CN104134184B - Image encryption method based on iteration cut fractional Fourier transform - Google Patents
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Abstract
A kind of image encryption method based on iteration cut fractional Fourier transform.One original image to be encrypted is encrypted to two pieces of phase blocks on the basis of iteration cut Fourier transform, and ciphering process has non-linear, is completed using numerical method；Decryption can use optical instrument to realize that the Optical Implementation device of decryption is fairly simple, and holographic technique record phase information need not be used in decrypting process；Iterative cryptographic method fast convergence rate proposed by the present invention, while the fractional order in ciphering process turns into the key needed for decryption, adds the security of system, and the problem of information leakage is not present in the result encrypted.
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, the research heat that safe handling is increasingly becoming information security field is carried out to information with optical means
Point.Nineteen ninety-five, P.R é fr é two expert designs of gier and B.Javidi of Connecticut universities of the U.S. have gone out to be based on optics
The Double random phase system of 4f systems.They on the input plane of 4f optical systems and fourier spectrum face by placing
The unrelated random phase plate of two pieces of statistics, finally realizes the encryption of information.The technology has obtained United States Patent (USP) protection.But due to
Double random phase technology needs to record the phase information of encrypted result with holographic technique, therefore its Optical Implementation
It is complex.What is more important, the research of recent years show, due to the linear feature of ciphering process, based on optics 4f systems
There is safety problem for the Double random phase system of system.2010, Peng Xiang of domestic Shenzhen University et al., which is proposed, to be based on
The optical image encryption system of cut Fourier transform, by introduce phase excision operation ciphering process be provided with it is non-linear
The characteristics of, so as to improve the security of system.It is to be noted that non-linear double random-phase encoding system needs repeatedly
Phase information is recorded, therefore its Optical Implementation device is more increasingly complex than linear double random-phase encoding system.In order to solve
The problem of most of encryption system Optical Implementation devices are excessively complicated, 2008, Zhang Yan of Capital Normal University et al. was transported first
With principle of optical interference, by piece image, method is encrypted to two pieces of phase-plates by numeric value analysis, and its ciphering process uses numerical value
Computational methods, decrypting process then use optical instrument.During decryption, as long as two pieces of phase-plates are properly placed at into visual decryption system
In system, with regard to correct decrypted result can be obtained in the output face of system.Decrypted result can directly be recorded by light intensity detector.
But research shows, one piece of phase-plate therein need to be only placed on optics by this encryption method there is information leakage problem
In decryption system, the most information of original image can be just obtained in output face.If eliminate the information leakage of phase-plate
Problem, then need to phase-plate carry out numerical value disorder processing, but this can cause decryption when can not in the output face of system directly
Original image is obtained, and to realize the also principle of original image and still need to further enter line number to visual decryption result by computer
Value processing.
【The content 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 above-mentioned technical problem and use following technical measures：Image encryption based on iteration cut fractional Fourier transform
Method is carried out as follows：
(1) encrypt：
(i) f (x, y) represents original image to be encrypted, R1(x, y) and R '1(u, v) is to be used as to add in first interative computation
Two pieces of random phase plates that key uses, exp [2 π r can be specifically expressed as respectively1(x, y)] and exp [2 π r2(u, v)], its
In (x, y) and (u, v) representation space domain and fraction Fourier frequency domain respectively coordinate, r1(x, y) and r2(u, v) represent two
On section [0,1] there is non-uniform probability to be distributed and count unrelated random matrix, become when with iteration cut fraction Fourier
Change when being encrypted, kth time (k=1,2,3 ...) interative computation process can generate two required for+1 interative computation of kth
Encryption key Rk+1(x, y) and R 'k+1(u, v), when carrying out kth time interative computation, first to f (x, y) and encryption key Rk(x,
Y) product makees fractional Fourier transform, then the complex amplitude obtained after conversion is carried out taking amplitude and takes phase operation, respectively
Obtain distribution of amplitudes gk(u, v) and phase distribution Pk(u, v), i.e.,
gk(u, v)=PT { Fα[f (x, y) Rk(x, y)] } (1)
Pk(u, v)=PR { Fα[f (x, y) Rk(x, y)] } (2)
Wherein PT { }, which is represented, takes amplitude computing, that is, removes the phase information of complex amplitude, and PR { } is represented and taken phase operation, that is, removes
Remove the amplitude information of complex amplitude, Fα[] represents fractional Fourier transform (Fractional Fourier of the exponent number as α
Transform, FRFT), product f (x, y) R of two functions in formula (1) and formula (2)kThe α rank fractional Fourier transforms of (x, y) are determined
Justice is
Wherein Kα(x, y；U, v) be two-dimentional fractional Fourier transform core, i.e.,
WhereinAnd φ=α pi/2s, α are the exponent numbers of fractional form；
(ii) to gk(u, v) and R 'kThe product of (u, v) carries out taking phase operation after making α rank fractional Fourier transforms, obtains
Phase distribution P 'k(x, y), i.e.,
P′k(x, y)=PR { Fα[gk(u, v) R 'k(u, v)] } (5)
(iii) to P 'k(x, y) obtains a COMPLEX AMPLITUDE after making (- α) rank fractional Fourier transform, and the distribution is carried out
Take amplitude and respectively obtain distribution of amplitudes g ' after taking phase operationk(u, v) and phase distribution R 'k+1(u, v) i.e.
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 PkThe product of (u, v) makees (- α) rank fractional Fourier transform, and answering of being obtained after conversion is shaken
Width carries out taking phase and takes amplitude to operate, and respectively obtains phase distribution Rk+1(x, y) and distribution of amplitudes f 'k(x, y), calculation formula
Respectively
Rk+1(x, y)=PR { F-α[g′k(u, v) Pk(u, v)] } (8)
f′k(x, y)=PT { F-α[g′k(u, v) Pk(u, v)] } (9)
Thus, during kth time interative computation, by using P 'k(x, y) and Pk(u, v) two pieces of phase-plates are calculated
The two encryption key R ' used required for+1 interative computation process of kthk+1(u, v) and Rk+1(x, y), it is also obtained shakes in addition
Width image f 'k(x, y), subsequently enter lower whorl interative computation process (i.e.+1 interative computation of kth)；
(iv) when iterations is completed n times altogether, interative computation is terminated, and two pieces are respectively obtained according to formula (2), formula (5)
Phase-plate P 'n(u, v) and Pn(x, y), i.e.,
Pn(u, v)=PR { Fα[f (x, y) Rn(x, y)] } (10)
P′n(x, y)=PR { Fα[gn(u, v) R 'n(u, v)] } (11)
Wherein gn(u, v) is generated in nth iteration calculating process, and its value is gn(u, v)=PT { Fα[f (x, y) Rn(x,
Y)] }, from formula (7) and formula (8), Rn(x, y) and R 'n(u, v) is generated during (n-1)th interative computation, in n-th
The encrypted result finally given after interative computation is two pieces of phase-plates, is represented respectively with function P (u, v) and P ' (x, y), its table
It is up to formula
P ' (x, y)=P 'n(x, y) (13)
Wherein " * " represents conjugation, R 'n+1(u, v) is generated in nth iteration calculating process, and its value is R 'n+1(u, v)=PR { F-α[P′n(x, y)] }；
(2) decrypt：
(i) (- α) rank fractional Fourier transform, the result F obtained after conversion are made to the phase-plate P ' (x, y) that encryption obtains-α
Another phase-plate P (u, v) that [P ' (x, y)] obtains with encryption makees (- α) rank fractional Fourier transform after being multiplied, and is obtained after conversion
Result be expressed as F-α[F-α[P ' (x, y)] P (u, v)]；
(ii) result obtained in previous step is carried out taking amplitude computing, finally gives decrypted image, with f ' (x, y) table
Show then there are f ' (x, y)=PT { F-α[F-α[P ' (x, y)] P (u, v)] }, by formula (6), formula (7), formula (9), formula (12), formula (13)
It can prove：
Therefore, the image for decrypting to obtain is exactly amplitude image that ciphering process nth iteration computing obtains as f 'n(x, y).
The beneficial effects of the present invention are：First, ciphering process uses numerical computation method, and decrypting process then uses optics
Method, it is not necessary to carry out the holographic recording of phase；Secondly, the iterative cryptographic method convergence speed based on cut fractional Fourier transform
Degree is fast, while fractional order turns into the key needed for decryption, adds security；Finally, encryption ultimately generates two pieces of phase-plates,
Generating amplitude plate is not needed, and the problem of information leakage is not present in two pieces of phase-plates.
【Brief description of the drawings】
Fig. 1 is kth time iteration cut fractional Fourier transform ciphering process flow chart.
Fig. 2 decrypting process flow charts.
Fig. 3 is visual decryption schematic diagram.
Fig. 4 (a) image f (x, y) (Cameraman) to be encrypted；(b) obtained after interative computation 50 times phase-plate P (u,
v)；(c) the phase-plate P ' (x, y) obtained after interative computation 50 times；(d) decrypted result obtained by P (u, v) and P ' (x, y).
Fig. 5 be during interative computation obtained amplitude image as fkMSE values between (x, y) and original image f (x, y) are with changing
The graph of a relation of generation number.
Decrypted result corresponding to Fig. 6 difference iterationses：(a)3；(b)5；(c)10.
The decrypted result that Fig. 7 (a) is obtained after being only decrypted with P (u, v)；(b) obtained after being only decrypted with P ' (x, y)
Decrypted result；(c) decrypted result obtained after being decrypted with P (u, v) and one piece of phase-plate generated at random；(d) P ' is used
The decrypted result that (x, y) and one piece of phase-plate generated at random obtain after being decrypted.
The decrypted result that Fig. 8 (a) is obtained using fractional order-α=- 0.75；(b) obtained using fractional order-α=- 0.65
The decrypted result arrived.
【Embodiment】
The embodiment of the method for the invention is as follows：
(1) ciphering process (as shown in Figure 1) of image divides the following steps：
(i) f (x, y) represents original image to be encrypted, R1(x, y) and R '1(u, v) is to be used as to add in first interative computation
Two pieces of random phase plates that key uses, exp [2 π r can be specifically expressed as respectively1(x, y)] and exp [2 π r2(u, v)], its
In (x, y) and (u, v) representation space domain and fraction Fourier frequency domain respectively coordinate, r1(x, y) and r2(u, v) represent two
On section [0,1] there is non-uniform probability to be distributed and count unrelated random matrix, become when with iteration cut fraction Fourier
Change when being encrypted, kth time (k=1,2,3 ...) interative computation process can generate two required for+1 interative computation of kth
Encryption key Rk+1(x, y) and R 'k+1(u, v), when carrying out kth time interative computation, first to f (x, y) and encryption key Rk(x,
Y) product makees fractional Fourier transform, then the complex amplitude obtained after conversion is carried out taking amplitude and takes phase operation, respectively
Obtain distribution of amplitudes gk(u, v) and phase distribution Pk(u, v), i.e. gk(u, v)=PT { Fα[f (x, y) Rk(x, y)] }, Pk(u, v)=
PR{Fα[f (x, y) Rk(x, y)] }, wherein PT { }, which is represented, takes amplitude computing, that is, removes the phase information of complex amplitude, and PR { } is represented and taken
Phase operation, that is, remove the amplitude information of complex amplitude, Fα[] represents fractional Fourier transform (Fractional of the exponent number as α
Fourier Transform, FRFT)；
(ii) to gk(u, v) and R 'kThe product of (u, v) carries out taking phase operation after making α rank fractional Fourier transforms, obtains
Phase distribution P 'k(x, y), i.e. P 'k(x, y)=PR { Fα[gk(u, v) R 'k(u, v)] }；
(iii) to P 'k(x, y) obtains a COMPLEX AMPLITUDE after making (- α) rank fractional Fourier transform, and the distribution is carried out
Take amplitude and respectively obtain distribution of amplitudes g ' after taking phase operationk(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 PkThe product of (u, v) makees (- α)
Rank fractional Fourier transform, the complex amplitude obtained after conversion is carried out taking phase and takes amplitude to operate, respectively obtains phase distribution
Rk+1(x, y) and distribution of amplitudes f 'k(x, y), calculation formula are respectively Rk+1(x, y)=PR { F-α[g′k(u, v) Pk(u, v)] }, f 'k
(x, y)=PT { F-α[g′k(u, v) Pk(u, v)] }, thus, during kth time interative computation, by using P 'k(x, y) and Pk
The two encryption key R ' used required for+1 interative computation process of kth are calculated in (u, v) two pieces of phase-platesk+1(u, v) and
Rk+1(x, y), amplitude image is also obtained in addition as f 'k(x, y), subsequently enter lower whorl interative computation process (i.e.+1 iteration of kth
Computing)；
(iv) when iterations is completed n times altogether, interative computation terminates, and respectively obtains two pieces of phase-plate P 'n(u, v) and
Pn(x, y), i.e. Pn(u, v)=PR { Fα[f (x, y) Rn(x, y)] }, P 'n(x, y)=PR { Fα[gn(u, v) R 'n(u, v)] }, wherein
gn(u, v) is generated in nth iteration calculating process, and its value is gn(u, v)=PT { Fα[f (x, y) Rn(x, y)] }, Rn(x, y) and
R′n(u, v) is generated during (n-1)th interative computation, and the encrypted result finally given after nth iteration computing is
Two pieces of phase-plates, represent, its expression formula is with function P (u, v) and P ' (x, y) respectively
P ' (x, y)=P 'n(x, y), wherein " * " represents conjugation, R 'n+1(u, v) is generated in nth iteration calculating process, and its value is
R′n+1(u, v)=PR { F-α[P′n(x, y)] }；
(2) decrypt：
(i) (- α) rank fractional Fourier transform, the result F obtained after conversion are made to the phase-plate P ' (x, y) that encryption obtains-α
Another phase-plate P (u, v) that [P ' (x, y)] obtains with encryption makees (- α) rank fractional Fourier transform after being multiplied, and is obtained after conversion
Result be expressed as F-α[F-α[P ' (x, y)] P (u, v)]；
(ii) result obtained in previous step is carried out taking amplitude computing, finally gives decrypted image, with f ' (x, y) table
Show then there are f ' (x, y)=PT { F-α[F-α[P ' (x, y)] P (u, v)] }, it can prove：F ' (x, y)=f 'n(x, y), therefore, solution
Close obtained image is exactly amplitude image that ciphering process nth iteration computing obtains as f 'n(x, y).
The ciphering process of image encryption method proposed by the present invention based on iteration cut fractional Fourier transform has non-
The characteristics of linear, its fast convergence rate, the exponent number of fractional Fourier transform turn into the key needed for decryption；The result of encryption is two
Width phase-plate, the problem of in the absence of information leakage；Decrypting process is linear, can both be completed by numerical computations, can also
Realized using optical instrument, will two pieces of phase-plate P ' (x, y) and P (u, v) be placed in the light path of decryption system, in output face
It is upper to obtain decrypted image using CCD directly records.
The visual decryption mode used in the present invention is specifically described below：
The process reference picture 3 of visual decryption, fraction is completed using the simple lens structure (type I types) of Lohmann propositions
Fourier transform.Spatial light modulator (Spatial Light Modulator, SLM) has the ability of display phase signal.Add
When close, P ' (x, y) and P (u, v) are shown by computer controllable SLM1 and SLM2 respectively, incidence wave is unit Amplitude Plane light
Ripple, two pieces of lens realize the fractional Fourier transform of (- α) rank twice in succession, and the result in the output face of system is complex amplitude, but
Only need to use light intensity detector, such as the amplitude components information in CCD record output faces, obtain amplitude information f ' after record
(x, y).Therefore, whole visual decryption process can be expressed as f ' (x, y)=PT { F-α[F-α[P ' (x, y)] P (u, v)] }.
In computing using mean square deviation (Mean Square Error, MSE) as measurement two images quality on difference,
The image that known f (x, y) and f ' (x, y) represent original image respectively and decryption obtains, MSE between the two can be expressed as
Wherein M, N are the sizes of image, and f (x, y) and f ' (x, y) represent two width amplitude image pictures at pixel (x, y) respectively
Value, the convergence of the interative computation that this method carried out can be reflected by MSE.
Present disclosure is further explained with reference to embodiment and accompanying drawing.
Select size for 256 × 256 gray-scale map " Cameraman " as original image to be encrypted, after normalization such as
Shown in Fig. 4 (a), it is encrypted according to flow chart Fig. 1, the exponent number for the fractional Fourier transform that ciphering process uses is α in emulation
=0.7, the encrypted result P (x, y) and P ' (u, υ) phase distribution obtained after interative computation 50 times is then respectively such as Fig. 4 (b) and 4
(c) shown in.With obtained two pieces of phase-plate P (x, y) and P ' (u, υ) according to decryption flow chart Fig. 2 decryption after, obtained decryption
As a result as shown in Fig. 4 (d), the MSE values corresponding to it are 2.20 × 10-11.It can be seen by Fig. 4 (d) and Fig. 4 (a) contrast
Go out, we are difficult to visually distinguish image and original image that decryption obtains.When using different iteration to transport in ciphering process
When calculating number, corresponding decrypted result f ' (x, y) simultaneously differs, and the MSE values between f ' (x, y) and original image f (x, y) are with changing
Relation between generation number is as shown in Figure 5.As can be seen that when interative computation number reaches 10 times, f ' (x, y) and f (x, y) it
Between MSE values become very little, continue to increase interative computation number, MSE occurs the change of very little, be difficult to show in figure.For example, work as
When iterations is respectively 20,30 and 40 times, MSE corresponding to each of which respectively may be about 1.06 × 10-6、2.62×10-8With
7.42×10-10.Interative computation number be decrypted result corresponding to 3,5,10 respectively as shown in Fig. 6 (a), 6 (b) and 6 (c), it
Each self-corresponding MSE respectively may be about 3.80 × 10-3、8.55×10-4With 5.59 × 10-5, it is seen that it is proposed by the present invention based on repeatedly
For cut fractional Fourier transform encryption method convergence rate quickly, from Fig. 6 (b) as can be seen that with after interative computation 10 times
The image that two pieces of obtained phase-plates are decrypted to obtain has extraordinary visual effect.
Because the encrypted result of encryption method proposed by the present invention is two pieces of phase-plates, therefore even if attacker obtains wherein
One piece of phase-plate, the phase distribution of another piece of phase-plate can not be being calculated by the phase-plate.Two pieces of phase-plates are investigated below
The problem of with the presence or absence of information leakage, it is used alone shown in the result such as Fig. 7 (a) obtained after P (x, y) is decrypted, individually makes
Shown in the result obtained after being decrypted with P ' (u, v) such as Fig. 7 (b), decrypted result does not all show the profile of original image
Information, it is seen that encryption method proposed by the present invention not existence information leakage problem.When decryption two pieces of phase-plates using its
In one piece it is correct, in addition during one piece of mistake, obtained decrypted result is noise pattern.Use P (x, y) and one piece of random phase plate
Shown in the result obtained after being decrypted such as Fig. 7 (c), obtained after being decrypted using P ' (u, v) and one piece of random phase plate
As a result as shown in Fig. 7 (d).If the fractional order used in decrypting process makes a mistake, decryption also can not be successful.Decryption
When the use of fractional order is-α=- 0.75, the use of fractional order is-α during decryption shown in obtained decrypted result such as Fig. 8 (a)
=-0.65, obtained decrypted result is then as shown in Fig. 8 (b).
Claims (1)
- A kind of 1. image encryption method based on iteration cut fractional Fourier transform, it is characterized in that carrying out as follows：(1) encrypt：(i) f (x, y) represents original image to be encrypted, R1(x, y) and R '1(u, v) is close as encrypting in first interative computation Two pieces of random phase plates that key uses, exp [2 π r can be specifically expressed as respectively1(x, y)] and exp [2 π r2(u, v)], wherein (x, y) and (u, v) difference representation space domain and the coordinate of fraction Fourier frequency domain, r1(x, y) and r2(u, v) represents Liang Ge areas Between there is non-uniform probability to be distributed on [0,1] and count unrelated random matrix, when using iteration cut fractional Fourier transform When being encrypted, kth time interative computation process can generate two encryption key R required for+1 interative computation of kthk+1(x, And R ' y)k+1(u, v), wherein k=1,2,3 ..., when carrying out kth time interative computation, first to f (x, y) and encryption key Rk The product of (x, y) makees fractional Fourier transform, then the complex amplitude obtained after conversion is carried out taking amplitude and takes phase operation, point Distribution of amplitudes g is not obtainedk(u, v) and phase distribution Pk(u, v), i.e.,gk(u, v)=PT { Fα[f (x, y) Rk(x, y)] } (1)Pk(u, v)=PR { Fα[f (x, y) Rk(x, y)] } (2)Wherein PT { }, which is represented, takes amplitude computing, that is, removes the phase information of complex amplitude, and PR { } is represented and taken phase operation, that is, removes multiple The amplitude information of amplitude, Fα[] represent exponent number as α fractional Fourier transform (Fractional Fourier Transform, FRFT), in formula (1) and formula (2) two functions product f (x, y) RkThe α rank fractional Fourier transforms of (x, y) are defined as<mrow> <msup> <mi>F</mi> <mi>&alpha;</mi> </msup> <mrow> <mo>&lsqb;</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>R</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>&infin;</mi> </mrow> <mrow> <mo>+</mo> <mi>&infin;</mi> </mrow> </msubsup> <msub> <mi>K</mi> <mi>&alpha;</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>R</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>Wherein Kα(x, y；U, v) be two-dimentional fractional Fourier transform core, i.e.,<mrow> <msub> <mi>K</mi> <mi>&alpha;</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>;</mo> <mi>y</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mi> </mi> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>&pi;</mi> <mfrac> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> <mrow> <mi>&lambda;</mi> <mi>f</mi> <mi> </mi> <mi>tan</mi> <mi>&phi;</mi> </mrow> </mfrac> <mo>-</mo> <mn>2</mn> <mi>i</mi> <mi>&pi;</mi> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> <mi>u</mi> <mi>v</mi> </mrow> <mrow> <msup> <mi>&lambda;</mi> <mn>2</mn> </msup> <msup> <mi>f</mi> <mn>2</mn> </msup> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>WhereinAnd φ=α pi/2s, α are the exponent numbers of fractional form；(ii) to gk(u, v) and R 'kThe product of (u, v) carries out taking phase operation after making α rank fractional Fourier transforms, obtains phase It is distributed P 'k(x, y), i.e.,P′k(x, y)=PR { Fα[gk(u, v) R 'k(u, v)] } (5)(iii) to P 'k(x, y) obtains a COMPLEX AMPLITUDE after making (- α) rank fractional Fourier transform, and the distribution is carried out to take amplitude With take phase operation after respectively obtain distribution of amplitudes g 'k(u, v) and phase distribution R 'k+1(u, v), i.e.,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 PkThe product of (u, v) makees (- α) rank fractional Fourier transform, and the complex amplitude obtained after conversion is entered Row takes phase and takes amplitude to operate, and respectively obtains phase distribution Rk+1(x, y) and distribution of amplitudes f 'k(x, y), calculation formula difference ForRk+1(x, y)=PR { F-α[g′k(u, v) Pk(u, v)] } (8)f′k(x, y)=PT { F-α[g′k(u, v) Pk(u, v)] } (9)Thus, during kth time interative computation, by using P 'k(x, y) and Pk(u, v) two pieces of phase-plates be calculated kth+ The two encryption key R ' used required for 1 interative computation processk+1(u, v) and Rk+1(x, y), amplitude image is also obtained in addition As f 'k(x, y), subsequently enter lower whorl interative computation process (i.e.+1 interative computation of kth)；(iv) when iterations is completed n times altogether, interative computation is terminated, and two pieces of phases are respectively obtained according to formula (2), formula (5) Plate P 'n(u, v) and Pn(x, y), i.e.,Pn(u, v)=PR { Fα[f (x, y) Rn(x, y)] } (10)P′n(x, y)=PR { Fα[gn(u, v) R 'n(u, v)] } (11)Wherein gn(u, v) is generated in nth iteration calculating process, and its value is gn(u, v)=PT { Fα[f (x, y) Rn(x, y)] }, From formula (7) and formula (8), Rn(x, y) and R 'n(u, v) is generated during (n-1)th interative computation, in nth iteration The encrypted result finally given after computing is two pieces of phase-plates, is represented respectively with function P (u, v) and P ' (x, y), its expression formula For<mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>R</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mo>*</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>P ' (x, y)=P 'n(x, y) (13)Wherein " * " represents conjugation, R 'n+1(u, v) is generated in nth iteration calculating process, and its value is R 'n+1(u, v)=PR { F-α [P′n(x, y)] }；(2) decrypt：(i) (- α) rank fractional Fourier transform, the result F obtained after conversion are made to the phase-plate P ' (x, y) that encryption obtains-α[P′ (x, y)] with encrypting work (- α) rank fractional Fourier transform after obtained another phase-plate P (u, v) is multiplied, the knot obtained after conversion Fruit is expressed as F-α[F-α[P ' (x, y)] P (u, v)]；(ii) result obtained in previous step is carried out taking amplitude computing, finally gives decrypted image, represented with f ' (x, y), Then there are f ' (x, y)=PT { F-α[F-α[P ' (x, y)] P (u, v)] }, can be with by formula (6), formula (7), formula (9), formula (12), formula (13) Prove：<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>f</mi> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mi>T</mi> <mrow> <mo>{</mo> <mrow> <msup> <mi>F</mi> <mrow> <mo>-</mo> <mi>&alpha;</mi> </mrow> </msup> <mrow> <mo>&lsqb;</mo> <mrow> <msup> <mi>F</mi> <mrow> <mo>-</mo> <mi>&alpha;</mi> </mrow> </msup> <mrow> <mo>&lsqb;</mo> <mrow> <msup> <mi>P</mi> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>P</mi> <mi>T</mi> <mrow> <mo>{</mo> <mrow> <msup> <mi>F</mi> <mrow> <mo>-</mo> <mi>&alpha;</mi> </mrow> </msup> <mrow> <mo>&lsqb;</mo> <mrow> <msup> <mi>F</mi> <mrow> <mo>-</mo> <mi>&alpha;</mi> </mrow> </msup> <mrow> <mo>&lsqb;</mo> <mrow> <msubsup> <mi>P</mi> <mi>n</mi> <mo>&prime;</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <msubsup> <mi>R</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mo>*</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>P</mi> <mi>T</mi> <mrow> <mo>{</mo> <mrow> <msup> <mi>F</mi> <mrow> <mo>-</mo> <mi>&alpha;</mi> </mrow> </msup> <mrow> <mo>&lsqb;</mo> <mrow> <msubsup> <mi>g</mi> <mi>n</mi> <mo>&prime;</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>R</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>R</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mo>*</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mo>&prime;</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>Therefore, the image for decrypting to obtain is exactly amplitude image that ciphering process nth iteration computing obtains as f 'n(x, y)；The process of visual decryption, fractional Fourier transform is completed using the simple lens structure of Lohmann propositions；Space light modulation Device (Spatial Light Modulator, SLM) has the ability of display phase signal；It is controllable by computer during encryption SLM1 and SLM2 shows P ' (x, y) and P (u, v) respectively, and incidence wave is unit Amplitude Plane light wave, and two pieces of lens are real twice in succession The fractional Fourier transform of existing (- α) rank, the result in the output face of system be complex amplitude, it is only necessary to light intensity detector C CD notes The amplitude components information in output face is recorded, amplitude information f ' (x, y) is obtained after record；Whole visual decryption process can be expressed as F ' (x, y)=PT { F-α[F-α[P ' (x, y)] P (u, v)] }；Using mean square deviation (Mean Square Error, MSE) as the difference weighed in two images quality in computing, it is known that f The image that (x, y) and f ' (x, y) represent original image respectively and decryption obtains, MSE between the two can be expressed as<mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>,</mo> <msup> <mi>f</mi> <mo>&prime;</mo> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>M</mi> <mi>N</mi> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <munderover> <mi>&Sigma;</mi> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <mo>|</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>f</mi> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>Wherein M, N are the sizes of image, and f (x, y) and f ' (x, y) represent value of the two width amplitude image pictures in pixel (x, y) respectively, Reflect the convergence for the interative computation that this method carried out by MSE.
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