Optical encryption using a liquid phase mask

David R. Schipf; Wei-Chih Wang

doi:10.1364/OSAC.1.001026

1. Introduction

Intrinsically optical applications that require secure transfer of information contained in images are potential applications for optical image cryptography. The use of optical processes allows for alteration of polarization, phase, and intensity for encoding [1]; even bit entanglement is possible [2,3]. Mathematical transforms, such as the Fourier transform, can be performed on wave-fields at the speed of light for near instantaneous encoding of images. Optical cryptography continues to evolve toward quicker, safer transfer of information.

Random phase mask encryption is an extensively studied optical cryptography technique that uses optical Fourier transforms to encode wave-fields. One of the simplest, the double random phase encryption (DRPE) system, has been shown to be formidable against brute force attacks [4,5]. In the DRPE, two random phase masks constitute the keyspace of a DRPE system, with one phase mask placed between two lenses in the Fourier plane.

The DRPE can be exploited by chosen-plaintext and known-plaintext attacks, and in certain circumstances by ciphertext-only attacks. A chosen-plaintext attack on the DRPE can be performed with a point light source plaintext image as an approximate spatial Dirac delta function, yielding the impulse response of the system in the resulting encrypted image. This type of attack may be too suspicious for security systems if only one chosen plaintext image is used. Ciphertext-only attacks take advantage of purely phase modulation in the Fourier plane to retrieve the phase mask keys from a captured encrypted image. Given only one encrypted image, this attack has been successfully demonstrated for plaintext images with less than 1/4 of the pixels containing non-zero values [6]. This pixel limitation, however, may change with multiple encrypted images available to the adversary [6]. When multiple plaintext-encrypted image pairs from a DRPE system fall into the hands of an adversary, the phase mask keys can also be recovered by a known-plaintext attacks [7,8]. If the image intensity is the plaintext, the system is vulnerable to a known-plaintext attack using a single or multiple binary intensity images. If the plaintext is a phase image, constituting what is herein called fully-phase DRPE, the system is vulnerable with multiple pairs of binary phase images [8]. While potentially requiring multiple images or pairs of images, known-plaintext and ciphertext-only attacks may be less recognizable by system security than the single image chosen-plaintext attack.

A solution to address vulnerabilities exploited with multiple encrypted images or image pairs is to have the optical transfer function of a DRPE system change over time or periodically between uses [7]. Most experimental DRPE systems use spatial light modulators (SLMs), which means that an SLM would consistently need to be refreshed with a new random phase grid. We propose to change the optical transfer function of a random phase mask encryption system by adding an additional analog element that changes over time: a thin liquid system with a controllable surface topology. With the use of a liquid system, random phase mask encryption can use fixed, static phase masks without the need to change, while the system can have a time-dependent transfer function by the user-defined liquid surface. A liquid system augmenting the use of random phase masks can prevent adversarial attacks that depend on multiple ciphertext images or plaintext-ciphertext image pairs.

Fluid systems can make effective tunable optical elements that will exhibit oscillatory behavior. Liquid deformable mirrors have been demonstrated to be a suitable approach to phase correction for adaptive optics while also being cost effective [9]. Other research has shown the potential of using fluids as lenses with tunable optical power [10–14]. While it is most often desired that tunable liquid optical elements have stable geometries, liquid systems can exhibit low amplitude surface waves with spatially varying patterns that oscillate at distinguishable time-frequencies. Recently, commercially available electrowetting lenses driven by sinusoidal voltages have been shown to exhibit predictable, low amplitude axis symmetric surface wave patterns that match with a Bessel function mathematical model [15]. Surface wave patterns from electrowetting or vibration forced systems contain information regarding fluid properties, geometry, and forcing [15–17].

We present a numerical study on the security of using a liquid system with two random phase masks for image encryption/decryption. We use low amplitude, axis symmetric and asymmetric oscillating liquid surface patterns, in combination with random phase masks, for optical encryption and decryption. We use a simple liquid lens design, similar to the electrowetting lens designs in [11] and [14]. Even though many variations of random phase mask encryption exist [18–26], we first explore the integration of a liquid system with a phase-only two random phase mask encryption system, with one mask in the object plane and one in the Fourier plane. Due to the similarity of the two random phase mask system with many of the more recently proposed optical encryption systems, subsequent studies can be performed using different combinations of random phase masks, lenses, and other optical systems. We use only one oscillating liquid system in this study to focus on changing the transfer function over time for the prevention of multiple image or image pair adversarial attacks.

2. Methods

We use a simple and practical placement of the liquid system for optical encryption. The liquid system can be placed after the plaintext image and first phase mask, but before the first lens in the system. Another option, however, is to use the liquid system both as the first lens and as the time-dependent phase modulating mechanism, as shown in Fig. 1(a)

Fig. 1 (a) A modified double random phase mask encryption system with a tunable liquid lens. The liquid system has an oscillating convex surface, making it also the first lens L1. The red area is the light propagating through the system, while the yellow boxed area contains the cryptographic keys. The system includes lens L2, and phase masks D1 and D2. (b) The liquid system used for this study, with two immiscible liquids of differing refractive indices n₁ and n₂.

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. This choice reduces the amount of optical elements in the system, and enables a simpler expression for the encrypted image using Fresnel approximations. Placing the liquid system between the two random phase masks ensures concealment of the liquid surface behavior in the encrypted image.

In this study, we use a simple two-liquid lens geometry: two immiscible liquids, enclosed by optical windows and a dielectric side-wall, as shown in Fig. 1(b). This geometry has been used for electrowetting lens designs, with density matched liquids proving a liquid system with consistent performance over any orientation [14,15,27,28]. While the thickness of the liquid system remains constant, the meniscus profile between the two liquids can change over time. To model the thickness of the first liquid in the $(x, y)$ plane, we use the equation

Δ_{l} (x, y, t) = w - \frac{(x^{2} + y^{2})}{2 R} + h (x, y, t),

where

w

is the nominal width of the first liquid,

R

is the radius of curvature, and

h (x, y, t)

is the height of the liquid relative to the static parabolic shape. The paraxial approximation is used to model the liquid system as a thin lens. With the radius of curvature set as the focal length of a

4 f

system

(R = f)

, and the refractive indices of the two liquids given as

n_{1}

and

n_{2}

, the transmission function is given as [29]

t_{l} (x, y, t) = \exp (j k φ) \cdot \exp [j k h (x, y, t) (n_{1} - n_{2})] \cdot \exp [- j k \frac{x^{2} + y {}^{2}}{2 R} (n_{1} - n_{2})],

where wavenumber

k = 2 π / λ

with optical wavelength

λ

, and the constant phase retardation due to the nominal thickness of the liquids and the optical windows is

φ

.

For a liquid lens with uniform forcing at the contact line between the liquids and the side-wall, the shape of oscillations is a time-changing Bessel function. This has been observed in two electrode electrowetting lenses [15], and is the solution for unforced fluid sloshing modes in a circular container, as shown in the book by Ibrahim [30]. The liquid surface shown in Fig. 2

Fig. 2 (a) Top view of liquid surface mode shapes with (top) axis symmetric motion (m = 0) and (bottom) asymmetric motion (m = 1), with mode numbers $n = 2, 10, 20$ . (b) isometric projection of random liquid surface profile (along radial direction) with exaggerated vertical scale.

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is a fluid sloshing mode defined by

h_{m n} (r, θ, t) = A_{m n} \cos (m θ) J_{m} (\frac{ω_{m n}}{c} r) \cos (ω_{m n} t),

where

J_{m}

is the Bessel function of the

m

kind,

A_{m n}

is the mode dependent amplitude,

c

is the liquid wave speed, and

ω_{m n}

is the circular oscillation frequency. The coordinates can be easily transformed from

(r, θ, t)

to

(x, y, t)

for integration into the remaining equations. For this study, we assume initial conditions that excite either the first set of modes

m = 0

, or the second set

m = 1

. For the previously used example of electrowetting lenses,

m = 0

corresponds to one electrode around the side-wall, and

m = 1

corresponds to two. The two electrode arrangement has been used before for beam steering [31]. A top view of liquid profiles with modes

n = 2, 10, 20

for both the axis symmetric

m = 0

and asymmetric

m = 1

cases is shown in Fig. 2(a). An accurate calculation of the amplitude requires a more detailed model, but for now we set the amplitude as

A_{m n} = λ / (2 (n_{1} - n_{2}))

to give a maximum phase shift of

\pm π

for all

m

and

n

. Using only single mode shapes on a liquid profile allows for the study of the security of a liquid crypto-system before more complex surfaces are used, such as the surface shown in Fig. 2(b), which is composed of a combination of modes between

n = 1

and

n = 30

to construct a surface with random height in the radial direction.

In this study, we simulate the encryption and decryption of a phase image using monochromatic, coherent light. The plaintext $P (x, y) \in {0, 1}$ is stored in the phase of an intensity image $I (x, y) = \exp (j 2 π P (x, y))$ which is attached to a 16-bit phase mask $D 1 (x, y) = \exp (j D P_{1} (x, y))$ with phase altering values $D P_{1} \in {0, ..., 2 π}$ from a uniform distribution. The $(x, y)$ coordinates given here are in the direction of the object plane and image planes, while the perpendicular $z$ direction is the direction of propagation in the $4 f$ system. The complex Fourier distribution from the image, first phase mask, and liquid lens, according to the Fresnel diffraction approximation, is modeled as [29]

U (x_{f}, y_{f}, t) = \frac{1}{j λ f} \exp [\frac{j k (n_{3} - n_{2})}{2 f} (x_{f}^{2} + y_{f}^{2})] \int \int_{- \infty}^{\infty} F (x, y, t) \cdot \exp (- j \frac{k}{f} (x x_{f} + y y_{f})) d x d y,

where the double integral is a two-dimensional Fourier transform modeling the frequency spectrum of the light leaving the lens. The function

F (x, y, t)

is

F (x, y, t) = I (x, y) \cdot D 1 (x, y) \cdot \exp [j k h (x, y, t) (n_{2} - n_{3})] .

The spatial frequencies of the Fourier plane are

x_{f} = x / λ f

and

y_{f} = y / λ f

. The spatial frequency spectrum transmitted to the Fourier plane is [29]

S (x_{f}, y_{f}, t) = U (x_{f}, y_{f}, t) \cdot \exp [- j π λ f (x_{f}^{2} + y_{f}^{2})] .

With the canceling of the phase terms in front of the Fourier transform and with neglecting the spatially constant phase terms in front of the integration, the complex field at the Fourier plane becomes

S (x_{f}, y_{f}, t) = \int \int_{- \infty}^{\infty} F (x, y, t) \cdot \exp (- j \frac{k}{f} (x x_{f} + y y_{f})) d x d y,

which is a two-dimensional Fourier transform of the Image, the first phase mask, and the phase modulation from the liquid surface height changes. Given the second 16-bit phase mask

D 2

in the Fourier plane

(x_{f}, y_{f})

, the complex instantaneous encrypted image is given as

E (x, y, t) = \int \int_{- \infty}^{\infty} S (x_{f}, y_{f}, t) \cdot D 2 (x_{f}, y_{f}) \exp (j \frac{k}{f} (x x_{f} + y y_{f})) d x_{f} d y_{f} .

Decryption can be performed by undoing the operations on the plaintext image. The recovered image $R (x, y)$ can be found with

R (x, y, t) = W (x, y, t) \cdot \exp [j k h^{*} (x, y, t) (n_{2} - n_{3})] \cdot D 4 (x, y),

where

W (x, y, t) = \int \int_{- \infty}^{\infty} Q (x_{f}, y_{f}, t) \cdot D 3 (x_{f}, y_{f}) \exp (j \frac{k}{f} (x x_{f} + y y_{f})) d x_{f} d y_{f},

and

Q (x_{f}, y_{f}, t) = \int \int_{- \infty}^{\infty} E (x, y, t) \exp (- j \frac{k}{f} (x_{r} x_{f} + y_{r} y_{f}) d x d y .

When phase masks

D 3 (x_{f}, y_{f}) = D 2^{*} (x_{f}, y_{f})

and

D 4 (x, y) = D 1^{*} (x, y)

, and the liquid height is

h^{*} (x, y, t) = - h (x, y, t)

, the original image is recovered

R (x, y, t) = I (x, y, t)

.

For decryption to be possible, the liquid surface topology at the time of encryption must be known. The liquid surface component of the keyspace is four-dimensional (3-D space and time). This component is realized by either knowing the liquid excitation input and estimating the liquid surface profile with a sufficiently accurate model, or by estimating the liquid surface profile with measurements performed during encryption. We use a simple example of successful decryption with a single oscillating liquid mode is used for encryption at time $t_{e}$ and $h^{*} (x, y, t) = - h (x, y, t_{e} + π)$ , with a canceling constant phase retardation term $\exp {(j k φ)}^{*} = \exp (- j k φ)$ .

Modeling the physical characteristics of an optical system yields a more complex expression for the encrypted image. A complex encrypted image in systems that do not vary over time is usually calculated using digital interference holography [32,33]. Photons are collected over a finite exposure time τ from the beam propagating through the liquid-crypto system and a reference beam. To model a physical recording of a liquid-crypto system with the encrypted image changing over time, an adaptation to Eq. (8) is needed. The Integration of the complex encrypted image $E (x, y, t)$ and a reference beam $E_{r b} (x, y)$ over the exposure time τ from initial time $t_{0}$ can be shown as

E_{r c} (x, y, t_{0}, τ) = \int_{t_{0}}^{t_{0} + τ} | E (x, y, t) + E_{r b} (x, y) |^{2} d t .

We can write this encrypted image and reference beam intensity as a function of pixel number

(q, v)

in an array of size

Q x V

. If we discretize with

Δ t = τ / K

, with

K

being the total amount of time steps in one full exposure, we can use a trapezoidal integration method to give

E_{r c} (q, v, t_{0}, K) = \frac{Δ t}{2} \sum_{k = 1}^{K} [E (q, v, t_{0} + k Δ t) + E (q, v, t_{0} + (k - 1) Δ t)] + Δ t \cdot K \cdot E_{r b} (q, v) .

We present two options for decryption. If the exposure time $τ$ is much smaller than the fluctuations of intensity, the exposed image becomes

E_{r c} (q, v, t_{0}, K) \approx (E (q, v, t_{0}) + E_{r b} (q, v)) \cdot {(E (q, v, t_{0}) + E_{r b} (q, v))}^{*} \cdot Δ t \cdot K

Techniques in holographic interferometry can be used then to calculate the phase and amplitude of the encrypted image

E (q, v, t_{0})

. The decryption is then done using the process expressed by Eq. (9)-(11). With a larger exposure time

τ

, the recorded compound encrypted image

E_{r c} (q, v, t_{0}, τ)

can be thought of as a summation of images, each recorded over a sufficiently short duration of time. This situation is shown in Fig. 3

Fig. 3 The encryption and decryption of a quick response (QR) code at discrete times with an oscillating liquid lens system and two random phase masks. The CMOS camera is shown recording a typical encrypted image intensity, which is shown to the right for a single pixel over time. The blue shaded region in the intensity plot is the recorded encrypted image integrated over exposure time $τ$ , while the red region is the intensity of encrypted image of interest $E_{s}$ that is used for decryption.

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. The encrypted image of interest,

E_{s} (q, v, t_{s})

, can be found by first subtracting the unwanted encrypted images from

E_{r c}

, and then using an integration rule to separate

E_{s} (q, v, t_{s})

from

E_{s} (q, v, t_{s} + Δ t)

. If the encrypted image of interest is at the beginning of the exposure time, this can be shown as

Δ t \cdot E_{s} (q, v) \approx E_{r c} - K \cdot Δ t \cdot E_{r b} (q, v) - Δ t \sum_{k = 1}^{K - 1} [E (q, v, t_{0} + k Δ t) + E (q, v, t_{0} + (k + 1) Δ t)] .

The fringe stability requirements when choosing the exposure time for the encrypted image must be addressed. Considering the dynamics of the proposed system, a reference beam or shearing holographic interferometry technique can be used for recording [34]. Typically, an optical path variation of 1/4 to 1/10 of the wavelength $λ$ should not be exceeded during exposure [34]. Variations that exceed this amount will give too great of shift in the fringes. The greatest liquid height change for the axis symmetric case of $m = 0$ over time is at $r = 0$ . This means that if we use an amplitude of $A_{m n} = λ / (2 (n_{1} - n_{2}))$ , we need

\frac{λ}{2} | \cos (ω_{0 n} (τ + t_{0})) - \cos (ω_{0 n} t_{0}) | \leq \frac{λ}{10} .

At the point in oscillation when the left hand quantity is maximized

(ω_{m n} t_{0} = \pm b π / 2, b = 1, 3, 5, \dots)

, the exposure time should be approximately

τ \leq 1 / (5 ω_{0 n})

. For commercial liquid lenses, oscillations may have frequencies from 50 –1000 Hz [15,27], which would require an exposure time less than 0.6 ms – 0.03 ms in the worst case. This is well within the capabilities of high speed cameras. The exposure time can be, of course, larger at other points in the oscillation, or if the amplitude of the oscillations is lessened.

As shown on the right side of Fig. 3, the intensity of the encrypted image at each pixel will vary over ½ of the liquid oscillation period $π / ω_{m n}$ . If the image is recorded for a full period $τ = 2 π / ω_{m n}$ , the intensity is doubled. Recording over $π / ω_{m n}$ enables pixels with little variation over time to increase in intensity at a different rate than pixels without much variation over time, which may give rise to a signature of the liquid system in the encrypted image. A vulnerability also arises if the exposure time is much smaller than the intensity fluctuations, and multiple plaintext-encrypted image pairs are recorded successively. This would produce a temporal frequency signature that can be traced back to the oscillation frequency of the liquid system. For these reasons, we use an exposure time of ½ of the liquid oscillation period $π / ω_{m n}$ .

To evaluate the security of the proposed liquid-crypto system, we first simulate encrypted images and calculate the entropy. The entropy of an image can be numerically calculated by summing bins in a histogram. For a histogram with 256 bins, the numerical calculation of entropy is then

E n t r o p y = \sum_{i = 1}^{256} p (α_{i}) \log_{2} p (α_{i}),

where

p (α_{i})

is the normalized histogram count of bin

i

. An image with maximum entropy will yield an entropy calculation of 8 when using 256 bins. This is reached with a phase-only random phase mask system without a liquid system present [32,33].

To measure how much encrypted images from the liquid-crypto system correlate to the liquid pattern used for encryption, we compared phase and intensity values across the midsection of the pattern/image. The midsection of the liquid surface pattern with a length $L$ at $t = 0$ is defined as $u (x = L / 2, y)$ , while the midsection of the encrypted image is defined as $v (x = L / 2, y)$ . These two data series are taken from the pixels along a line through the center of the two images, as shown by arrows in Fig. 4

Fig. 4 The phase values at the centerline of (top) an encrypted image using the liquid system and the random phase masks, and (bottom) the relevant liquid profile. A correlation coefficient is calculated using these two data series for each liquid surface mode number n.

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. We use the correlation coefficient ρ for the comparison. The correlation coefficient can be calculated by [35]

ρ = \frac{C_{u v}}{σ_{u} σ_{v}},

with

C_{u v}

being the covariance of the two data series

u

and

v

, while

σ_{u}

and

σ_{v}

are the square root of the variance of

u

and

v

, respectively.

For comparisons between the phase of a recovered image $R_{p} (q, v, t)$ with pixels $(q, v)$ in the $(x, y)$ plane and the phase of the original QR code image $P (q, v)$ , we use a mean squared error (MSE) defined as

M S E = \frac{1}{Q V} \sum_{q}^{Q} \sum_{v}^{V} | R_{p} (q, v, t) - P (q, v) |^{2} .

For these calculations, we use encrypted images exposed over half of the oscillation period $τ = π / ω_{m n}$ .

In addition to evaluating the security of a liquid system double random phase mask encryption method, we also estimate the expected contributions of a liquid system to the noise in recovered images. An often cited disadvantage of double random phase encryption is the noise found in the recovered images [32,36]. To understand the contribution of noise by the liquid system, we add noise to the encrypted image and error to the model of the liquid surface patterns over time. An expected amount of noise in a recovered image given additive Gaussian error for the encrypted image was formulated by Javidi et al. [32]. However, to model noise from the liquid system, we add error before the first Fourier transforms in Eq. (8) and after the inverse Fourier transforms to recover an image [Eq. (9)]. Therefore we use numerical simulations to test the random process in order to understand the expected amount of noise in the recovered image.

We use a modified model for the height of the liquid meniscus profile $h_{e} (x, y, t)$ , which for the amplitude error is

h_{e} (x, y, t) = h (x, y, t) (1 + η),

and for the horizontal positioning error is

h_{e} (x, y, t) = h (x, y - δ, t) .

Gaussian noise is added to the encrypted image phase, giving a modified encrypted image of

E_{e} (x, y, t) = E (x, y, t) \exp [j γ (x, y)]

The amplitude error

η

, horizontal error

δ

, and encrypted image noise

γ

are centered at zero and have standard deviations

σ_{η}, σ_{δ}, σ_{γ}

respectively. A mean MSE for recovered images from multiple simulations is non-zero, because the amplitude and horizontal position error are present during both the encryption and decryption.

3. Results and discussion

Simulations were conducted for different liquid oscillation modes $m = 0, 1$ and $n = 2, 3, \dots, 30$ using Matlab [37] to calculate fast Fourier transforms (FFTs) for the Fresnel approximation equations. A digital pseudo-random number generator was used to construct the random phase masks. There was some variation in the results for entropy, correlation coefficient, and MSE, so the simulations were repeated until the mean ceased to change by less than the three significant digits used. For the entropy and correlation coefficients, this was at 1328 simulations, while for the MSE calculations this was at 244 simulations. Parameters of focal length $f = 100$ mm, wavelength $λ = 632.8$ nm, image length $L = 4$ mm were used to give a low critical sampling number, and to reflect a potential physical system.

The mean entropy values for encrypted images at an instant of time were different for the intensity and phase, as shown in Fig. 5(a) and 5(b)

Fig. 5 Entropy of phase and intensity (int.) of encrypted images for axis symmetric and asymmetric liquid surface mode shapes of mode number n. Encrypted images taken at an instant of time $t = 0$ (inst.) and exposed over a full liquid surface oscillation period (exposed) were used.

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. The phase of encrypted images was consistently at the maximum of 8 (to three significant digits) for both axis symmetric

m = 0

and asymmetric

m = 1

surface patterns. This is consistent with the entropy of the phase without the liquid system, which corresponds to documented DRPE systems [32]. The liquid surface changes the phase of the wavefront at an instant of time after it has already been changed by random amounts by the first phase mask, which makes the liquid phase change not deleterious to the entropy of encrypted image phase.

The entropy of the intensity of encrypted images was higher with an exposure time of half of the liquid oscillation period $τ = π / ω_{m n}$ than it was at an instant, as shown in Fig. 5(a) and 5(b). The entropy at an instant of time is consistently at 5.93 for all mode numbers of axis symmetric and asymmetric liquid surface shapes. The entropy of the exposed images, however, shows an inverse relationship with increasing mode number $n$ , as seen in Fig. 5(a). This may be related to the relationship between the ratio of mean liquid wavelength over phase mask pixel width $λ_{f} / Δ x$ , which is an inverse relationship over liquid mode number $n$ . Though this relationship points to the possibility of an adversary using the entropy of the intensity to guess the liquid profile, it should be noted that given only a few encrypted images, the entropy is statistically not a reliable indicator of what the liquid mode number is, or whether the liquid profile is axis symmetric or not. Though information should be contained in the phase of the encrypted image, it would be of note if the entropy of the intensity of encrypted images was lower when a liquid system is used.

The mean correlation coefficient between encrypted images and the liquid surface mode pattern was near zero, with a variance that increased when one of the random phase masks wasn’t used. As shown in Fig. 4, the phase values from the mid-line of the axis symmetric $(m = 0)$ liquid surface pattern and QR code, and the instantaneous encrypted image were taken as two data series and compared. An encrypted image resulting from each liquid surface mode shape was compared with the corresponding liquid mode shape. The mean correlation coefficient when one or both of the random phase masks was used was on the order of 10⁻³. When no random phase masks were used, the mean correlation coefficient ranged from + 0.53 to −0.28, as shown in Fig. 6(a)

Fig. 6 The (a) mean and (b) variance of the correlation coefficient for $n = 2, 3, \dots, 30$ , calculated between centerlines of recovered phase images and the original plaintext phase image. Both the liquid system and the two random phase masks were used for the data series “Phase” and “Exposed”, while only one random phase mask was used for data series labeled “One Mask”. No random phase masks were used for series “No Masks”.

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. This is to be expected, as the resulting image is just the QR code multiplied pixel by pixel with the liquid surface profile. The variance was below 10⁻² when both random phase masks were used. However, when only one of the random phase masks was used, the variance was 2–10x higher (phase) and showed an inverse relationship with increasing liquid mode number

n

, as shown in Fig. 6(b). This could be due to the frequencies of the liquid surface profile not being completely concealed by the second random phase mask when it is absent. Although the lack of a statistically meaningful correlation coefficient when the liquid system and both random phase masks are used does not prove there is no correlation between encrypted images and the relevant liquid surface profiles [35], it does provide a strong case for there being no easily found relationship.

A mismatched liquid surface profile produced significant MSE values in the recovered images, increasing with lower mode number $n$ .The MSE between recovered phase images and the original QR code phase image was calculated with each liquid mode shape $n = 2, 3, \dots, 30$ used for decryption. For a particular mode shape used during encryption, this creates a zero MSE value when that same mode shape was used for decryption, as seen in the sharp drops in Fig. 7(a)

Fig. 7 (a) The mean squared error (MSE) when comparing the phase of recovered images and the original QR code phase image when different liquid surface mode shapes are used for decryption (mismatched liquid surfaces). (b) The original QR code used for the simulations. Recovered phase images for (c) a mismatched liquid surface, (d) A mistiming of the liquid system used during decryption of 1/10 liquid oscillation period and (e) for correct timing but one incorrect random phase mask.

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. From Fig. 7(a), it can be seen that the MSE ranges from 0.28 to 0.62 when the liquid profile is mismatched for decryption. This can be compared to

M S E = 0.42

, found when an incorrect random phase mask is used with no liquid system [Fig. 7(e)]. Figure 7(a) shows the max

M S E = 0.37

for

n = 20

, while the max

M S E = 0.62

for

n = 2

used during encryption. The

M S E

declines with an increase in the liquid mode number used for encryption. It should be noted that although the

M S E

values from a mismatched liquid surface approach the

M S E = 0.42

mark for an incorrect random phase mask, and in some cases pass this mark, the recovered images still show signs of an axis symmetric phase element in the system, as shown by the example in Fig. 7(c). The high

M S E

values, meaning a significant different in the recovered image from the original image, when a mismatched liquid surface pattern is used for decryption demonstrates the ability of the liquid system to vary the encryption and decryption of images with a tunable surface, even if the same random phase masks are used.

Significant $M S E$ values were also found when the timing of the liquid surface profile was altered during decryption. The $M S E$ between recovered image phase and the original image phase was calculated for a time mismatch of $t_{d} = {0.1, 0.3, 0.5} \cdot 2 π / ω_{m n}$ . The $M S E$ values ranged from 0.20 to 0.46, with the largest $M S E$ values coming from low liquid mode numbers and a mismatch fraction of $1 / 10$ $(t_{d} = (1 / 10) \cdot 2 π / ω_{m n})$ . An example is shown in Fig. 7(d). This study shows the need for precision in the user’s estimation of the liquid surface profile over time. A time mismatch contributing to high $M S E$ values further shows the difficulty an adversary would have in estimating an unknown liquid surface profile.

Simulations show the amount of noise that can be expected in recovered images from additive noise in the encrypted image and error in the understanding of the liquid system. A Gaussian distribution with a standard deviation of $σ_{η} = {10, 50, 100}$ % was used to model error in the estimation of the liquid oscillation amplitude. With an amplitude tuned to provide a $π$ phase shift, the oscillation height is $\max [h (x, y, t)] = 2.611$ μm. Similarly, a Gaussian distribution to model horizontal position error a standard deviation of $σ_{δ} = {4, 8, 12}$ pixels was used. In this study, this corresponds to a physical shift standard deviation of $σ_{δ} = {72, 144, 216}$ μm. Gaussian noise with a standard deviation of $σ_{γ} = {π / 10, \dots, π}$ was added to the phase of encrypted images at an instant of time for each simulation. Numerical simulations were performed 100 times for each error case. The mean MSE of recovered images with oscillation amplitude error and horizontal position error are shown in Fig. 8(a)

Fig. 8 (a) The mean squared error (MSE) when comparing the phase of recovered images and the original QR code phase image when noise is added to the encrypted image phase and there is error with varying standard deviation in the (a) amplitude of the liquid oscillations, or (b) the horizontal positioning of the liquid system. Recovered phase images with Gaussian error in the encrypted image $σ_{γ} = π / 10$ , and amplitude error $η = 0.05$ for (c) an instantaneous encryption and (d) exposed encryption. Recovered phase images with Gaussian error in the encrypted image $σ_{γ} = π / 10$ , and amplitude error $δ = 4$ pixels for (e) instantaneous encryption and (f) exposed encryption.

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and Fig. 8(b), respectively.

The error in recovered images increases when encryption is performed over an increasing amount time duration. This is intuitive, as accurate estimation of the liquid oscillation amplitude and horizontal position is needed to reconstruct an encrypted image at an instant of time from an exposed encrypted image. A comparison between recovered images when the encryption is instantaneous and when it is exposed over half of the oscillation period $τ = π / ω_{m n}$ is shown in Fig. 8(c)–8(f), with Fig. 8(c) and Fig. 8(e) coming from instantaneous encryption. The longer the encrypted image is exposed, the more error will be found in the recovered image.

4. Conclusions

An oscillating liquid system that modulates phase over time can be used to add a tunable and time-dependent element to a double random phase mask system. This can potentially be used to reduce vulnerability to multi-image ciphertext-only and known-plaintext attacks. The necessary assumption of an unchanging first phase mask in multiple image attacks will no longer be valid in a system with a time-dependent first lens. In order to use a liquid system as a phase modulating encryption device, the liquid system must be either predictable or measurable during encryption and decryption. During encryption, if the encrypted image is recorded over a sufficiently short period of time, the encrypted image can be directly decrypted by undoing the operations done during encryption. With a longer encryption exposure time, the encrypted image can be extracted from the recorded encrypted image by subtraction.

We found no significant correlation between the liquid system patterns and the resulting encrypted images, and the entropy of encrypted images did not decrease with the use of a liquid system. Simulations showed that entropy of the phase of instantaneous encrypted images is at a maximum, the same as with a traditional phase-only DRPE. The entropy of the intensity of an instantaneous and exposed encrypted images is slightly higher (0.30 to 0.31) than that found using a traditional phase-only DRPE [Fig. 5(b)], and increases with the use of multiple liquid mode numbers forced at once. It was found that the mean correlation coefficient is near zero between the liquid profiles and the encrypted images. The variance in the correlation coefficient between simulations is less than 0.01. However, when one of the random phase masks is removed, we found the variance to increase to levels between 0.06 and 0.01. The liquid system, in combination with two random phase masks gives way to no statistically significant correlation with the encrypted images.

The tunable and time-dependent nature of the liquid surface can add noise to the recovered image, but may not be as significant as error in the encrypted image from the other optics. A reasonable error in oscillation amplitude of 131 nm, and horizontal positioning of 72 μm does not make the image unrecoverable, even when simulated with additive Gaussian noise in the phase of the encrypted image with a standard deviation of $σ_{γ} = π / 10$ . This shows the feasibility of implementation an oscillating liquid surface in a double random phase mask system.

An incorrect liquid surface profile mode number, or mistiming significantly alters the recovered image. The mean squared error $(M S E)$ between the recovered image and the original image was shown to be between 0.28 to 0.62 when the liquid was mismatched during decryption. Mismatching of the liquid during decryption gave $M S E$ values from 0.20 to 0.46. This demonstrates the tunable and time-dependent, yet significant, change in encryption/decryption produced by a dynamic liquid lens system.

This tunable, time-dependent change in the system transfer function may be used to also prevent single image known-plaintext attacks. A vulnerability to Dirac delta known-plaintext attacks can be overcome if the second lens in the system [labeled L2 in Fig. 1(a)] is also a liquid system with the capability of tunable oscillations. If the lens oscillations have radially random oscillation amplitude, such as the example in Fig. 2(b), a physical system similar to the one proposed in Kumar et al. [38] to prevent chosen-plaintext attacks would be realized.

Future studies will involve experiments and multi-frequency forcing of liquid systems for randomly varying phase modulation. After generating a random map of liquid surface heights, multi-frequency forcing could then be used to produce the desired heights. An example of this with axis symmetric surface modes is shown in Fig. 2(b). Further study is needed to determine if Multi-frequency forced liquid surface could potentially be used to replace one of the random phase masks in a double random phase mask system. Experiments demonstrating the added security from a dynamic liquid lens are currently being prepared, and should demonstrate for the first time a time-dependent optical image encryption.

Funding

Boren Fellowship for International Study.

Acknowledgements

The authors would like to acknowledge the use of the University of Washington (UW) Hyak Computing Server in the making of this paper. The authors would like to thank Amir Amini (Oculus, Redmond WA) and Nicholas Boechler (University of California, San Diego) for relevant discussions. D. Schipf would like to acknowledge the support of the Boren Fellowship for International Study for partial funding while this work was put together.

References

1. B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016). [CrossRef]

2. F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16(7), 073019 (2014). [CrossRef]

3. A. Luis, “Coherence, polarization, and entanglement for classical light fields,” Opt. Commun. 282(18), 3665–3670 (2009). [CrossRef]

4. B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36(5), 1054–1058 (1997). [CrossRef] [PubMed]

5. B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994). [CrossRef]

6. X. Liu, J. Wu, W. He, M. Liao, C. Zhang, and X. Peng, “Vulnerability to ciphertext-only attack of optical encryption scheme based on double random phase encoding,” Opt. Express 23(15), 18955–18968 (2015). [CrossRef] [PubMed]

7. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007). [CrossRef] [PubMed]

8. K. Nakano, M. Takeda, H. Suzuki, and M. Yamaguchi, “Security analysis of phase-only DRPE based on known-plaintext attack using multiple known plaintext-ciphertext pairs,” Appl. Opt. 53(28), 6435–6443 (2014). [CrossRef] [PubMed]

9. E. S. ten Have and G. Vdovin, “Characterization and closed-loop performance of a liquid mirror adaptive optical system,” Appl. Opt. 51(12), 2155–2163 (2012). [CrossRef] [PubMed]

10. B. Berge, “Liquid lens technology: principle of electrowetting based lenses and applications to imaging,” in 18th IEEE International Conference on Micro Electro Mechanical Systems 2005 (IEEE, 2005), pp. 227–230. [CrossRef]

11. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000). [CrossRef]

12. C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26(12-17), 1–16 (2012). [CrossRef]

13. G.-H. Feng and J.-H. Liu, “Simple-structured capillary-force-dominated tunable-focus liquid lens based on the higher-order-harmonic resonance of a piezoelectric ring transducer,” Appl. Opt. 52(4), 829–837 (2013). [CrossRef] [PubMed]

14. S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004). [CrossRef]

15. M. Strauch, Y. Shao, F. Bociort, and H. P. Urbach, “Study of surface modes on a vibrating electrowetting liquid lens,” Appl. Phys. Lett. 111(17), 171106 (2017). [CrossRef]

16. J. M. Oh, S. H. Ko, and K. H. Kang, “Shape oscillation of a drop in AC electrowetting,” Langmuir 24(15), 8379–8386 (2008). [CrossRef] [PubMed]

17. Y. S. Nanayakkara, S. Perera, S. Bindiganavale, E. Wanigasekara, H. Moon, and D. W. Armstrong, “The effect of AC frequency on the electrowetting behavior of ionic liquids,” Anal. Chem. 82(8), 3146–3154 (2010). [CrossRef] [PubMed]

18. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed]

19. G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185(1-3), 25–31 (2000). [CrossRef]

20. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef] [PubMed]

21. S. Liu and J. T. Sheridan, “Optical encryption by combining image scrambling techniques in fractional fourier domains,” Opt. Commun. 287, 73–80 (2013). [CrossRef]

22. W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120 (2014). [CrossRef]

23. O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009). [CrossRef]

24. H. E. Hwang, H. T. Chang, and W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009). [CrossRef] [PubMed]

25. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010). [CrossRef] [PubMed]

26. E. Pérez-Cabré, H. C. Abril, M. S. Millán, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14(9), 094001 (2012). [CrossRef]

27. Corning liquid lens Product Brochure, “Corning Varioptic Lenses,” (27 June 2018). https://www.corning.com/media/worldwide/Innovation/documents/FINAL_CorningVariopticLenses_productbrochure_5.4.18_revised.6.28.18_lowres.pdf.

28. M. Maillard, J. Legrand, and B. Berge, “Two liquids wetting and low hysteresis electrowetting on dielectric applications,” Langmuir 25(11), 6162–6167 (2009). [CrossRef] [PubMed]

29. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co, 2005).

30. R. A. Ibrahim, Liquid Sloshing Dynamics: Theory and Applications, 2nd ed. (Cambridge University Press, 2013).

31. S. Terrab, A. M. Watson, C. Roath, J. T. Gopinath, and V. M. Bright, “Adaptive electrowetting lens-prism element,” Opt. Express 23(20), 25838–25845 (2015). [CrossRef] [PubMed]

32. 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(23), 4117–4130 (2000). [CrossRef] [PubMed]

33. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000). [CrossRef] [PubMed]

34. U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, “Digital holography,” in Digital Holography and Wavefront Sensing (Springer, 2015), pp. 39–68.

35. J. W. Goodman, Statistical Optics, 2nd ed. (John Wiley and Sons, 2013).

36. N. Towghi, B. Javidi, and Z. Luo, “Fully phase encrypted image processor,” J. Opt. Soc. Am. A 16(8), 1915–1927 (1999). [CrossRef]

37. Matlab, “The Mathworks, Inc.” (2017).

38. P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34(3), 331–333 (2009). [CrossRef] [PubMed]

Optical encryption using a liquid phase mask

Abstract

1. Introduction

2. Methods

3. Results and discussion

4. Conclusions

Funding

Acknowledgements

References

Cited By

Figures (8)

Equations (22)

OSA Continuum