [ d ] ) 2 (blue with triangle), (ii) E R [ d 2 ] (red with circle), and (iii) V R [ d ] (green with square). These curves are plotted by using the same input data and system as was used to generate Fig. 3. For all possible keys, i.e., over all R, V R [ d ] = 5.4 × 10 7 . It should be noted that, for any case where the expected value for λ is not 0, the expectation has been calculated over a subset of R.

Fig. 6
Fig. 6

The diagonal line is made up of 100 runs for each percentage correct pixel case (in total 10 4 points). The cross sections shown are for the cases when (a) 10% and (b) 90% of the correct decryption key is present. Insets, histograms showing the distribution of λ values for the specific percentages of pixels in error (1000 runs).

Fig. 7
Fig. 7

Following perfect encryption of a 256 × 256 pixel , 16 quantization level Lena test image, 10 6 decrypting phase keys, R 2 , d , are randomly generated. It can be seen that the expected value of NRMS 1 . Furthermore, even after attempting decryption with 10 6 different phase keys, the lowest NRMS value calculated was 0.9998. The calculated V R [ d ] value of this curve, with R = 10 6 , is 3.275 × 10 11 .

Equations (41)

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E [ f ] = μ = 1 K k = 1 k = K f ( k ) .
V [ f ] = σ 2 = 1 K k = 1 k = K { f ( k ) E [ f ] } 2 .
V [ f ] = E [ f 2 ] E [ f ] 2 .
N ( μ , σ ; x ) = 1 2 π σ 2 exp [ ( x μ ) 2 2 σ 2 ] ,
M n , z = + ( x z ) n N ( μ , σ ; x ) d x ,
E [ g 2 ] = μ 2 + σ 2 .
E [ n ] = E [ n r , i ] = 0 ,
V [ n r , i ] = σ 2 = E [ n r , i 2 ] .
n n * = n * n = | n | 2 = | n r | 2 + | n i | 2 ;
E [ | n | 2 ] = E [ | n r | 2 + | n i | 2 ] = E [ | n r | 2 ] + E [ | n i | 2 ] = 2 σ 2 .
p = 1 p = N | I ( p ) I d ( p ) | 2 p = 1 p = N | I ( p ) | 2 ,
d = p = 1 p = N | I ( p ) I d ( p ) | 2 .
A d = I 1 ( I 1 { I [ I ( f × R 1 , e ) ] } ) I 1 { I 1 [ I ( R 2 , e ) ] } I 1 ( R 2 , d )
= [ f × R 1 , e ] [ I ( R 2 , d ) I ( R 2 , e ) ] ,
A d ( ) λ f ( ) + n ( ) ,
I d = | λ f + n | 2 = λ 2 f 2 + λ f [ 2 n r ] + | n | 2 .
d = p = 1 p = N | ( λ 2 1 ) f 2 ( p ) + 2 λ f ( p ) n r ( p ) + | n ( p ) | 2 | 2 .
d = ( λ 2 1 ) 2 p = 1 p = N { f 4 ( p ) } + 4 λ ( λ 2 1 ) p = 1 p = N { f 3 ( p ) n r ( p ) } + 2 ( λ 2 1 ) p = 1 p = N { f 2 ( p ) | n ( p ) | 2 } + 4 λ 2 p = 1 p = N { f 2 ( p ) n r 2 ( p ) } + 4 λ p = 1 p = N { f ( p ) n r ( p ) | n ( p ) | 2 } + p = 1 p = N { | n ( p ) | 4 } .
d = ( λ 2 1 ) 2 E [ f 4 ( p ) ] N + 4 λ ( λ 2 1 ) E [ f 3 ( p ) n r ( p ) ] N + 2 ( λ 2 1 ) E [ f 2 ( p ) | n ( p ) | 2 ] N + 4 λ 2 E [ f 2 ( p ) n r 2 ( p ) ] N + 4 λ E [ f ( p ) n r ( p ) | n ( p ) | 2 ] N + E [ | n ( p ) | 4 ] N .
E [ f k n r l ] E [ f k ] × E [ n r l ] ,
E [ f k | n | 2 l ] E [ f k ] × E [ | n | 2 l ] .
d N = ( λ 2 1 ) 2 E [ f 4 ( p ) ] + 4 λ ( λ 2 1 ) E [ f 3 ( p ) ] E [ n r ( p ) ] + 2 ( λ 2 1 ) E [ f 2 ( p ) ] E [ | n ( p ) | 2 ] + 4 λ 2 E [ f 2 ( p ) ] E [ n r 2 ( p ) ] + 4 λ E [ f ( p ) ] E [ n r ( p ) | n ( p ) | 2 ] + E [ | n ( p ) | 4 ] .
f m = E [ f m ( p ) ] 1 N p = 1 p = N f m ( p ) ,
( n p ) m = E [ n m ( p ) ] 1 N p = 1 p = N n m ( p ) ,
| n p | m = E [ | n ( p ) | m ] 1 N p = 1 p = N | n ( p ) | m .
d = d N = ( λ 2 1 ) 2 f 4 + 4 λ ( λ 2 1 ) f 3 n r , p + 2 ( λ 2 1 ) f 2 | n p | 2 + 4 λ 2 f 2 n r , p 2 + 4 λ f 1 n r , p | n p | 2 + | n p | 4 .
E R [ d ] = E R [ ( λ 2 1 ) 2 ] f 4 + 4 f 3 E R [ λ ( λ 2 1 ) n r , p ] + 2 f 2 E R [ ( λ 2 1 ) | n p | 2 ] + 4 f 2 E R [ λ 2 n r , p 2 ] + 4 f 1 E R [ λ n r , p | n p | 2 ] + E R [ | n p | 4 ] .
E R [ d ] = f 4 ( E R [ λ 4 ] 2 E R [ λ 2 ] + 1 ) + 4 f 3 E R [ λ ] ( E R [ λ 2 ] 1 ) E R [ n r , p ] + 2 f 2 ( E R [ λ 2 ] 1 ) E R [ | n p | 2 ] + 4 f 2 E R [ λ 2 ] E R [ n r , p 2 ] + 4 f 1 E R [ λ ] E R [ n r , p ] E R [ | n p | 2 ] + E R [ | n p | 4 ] .
E [ | n | 4 ] = 2 E [ | n | 2 ] 2 ,
E [ | n | 2 n ] = 0 , E [ n 2 ] 0 .
E R [ d ] = ( lim R 1 R r ̃ = 1 R { λ 4 ( r ̃ ) } 2 lim R 1 R r ̃ = 1 R { λ 2 ( r ̃ ) } + 1 ) f 4 + 2 f 2 ( lim R 1 R r ̃ = 1 R { λ 2 ( r ̃ ) } 1 ) 2 σ 2 + 4 f 2 lim R 1 R r ̃ = 1 R { λ 2 ( r ̃ ) } σ 2 + 8 σ 4 .
1 N p = 1 N | λ f + n | 2 = 1 N p = 1 N | f | 2 .
σ 2 = [ lim R 1 R r ̃ = 1 R ( 1 λ ( r ̃ ) 2 ) ] f 2 2 .
E R [ d ] = { 1 E R [ λ ( r ̃ ) 2 ] } { 2 f 2 2 E R [ λ ( r ̃ ) 2 ] + f 4 ( 1 E R [ λ ( r ̃ ) 2 ] ) } f 4 ,
E [ n r , i q ] = σ q π 2 ( q 2 1 ) [ 1 + ( 1 ) q ] Γ ( 1 + q 2 ) .
E R [ d 2 ] = 4 f 2 2 f 4 E R [ λ 2 ] ( 1 + E R [ λ 2 ] ) 3 + f 4 2 ( 1 + E R [ λ 2 ] ) 4 + 4 f 2 4 E R [ λ 2 ] ( 4 5 E R [ λ 2 ] + 2 E R [ λ 4 ] ) .
V R [ d ] = [ 4 f 2 2 f 4 E R [ λ 2 ] ( 1 + E R [ λ 2 ] ) 3 + f 4 2 ( 1 + E R [ λ 2 ] ) 4 + 4 f 2 4 E R [ λ 2 ] ( 4 5 E R [ λ 2 ] + 2 E R [ λ 4 ] ) ] [ { 1 E R [ λ ( r ̃ ) 2 ] } { 2 f 2 2 E R [ λ ( r ̃ ) 2 ] + f 4 ( 1 E R [ λ ( r ̃ ) 2 ] ) } ] 2 f 4 2 f 4 2 0 .
Re [ A d ( p ) ] λ Re [ f ( p ) ] + Re [ n ( p ) ] ,
Im [ A d ( p ) ] λ Im [ f ( p ) ] + Im [ n ( p ) ] .
λ = p = 1 N Re [ A d ( p ) ] p = 1 N f ( p ) ,
( N N x ) × ( q 1 ) x .

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