CN104134184A - Image encryption method based on iteration phase cutting fractional Fourier transform - Google Patents
Image encryption method based on iteration phase cutting fractional Fourier transform Download PDFInfo
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Abstract
The invention provides an image encryption method based on iteration phase cutting fractional Fourier transform. According to the method, an original image to be encrypted is encrypted into two phase blocks on the basis of the iteration phase cutting fractional Fourier transform, and the encryption process realizes the nonlinearity and is completed by using a numerical method; and the decryption can be realized by adopting an optical measure, wherein an optical realization device for the decryption is simple, and a holographic technique does not need for recording phase information in the decryption process. The iteration encryption method provided by the invention has the advantages that the convergence speed is high; meanwhile, the fractional exponent number in the encryption process becomes a secret key required by the decryption; the security of a system is enhanced; and an information leakage problem of an encryption result is avoided.
Description
[technical field]
The present invention relates to the encryption method of a kind of field of information security technology, particularly image.
[background technology]
In the last few years, use optical means to carry out safe handling to information and become gradually the study hotspot of information security field.Nineteen ninety-five, the P.R é fr é gier of Connecticut university of the U.S. and two experts of B.Javidi have designed the double random phase coded system based on optics 4f system.They,, by the random phase plate of placing two statistical independences on the input plane in 4f optical system and fourier spectrum face, have finally realized the encryption of information.This technology has obtained United States Patent (USP) protection.But because double random phase coding techniques need to use holographic technique, the phase information of encrypted result is carried out to record, therefore its Optical Implementation is comparatively complicated.What is more important, the research of recent years shows, due to the linear feature of ciphering process, the double random phase coded system based on optics 4f system exists safety problem.2010, the people such as the Peng Xiang of domestic Shenzhen University proposed the optical image encryption system based on cut Fourier transform, make ciphering process have nonlinear feature, thereby improved the security of system by introducing phase place excision operation.But it is pointed out that non-linear double random phase encryption system need to repeatedly record phase information, therefore its Optical Implementation device is more more complicated than linear double random phase encryption system.In order to solve the too complicated problem of most of encryption system Optical Implementation devices, 2008, first the people such as the Zhang Yan of Capital Normal University use principle of optical interference that piece image is encrypted to two phase-plates by numeric value analysis ground method, its ciphering process uses numerical computation method, and decrypting process adopts optical instrument.When deciphering, as long as two phase-plates are correctly placed in optics decryption system, in the output face of system, just can obtain correct decrypted result.Decrypted result can pass through the direct record of light intensity detector.But research shows, this encryption method exists information leakage problem, only a phase-plate wherein need be placed in optics decryption system, just can in output face, obtain the most information of original image.If eliminate the information leakage problem of phase-plate, need phase-plate to carry out the processing of numerical value scramble, but when this can cause deciphering, cannot in the output face of system, directly obtain original image, and the also principle that will realize original image still needs further by computing machine, optics decrypted result to be carried out to numerical value processing.
[summary of the invention]
The technical problem to be solved in the present invention is to provide the image encryption method based on iteration cut fractional Fourier transform.
Solve the problems of the technologies described above and adopt following technical measures: the image encryption method based on iteration cut fractional Fourier transform carries out as follows:
(1) encrypt:
(i) f (x, y) represents original image to be encrypted, R
1(x, y) and R '
1(u, v) is two random phase plate that use as encryption key in first interative computation, can specifically be expressed as respectively exp[2 π r
1(x, y)] and exp[2 π r
2(u, v)], the wherein coordinate of (x, y) and (u, v) difference representation space territory and fraction Fourier frequency domain, r
1(x, y) and r
2(u, v) represent that two interval [0,1] on, there is the stochastic matrix of even probability distribution and statistical independence, in the time using iteration cut fractional Fourier transform to be encrypted, the k time (k=1,2,3 ...) interative computation process can generate needed two the encryption key R of interative computation the k+1 time
k+1(x, y) and R '
k+1(u, v), in the time carrying out the k time interative computation, first to f (x, y) and encryption key R
kthe product of (x, y) is made fractional Fourier transform, then the complex amplitude obtaining after conversion is got amplitude and is got phase operation, obtains respectively distribution of amplitudes g
k(u, v) and PHASE DISTRIBUTION P
k(u, v),
g
k(u，v)＝PT{F
α[f(x，y)R
k(x，y)]} (1)
P
k(u，v)＝PR{F
α[f(x，y)R
k(x，y)]} (2)
Wherein amplitude computing is got in PT{} representative, the phase information of removing complex amplitude, and phase bit arithmetic is got in PR{} representative, removes the amplitude information of complex amplitude, F
α[] represents that exponent number is the fractional Fourier transform (Fractional Fourier Transform, FRFT) of α, product f (x, the y) R of two functions in formula (1) and formula (2)
kthe α rank fractional Fourier transform of (x, y) is defined as
Wherein K
α(x, y; U, v) be the core of two-dimentional fractional Fourier transform,
Wherein
and φ=α pi/2, α is the exponent number of fractional form;
(ii) to g
k(u, v) and R '
kthe product of (u, v) is done to get phase operation after the fractional Fourier transform of α rank, obtains PHASE DISTRIBUTION P '
k(x, y),
P′
k(x，y)＝PR{F
α[g
k(u，v)R′
k(u，v)]} (5)
(iii) to P '
k(x, y) does to obtain a COMPLEX AMPLITUDE after the fractional Fourier transform of (α) rank, after this distribution is got amplitude and got phase operation, obtains respectively distribution of amplitudes g '
k(u, v) and PHASE DISTRIBUTION R '
k+1(u, v)
g′
k(u，v)＝PT{F
-α[P′
k(x，y)]} (6)
R′
k+1(u，v)＝PR{F
-α[P′
k(x，y)]} (7)
Then to g '
k(u, v) and P
kthe product of (u, v) is made (α) rank fractional Fourier transform, and the complex amplitude obtaining after conversion is got phase place and got amplitude operation, obtains respectively PHASE DISTRIBUTION R
k+1(x, y) and distribution of amplitudes f '
k(x, y), computing formula is respectively
R
k+1(x，y)＝PR{F
-α[g′
k(u，v)P
k(u，v)]} (8)
f′
k(x，y)＝PT{F
-α[g′
k(u，v)P
k(u，v)]} (9)
Thus, in the k time interative computation process, by using P '
k(x, y) and P
k(u, v) two phase-plates calculate required two the encryption key R ' that use of interative computation process the k+1 time
k+1(u, v) and R
k+1(x, y), has also obtained amplitude image in addition as f '
k(x, y), enters lower whorl interative computation process (i.e. the k+1 time interative computation) subsequently;
(iv), in the time that iterations completes n time altogether, interative computation stops, and obtains respectively two phase-plate P ' according to formula (2), formula (5)
n(u, v) and P
n(x, y),
P
n(u，v)＝PR{F
α[f(x，y)R
n(x，y)]} (10)
P′
n(x，y)＝PR{F
α[g
n(u，v)R′
n(u，v)]} (11)
Wherein g
n(u, v) generates in the n time interative computation process, and its value is g
n(u, v)=PT{F
α[f (x, y) R
n(x, y)] }, from formula (7) and formula (8), R
n(x, y) and R '
n(u, v) all generates in the n-1 time interative computation process, and the encrypted result finally obtaining after the n time interative computation is two phase-plates, uses respectively function P (u, v) and P ' (x, y) to represent, its expression formula is
P ' (x, y)=P '
n(x, y) (13) wherein " * " represent conjugation, R '
n+1(u, v) generates in the n time interative computation process, and its value is R '
n+1(u, v)=PR{F
-α[P '
n(x, y)] };
(2) deciphering:
(i) make (α) rank fractional Fourier transform, the result F obtaining after conversion to encrypting the phase-plate P ' (x, y) obtaining
-αafter another phase-plate P (u, v) that [P ' (x, y)] and encryption obtain multiplies each other, make (α) rank fractional Fourier transform, the result obtaining after conversion is expressed as F
-α[F
-α[P ' (x, y)] P (u, v)];
(ii) result obtaining in previous step is got to amplitude computing, finally obtain deciphering image, with f ' (x, y) expression, have f ' (x, y)=PT{F
-α[F
-α[P ' (x, y)] P (u, v)] }, can be proved by formula (6), formula (7), formula (9), formula (12), formula (13):
Therefore, the image that obtains of deciphering is exactly that the amplitude image that obtains of the n time interative computation of ciphering process is as f '
n(x, y).
Beneficial effect of the present invention is: first, ciphering process uses numerical computation method, and decrypting process adopts optical means, does not need to carry out the holographic recording of phase place; Secondly, the iterative cryptographic method fast convergence rate based on cut fractional Fourier transform, fractional order becomes the required key of deciphering simultaneously, has increased security; Finally, encrypt final two phase-plates that generate, do not need generating amplitude plate, and there is not the problem of information leakage in two phase-plates.
[brief description of the drawings]
Fig. 1 is the k time iteration cut fractional Fourier transform ciphering process process flow diagram.
Fig. 2 decrypting process process flow diagram.
Fig. 3 is optics deciphering schematic diagram.
Fig. 4 (a) image f to be encrypted (x, y) (Cameraman); (b) the phase-plate P (u, v) obtaining after interative computation 50 times; (c) the phase-plate P ' (x, y) obtaining after interative computation 50 times; (d) decrypted result being obtained by P (u, v) and P ' (x, y).
Fig. 5 is that the amplitude image that obtains in interative computation process is as f
kmSE value between (x, y) and original image f (x, y) and the graph of a relation of iterations.
The corresponding decrypted result of the different iterationses of Fig. 6: (a) 3; (b) 5; (c) 10.
The decrypted result that Fig. 7 (a) obtains after only using P (u, v) to be decrypted; (b) decrypted result obtaining after only using P ' (x, y) to be decrypted; (c) with P (u, v) and a decrypted result that the phase-plate of generation obtains after being decrypted at random; (d) with P ' (x, y) and a decrypted result that the phase-plate of generation obtains after being decrypted at random.
The decrypted result that Fig. 8 (a) uses fractional order-α=-0.75 to obtain; (b) decrypted result that uses fractional order-α=-0.65 to obtain.
[embodiment]
The embodiment of the method for the invention is as follows:
(1) ciphering process of image (as shown in Figure 1) point following several steps:
(i) f (x, y) represents original image to be encrypted, R
1(x, y) and R '
1(u, v) is two random phase plate that use as encryption key in first interative computation, can specifically be expressed as respectively exp[2 π r
1(x, y)] and exp[2 π r
2(u, v)], the wherein coordinate of (x, y) and (u, v) difference representation space territory and fraction Fourier frequency domain, r
1(x, y) and r
2(u, v) represent that two interval [0,1] on, there is the stochastic matrix of even probability distribution and statistical independence, in the time using iteration cut fractional Fourier transform to be encrypted, the k time (k=1,2,3 ...) interative computation process can generate needed two the encryption key R of interative computation the k+1 time
k+1(x, y) and R '
k+1(u, v), in the time carrying out the k time interative computation, first to f (x, y) and encryption key R
kthe product of (x, y) is made fractional Fourier transform, then the complex amplitude obtaining after conversion is got amplitude and is got phase operation, obtains respectively distribution of amplitudes g
k(u, v) and PHASE DISTRIBUTION P
k(u, v), i.e. g
k(u, v)=PT{F
α[f (x, y) R
k(x, y)] }, P
k(u, v)=PR{F
α[f (x, y) R
k(x, y)] }, wherein amplitude computing is got in PT{} representative, the phase information of removing complex amplitude, phase bit arithmetic is got in PR{} representative, removes the amplitude information of complex amplitude, F
α[] represents that exponent number is the fractional Fourier transform (Fractional Fourier Transform, FRFT) of α;
(ii) to g
k(u, v) and R '
kthe product of (u, v) is done to get phase operation after the fractional Fourier transform of α rank, obtains PHASE DISTRIBUTION P '
k(x, y), i.e. P '
k(x, y)=PR{F
α[g
k(u, v) R '
k(u, v)] };
(iii) to P '
k(x, y) does to obtain a COMPLEX AMPLITUDE after the fractional Fourier transform of (α) rank, after this distribution is got amplitude and got phase operation, obtains respectively distribution of amplitudes g '
k(u, v) and PHASE DISTRIBUTION R '
k+1(u, v) is g '
k(u, v)=PT{F
-α[P '
k(x, y)] }, R '
k+1(u, v)=PR{F
-α[P '
k(x, y)] }, then to g '
k(u, v) and P
kthe product of (u, v) is made (α) rank fractional Fourier transform, and the complex amplitude obtaining after conversion is got phase place and got amplitude operation, obtains respectively PHASE DISTRIBUTION R
k+1(x, y) and distribution of amplitudes f '
k(x, y), computing formula is respectively R
k+1(x, y)=PR{F
-α[g '
k(u, v) P
k(u, v)] }, f '
k(x, y)=PT{F
-α[g '
k(u, v) P
k(u, v)] }, thus, in the k time interative computation process, by using P '
k(x, y) and P
k(u, v) two phase-plates calculate required two the encryption key R ' that use of interative computation process the k+1 time
k+1(u, v) and R
k+1(x, y), has also obtained amplitude image in addition as f '
k(x, y), enters lower whorl interative computation process (i.e. the k+1 time interative computation) subsequently;
(iv), in the time that iterations completes n time altogether, interative computation stops, and obtains respectively two phase-plate P '
n(u, v) and P
n(x, y), i.e. P
n(u, v)=PR{F
α[f (x, y) R
n(x, y)] }, P '
n(x, y)=PR{F
α[g
n(u, v) R '
n(u, v)] }, wherein g
n(u, v) generates in the n time interative computation process, and its value is g
n(u, v)=PT{F
α[f (x, y) R
n(x, y)] }, R
n(x, y) and R '
n(u, v) all generates in the n-1 time interative computation process, and the encrypted result finally obtaining after the n time interative computation is two phase-plates, uses respectively function P (u, v) and P ' (x, y) to represent, its expression formula is
p ' (x, y)=P '
n(x, y), wherein " * " represents conjugation, R '
n+1(u, v) generates in the n time interative computation process, and its value is R '
n+1(u, v)=PR{F
-α[P '
n(x, y)] };
(2) deciphering:
(i) make (α) rank fractional Fourier transform, the result F obtaining after conversion to encrypting the phase-plate P ' (x, y) obtaining
-αafter another phase-plate P (u, v) that [P ' (x, y)] and encryption obtain multiplies each other, make (α) rank fractional Fourier transform, the result obtaining after conversion is expressed as F
-α[F
-α[P ' (x, y)] P (u, v)];
(ii) result obtaining in previous step is got to amplitude computing, finally obtain deciphering image, with f ' (x, y) expression, have f ' (x, y)=PT{F
-α[F
-α[P ' (x, y)] P (u, v)] }, can prove: f ' (x, y)=f '
n(x, y), therefore, the image that obtains of deciphering is exactly that the amplitude image that obtains of the n time interative computation of ciphering process is as f '
n(x, y).
The ciphering process of the image encryption method based on iteration cut fractional Fourier transform that the present invention proposes has nonlinear feature, its fast convergence rate, and the exponent number of fractional Fourier transform becomes the required key of deciphering; The result of encrypting is two width phase-plates, does not have the problem of information leakage; Decrypting process is linear, both can complete by numerical evaluation, also can adopt optical instrument to realize, be about to two phase-plate P ' (x, y) and P (u, v) be placed in the light path of decryption system, in output face, utilize CCD directly record obtain deciphering image.
Below the optics manner of decryption adopting in the present invention is specifically described:
The process of optics deciphering is with reference to Fig. 3, and the simple lens structure (type I type) of utilizing Lohmann to propose completes fractional Fourier transform.Spatial light modulator (Spatial Light Modulator, SLM) has the ability that shows phase signal.When encryption, show respectively P ' (x by the controlled SLM1 of computing machine and SLM2, y) with P (u, v), incident wave is unit amplitude plane light wave, the fractional Fourier transform on two double realizations of lens (α) rank, result in the output face of system is complex amplitude, but only need to use light intensity detector, as CCD records the amplitude partial information in output face, after record, obtain amplitude information f ' (x, y).Therefore, whole optics decrypting process can be expressed as f ' (x, y)=PT{F
-α[F
-α[P ' (x, y)] P (u, v)] }.
In computing, use mean square deviation (Mean Square Error, MSE) as the difference of weighing on two width image qualities, known f (x, y) and f ' (x, y) represent respectively the image that original image and deciphering obtain, MSE between the two can be expressed as
Wherein M, N is the size of image, f (x, y) and f ' (x, y) represent respectively the value of two width amplitude image pictures at pixel (x, y), can reflect the convergence of the interative computation that this method carries out by MSE.
Below in conjunction with embodiment and accompanying drawing, content of the present invention is further explained.
The gray-scale map " Cameraman " that selection size is 256 × 256 is as original image to be encrypted, after normalization as shown in Fig. 4 (a), according to process flow diagram, Fig. 1 is encrypted, the exponent number of the fractional Fourier transform that in emulation, ciphering process uses is α=0.7, encrypted result P (the x obtaining after interative computation 50 times, y) and the PHASE DISTRIBUTION of P ' (u, υ) respectively as shown in Fig. 4 (b) and 4 (c).After deciphering according to deciphering process flow diagram Fig. 2 with the two phase-plate P (x, y) that obtain and P ' (u, υ), the decrypted result obtaining is as shown in Fig. 4 (d), and its corresponding MSE value is 2.20 × 10
-11.Contrast by Fig. 4 (d) and Fig. 4 (a) can find out, we are difficult to the image and the original image that obtain from visually distinguishing deciphering.In the time adopting different interative computation number of times in ciphering process, corresponding decrypted result f ' (x, y) is not identical, f ' (x, y) relation between MSE value and iterations and between original image f (x, y) as shown in Figure 5.Can find out, in the time that interative computation number of times reaches 10 times, between f ' (x, y) and f (x, y), MSE value becomes very little, and continuing increases interative computation number of times, and MSE there will be very little variation, is difficult to show in figure.For example, in the time that iterations is respectively 20,30 and 40 times, their each self-corresponding MSE are about respectively 1.06 × 10
-6, 2.62 × 10
-8with 7.42 × 10
-10.Interative computation number of times be 3,5,10 corresponding decrypted results respectively as shown in Fig. 6 (a), 6 (b) and 6 (c), their each self-corresponding MSE are about respectively 3.80 × 10
-3, 8.55 × 10
-4with 5.59 × 10
-5the encryption method speed of convergence based on iteration cut fractional Fourier transform that visible the present invention proposes is very fast, can find out from Fig. 6 (b), be decrypted with two phase-plates that obtain after interative computation 10 times the image obtaining and there is extraordinary visual effect.
The encrypted result of encryption method proposing due to the present invention is two phase-plates, even if therefore assailant obtains a wherein phase-plate, in the PHASE DISTRIBUTION that cannot calculate by this phase-plate another piece phase-plate.Investigate the problem whether two phase-plates exist information leakage below, use separately P (x, y) result obtaining after being decrypted is as shown in Fig. 7 (a), use separately P ' (u, v) result obtaining after being decrypted is as shown in Fig. 7 (b), decrypted result does not all demonstrate the profile information of original image, and the encryption method that visible the present invention proposes does not exist information leakage problem.Two phase-plates that use when when deciphering wherein one correct, other one when wrong, the decrypted result obtaining is noise pattern.The result obtaining after using P (x, y) and a random phase plate to be decrypted is as shown in Fig. 7 (c), and the result obtaining after use P ' (u, v) and a random phase plate are decrypted is as shown in Fig. 7 (d).If the fractional order using in decrypting process makes a mistake, deciphering also cannot obtain successfully.When deciphering, use fractional order to be-α=-0.75, the decrypted result obtaining, as shown in Fig. 8 (a), uses fractional order to be-α=-0.65 when deciphering, and the decrypted result obtaining is as shown in Fig. 8 (b).
Claims (1)
1. the image encryption method based on iteration cut fractional Fourier transform, is characterized in that carrying out as follows:
(1) encrypt:
(i) f (x, y) represents original image to be encrypted, R
1(x, y) and R '
1(u, v) is two random phase plate that use as encryption key in first interative computation, can specifically be expressed as respectively exp[2 π r
1(x, y)] and exp[2 π r
2(u, v)], the wherein coordinate of (x, y) and (u, v) difference representation space territory and fraction Fourier frequency domain, r
1(x, y) and r
2(u, v) represent that two interval [0,1] on, there is the stochastic matrix of even probability distribution and statistical independence, in the time using iteration cut fractional Fourier transform to be encrypted, the k time (k=1,2,3 ...) interative computation process can generate needed two the encryption key R of interative computation the k+1 time
k+1(x, y) and R '
k+1(u, v), in the time carrying out the k time interative computation, first to f (x, y) and encryption key R
kthe product of (x, y) is made fractional Fourier transform, then the complex amplitude obtaining after conversion is got amplitude and is got phase operation, obtains respectively distribution of amplitudes g
k(u, v) and PHASE DISTRIBUTION P
k(u, v),
g
k(u，v)＝PT{F
α[f(x，y)R
k(x，y)]} (1)
P
k(u，v)＝PR{F
α[f(x，y)R
k(x，y)]} (2)
Wherein amplitude computing is got in PT{} representative, the phase information of removing complex amplitude, and phase bit arithmetic is got in PR{} representative, removes the amplitude information of complex amplitude, F
α[] represents that exponent number is the fractional Fourier transform (Fractional Fourier Transform, FRFT) of α, product f (x, the y) R of two functions in formula (1) and formula (2)
kthe α rank fractional Fourier transform of (x, y) is defined as
Wherein K
α(x, y; U, v) be the core of two-dimentional fractional Fourier transform,
Wherein
and φ=α pi/2, α is the exponent number of fractional form;
(ii) to g
k(u, v) and R '
kthe product of (u, v) is done to get phase operation after the fractional Fourier transform of α rank, obtains PHASE DISTRIBUTION P '
k(x, y),
P′
k(x，y)＝PR{F
α[g
k(u，v)R′
k(u，v)]} (5)
(iii) to P '
k(x, y) does to obtain a COMPLEX AMPLITUDE after the fractional Fourier transform of (α) rank, after this distribution is got amplitude and got phase operation, obtains respectively distribution of amplitudes g '
k(u, v) and PHASE DISTRIBUTION R '
k+1(u, v)
g′
k(u，v)＝PT{F
-α[P′
k(x，y)]} (6)
R′
k+1(u，v)＝PR{F
-α[P′
k(x，y)]} (7)
Then to g '
k(u, v) and P
kthe product of (u, v) is made (α) rank fractional Fourier transform, and the complex amplitude obtaining after conversion is got phase place and got amplitude operation, obtains respectively PHASE DISTRIBUTION R
k+1(x, y) and distribution of amplitudes f '
k(x, y), computing formula is respectively
R
k+1(x，y)＝PR{F
-α[g′
k(u，v)P
k(u，v)]} (8)
f′
k(x，y)＝PT{F
-α[g′
k(u，v)P
k(u，v)]} (9)
Thus, in the k time interative computation process, by using P '
k(x, y) and P
k(u, v) two phase-plates calculate required two the encryption key R ' that use of interative computation process the k+1 time
k+1(u, v) and R
k+1(x, y), has also obtained amplitude image in addition as f '
k(x, y), enters lower whorl interative computation process (i.e. the k+1 time interative computation) subsequently;
(iv), in the time that iterations completes n time altogether, interative computation stops, and obtains respectively two phase-plate P ' according to formula (2), formula (5)
n(u, v) and P
n(x, y),
P
n(u，v)＝PR{F
α[f(x，y)R
n(x，y)]} (10)
P′
n(x，y)＝PR{F
α[g
n(u，v)R′
n(u，v)]} (11)
Wherein g
n(u, v) generates in the n time interative computation process, and its value is g
n(u, v)=PT{F
α[f (x, y) R
n(x, y)] }, from formula (7) and formula (8), R
n(x, y) and R '
n(u, v) all generates in the n-1 time interative computation process, and the encrypted result finally obtaining after the n time interative computation is two phase-plates, uses respectively function P (u, v) and P ' (x, y) to represent, its expression formula is
P′(x，y)＝P′
n(x，y) (13)
Wherein " * " represents conjugation, R '
n+1(u, v) generates in the n time interative computation process, and its value is R '
n+1(u, v)=PR{F
-α[P '
n(x, y)] };
(2) deciphering:
(i) make (α) rank fractional Fourier transform, the result F obtaining after conversion to encrypting the phase-plate P ' (x, y) obtaining
-αafter another phase-plate P (u, v) that [P ' (x, y)] and encryption obtain multiplies each other, make (α) rank fractional Fourier transform, the result obtaining after conversion is expressed as F
-α[F
-α[P ' (x, y)] P (u, v)];
(ii) result obtaining in previous step is got to amplitude computing, finally obtain deciphering image, with f ' (x, y) expression, have f ' (x, y)=PT{F
-α[F
-α[P ' (x, y)] P (u, v)] }, can be proved by formula (6), formula (7), formula (9), formula (12), formula (13):
Therefore, the image that obtains of deciphering is exactly that the amplitude image that obtains of the n time interative computation of ciphering process is as f '
n(x, y).
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| CN110191251A (en) * | 2019-05-13 | 2019-08-30 | 四川大学 | A kind of scalability optical image encryption method based on cylinder diffraction and phase truncation |
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