A Data Concealing Technique with Random Noise Disturbance and A Restoring Technique for the Concealed Data by Stochastic Process Estimation

We propose a technique to conceal data on a physical layer by disturbing them with some random noises, and moreover, a technique to restore the concealed data to the original ones by using the stochastic process estimation. Our concealing-restoring system manages the data on the physical layer from the data link layer. In addition to these proposals, we show the simulation result and some applications of our concealing-restoring technique


Introduction
Micro-device technology in the near future realizes the remote control of microprocessor chips in several things such as household electric appliances, information-processing equipments, and even brain-computer interface from the outside through the wireless or the so-called IoT (i.e., Internet of Things). Moreover, it enables the automatic operation of such things with that remote control. They are going to infiltrate society and play several important roles in every area of society. We then have to establish the security for them. In particular, we have to stem the hacking of the remote control and the wiretapping of the data of communication. We are interested in a data concealing technique with disturbance on a physical layer and a restoring technique for those concealed data. Here, the physical layer is the lowest layer of the open systems interconnection (OSI) [13] (see Fig.1). OSI is a reference model to grasp and analyze how data are sent and received over a computation or communication network. Some methods using disturbance have been presented to conceal data for storage and communication. For instance, chaotic cryptology [6,9,16,28] uses chaos to make the disturbance. The method using cryptographic hash functions for the disturbance has lately been gaining a practical position [7,17,18,26]. Particularly, there have been some endeavors for the concealing technique on physical layers: the chaos multiple-input multiple-output [12,20,21,22,30]. Meanwhile, it is noteworthy that there have been many studies on the secured telecommunication using noises [8,11,19,27,29]. For the signal put on a carrier wave and sent from an antenna, we send some noises from interference antennas; we have the signal interfering with them and make it an interference wave.
We take interest in how to conceal data on a physical layer using some random noise disturbances and how to restore those concealed data applying a stochastic filtering theory to maintain the safety of data over a proper period of time, which is different from the interference wave method. Thus, our concealing-restoring system should be installed on the data link layer above the physical layer (see Fig.1). The idea of the concealing-restoring system that we propose in this paper is primarily originated in keeping security over a Figure 1: The left picture shows that the OSI consists of 7 layers. The encryption and decryption are usually done on one out of layers between Layer 3 and Layer 7, typically on the presentation layer. The right picture shows what we aim our concealing-restoring system at.
necessary period for the data processing on the physical layer of our developing quantumsensing device. This device detects and handles some ultimate personal information. Since we are required to remove several noises in the device, we make our concealing-restoring system coexist with the denoising system of the device. As some applications deriving therefrom, we establish a mathematical technique for concealing data by the disturbance with randomness of the noises, and moreover, a mathematical technique for restoring the concealed data by the stochastic process estimation. In addition to these establishments, we show the simulation result and some applications for the two techniques. The idea on our method to conceal data comes from an image of the scene when we conceal a treasure map, and it is so simple as follows: c1) we plaster over the treasure map at random and make it messy; c2) we repeat c1 and plaster it over repeatedly.
In this paper, we mathematically realize c1 and c2, and make their implementation. In addition to c1 and c2, we can consider that c3) we tear the muddled map by c1 and c2, and split it into several pieces though we do not make its implementation in this paper.
We are planning that we use the concealed data for saving them in a memory or for sending them for telecommunication. We expect to use our methods in the situation where the physical layer is under restrictions in the implementation space due to a small consumed electric power, a small arithmetic capacity, a small line capacity, and a bad access environment, and moreover, in the situation where it is too harsh to make a remote maintenance of the physical layer, for example, in outerspace development or seafloor development. We hope to apply the implementation of our techniques, for instance, to the remote control of drones and devices on them, and to the security of some data sent from those devices.

Mathematical Set-Ups
We first explain the outline of how to make our concealing-restoring system for data X t , t ∈ R, which is given by a simultaneous equation system (SES). This SES consists of some stochastic differential equations (SDEs), linear equations, and a nonlinear equation (NLE). The data X t is input as the initial data of the SES. We prepare N functionals F i , i = 1, 2, · · · , N , making the SDEs. We suppose that each form of the individual functional F i is known only by those who conceal the original data X t and restore their concealed data. We use the forms of the functionals as well as the composition of the SES for secret keys or common keys. We prepare 2N random noises W j,i t , j = 1, 2; i = 1, 2, · · · , N , for the SDEs, and a nonlinear bijection f for the NLE. The SDEs for processes X i t , i = 1, 2, · · · , N , and the NLE for the process X N +1 t are used to introduce the noise disturbance in our concealing-restoring system. We also use the means, variances, and distributions of the random noises as well as the nonlinear bijection as secret keys. As shown below, we obtain N + 1 concealed data, U i t , i = 1, 2, · · · , N, N + 1, using the SDEs and the NLE. We use them as the data for saving in a digital memory such as a semiconductor memory or an analogue memory such as a magnetic tape. We may also put the concealed data on a carrier wave and send them. This is the outline of the data concealing. Meanwhile, the data restoration is done in the following. Using the stochastic filtering theory and the inverse function f −1 , we remove the random noises from every concealed data U i t , and we estimate the process X i t . We denote the estimate by X i t , and call it estimated data for the process X i t . We think the estimate X 1 t the restoration of the original data X t . We denote it by X t . We here explain how to make the data X t from binary data. We use the low/high-signal for the binary data in this paper though there are many other ways. Thus, we represent 'low' by 0 and 'high' by 1. For n + 1 bits, a 0 , a 1 , · · · , a n ∈ {0, 1}, we concatenate them and make a word a 0 a 1 · · · a n . We employ the following linear interpolation as a simple digital-analogue (D/A) transformation. We first define X i by We connect X i and X i+1 with a straight line for each i = 0, 1, · · · , n − 1, and we have polygonal line X t , 0 ≤ t ≤ n. When the data X t are made from the binary word a 0 a 1 · · · a n , we call X t a binary pulse for the word a 0 a 1 · · · a n . As for the restoration of the word, we use the simple analogue-digital (A/D) transformation to seek the character a i ∈ {0, 1} for each i = 0, 1, · · · , n, and make a word a 0 a 1 · · · a n for the original word a 0 a 1 · · · a n in the following. We determine a threshold in advance between those who conceal the binary pulse and restore its concealed data to it. The threshold is basically determined taking into account the mean and variance of the random noises when used for concealing data. For each i = 0, 1, · · · , n, we define the character a i by We call the word a 0 a 1 · · · a n restored word from X t . We note that the mean and variance play an important role to define a threshold between 'low' and 'high' of signals, in particular, when we use ν-adic numbers such as octal numbers and hexadecimal numbers instead of binary numbers. From now on, we explain mathematical details for our data concealing technique and restoring technique. We give our secret SES by In the above system,Ẋ i Eqs.(2.1) and (2.3) are the mathematical realization of c1. The repetition of Eq.(2.1) from i = 1 to i = N with the help of Eq.(2.2) is for the mathematical realization of c2. We can mathematically realize c3 by taking numbers r , = 1, 2, · · · , M , with M =1 r = 0, and defining where i = j. Then, we can split the data U i t into the data U t , = 1, 2, · · · , M . For instance, we generate a random number r with r = 0, and set r 1 and r 2 as r 1 = r and r 2 = −r for M = 2. From the split data, U t , = 1, 2, · · · , M , we can restore the data U t to the data U i t and U j t by for an satisfying r = 0. We can use the sequence, r 1 , r 2 , · · · , r M , as a secret or common key. We note that the last stochastic process appearing in Eq.(2.3) has the form,

How to Conceal Data
We take the original signal X t as initial data, In the light of the stochastic filtering theory, Eqs.(2.1) and (2.2) are the state equation and the observation equation, respectively. That is, they make the system form the noise-filtering. Inputting the above X N +1 t into Eq.(2.2), and the concealed data U N t into Eq.(2.1), we have simultaneous equations to seek the data X N t , Since we cannot completely restore the noises to the original ones, W 1,N t and W 2,N t , we estimate the stochastic process X N t with the help of a proper stochastic filtering theory to remove the random noises. We then obtain the estimated data X N t . Inputting the estimated data X N t into the slot of X N t of Eq.(2.2), and the concealed data In the same way as in the above, the stochastic filtering theory gives us the next estimated data X N −1 , generally speaking, we have to know the concrete forms of the functionals, and the noise-filtering theory. Therefore, we hide the both for securing the original data.
Though it is actually supposed to be in secret, we give one of examples of the concrete definition of the functionals in this paper. We determine functions A i (t), v i (t) and non-zero Here v i (t) can be a random noise. For instance, we often makes v i (t) by the linear interpolation based on normal random numbers. Namely, we first assign a normal random number with N (0, σ 2 v ) to v i (k) for each i and k, and then, connect them by linear interpolation. Here N (0, σ 2 v ) means the normal distribution whose mean and standard deviation are respectively 0 and σ v . We give each functional F i such that it makes a SDE, for each i = 1, 2, · · · , N . That is, Here W 1,i t and W 2,i t are Gaussian white noises whose mean m j,i and variance V j,i are respectively 0 and (σ i j ) 2 . B i t is the Brownian motion given by W 1,i t = dB i t /dt, i = 1, 2, · · · , N . We assume that the noises W 1,i t and W 2,i t are independent for each i = 1, 2, · · · , N , but the noises W 2,i t , i = 1, 2, · · · , N , are not always independent. Thus, in the case where they are not independent, the linear combination of white noises appearing in Eq.(2.4) is not always white noise.
We regard the functions A i (t), the constants b i u , b i , and the mean m j,i and variance V j,i = (σ i j ) 2 of the white noises as secrete keys which are known only by the administrator of our concealing-restoring system. We use functions v i (t) as common keys which are used for both concealing and restoring the data. Since Eqs.

Discrete Version of Kalman Filtering
We seek the restoration X t by numerical analysis. Thus, we discretize Eq.(2.1); we approximate the differential by the forward difference, for t = k∆t with k = 0, 1, 2, · · · . Employing ∆t as a unit, the differential is approximated as dX i t /dt ≈ X i k+1 − X i k for k = 1, 2, · · · , and therefore, Eq.(2.1) is discretized as for each i = 1, 2, · · · , N . Here we respectively denote A i (k) and v i (k) by A i k and by v i k . With this discretization, Eqs.(2.2) and (2.3) respectively become In addition to the concealed data U N +1 k given by Eq.(3.3), using Eq.(3.2) we give the other concealed data U i k , i = 1, 2, · · · , N , by The concealed data are actually created by a computer with Alg.1.

Algorithm 1
Concealing Data X t Determine secret keys, A i k , b i u , b i , c i ; m j,i , σ i j . Determine a common key, v i k . Define white noises, W j,i k , j = 1, 2; i = 1, 2, · · · , N , with the individual mean m j,i and variance (σ i j ) 2 . Determine N , how many SDEs you want. Determine n, how many data you handle. for k = 0, 1, · · · , n do Set X 1 k := X k end for for i = 1, 2, · · · , N do for k = 0, 1, · · · , n do

end for
Conversely, we can estimate the data, X N k , X N −1 k , · · · , X 1 k , from the concealed data, U N k , U N −1 k , · · · , U 1 k . The linear Kalman filtering theory [4,10,14,15] says that we can make an algorithm to obtain the estimated data, X N k , X N −1 k , · · · , X 1 k , in the following. We denote the priori estimate by X −,i k , the variance by P i k , the priori variance by P −,i k , and the Kalman gain by g i k for each i = N, N − 1, · · · , 1. We repeat the procedure consisting of 'Prediction Step' and 'Filtering Step' from i = N to i = 1. We note that in the Kalman filtering theory the estimate X k represents an optimal estimate, and is called the posteriori estimate.
Before the Kalman filtering, the concealed data X N +1 k is obtained as

'Prediction Step' and 'Filtering
Step' of the Kalman filtering are as follows:

Prediction
Step: Filtering Step: , and X i 0 .
The stochastic-process estimation consisting of the prediction and filtering steps is done by a computer with Alg.2.

Algorithm 2 Restoration from Concealed Data
Get the common key, v i k . Obtain the concealed data, U i k . for k = 0, 1, · · · , n do Set X N +1 k end for end for for k = 0, 1, · · · , n do Set X k := X 1 k end for

Simulation of Concealing and Restoring Data on Physical Layer
In our simulation of concealing and restoring data on physical layer, we employ the message digest [5,23,24,25] to check the coincidence of the original word a 0 a 1 · · · a n and its restored word a 0 a 1 · · · a n though the message digest works on upper layers. Moreover, we can use the message digest to detect any falsification of the concealed data. We take the original word a 0 a 1 · · · a n as a message, and then, produce its digest. We also produce the digest for the restored word a 0 a 1 · · · a n . Comparing hash values of the two digests, we can make the check of the coincidence and the detection of the falsification at the same time. They should be done on one out of layers between Layer 3 and Layer 7. In our simulation, we employ SHA-256 to make the hash values [3].
To make the estimation in the simulation, we employ the linear Kalman filtering theory [15] under the following conditions. We make Eqs.(2.1)-(2.3) for N = 2 with A i (t) = 0.1 (constant function), b i = 1, b i u = 1, and c i = 1 for each i = 1, 2. We define the common key v i (t) by the linear interpolation based on a normal random number with N (0, 1 2 ). We assume that the mean of white noises are all 0. The standard deviation of the white noise W j,1 t is σ 1 j = 0.1, and that of the white noise W j,2 t is σ 2 j = 1. The length of the word a 0 a 1 · · · a n is 100, and therefore, n = 99.
Our original word a 1 a 2 · · · a 99 is given by Eq. (3.5). We here note that we remove the character a 0 because we cannot estimate the first bit in our concealing-restoring system.

1010011110111101100101010100010110111100110111001.
Then, we get its binary pulse X t as in Fig.2. The hash value of the digest made from the We make the concealed data, U 1 t and U 2 t , by Eq.(3.4) with the help of the linear equation given in Eq.(2.2). We finally make the concealed data U 3 t using the nonlinear equation given in Eq.(2. 3) with f (ξ) = ξ 3 . Their graphs are in Figs.3 & 4. Following the Kalman filtering theory, we remove the white noises, and estimate the binary pulse X t . Then, we obtain the restoration X t as in Fig.5. Let us take 0 as the threshold. Then, we obtain the restored word a 1 a 2 · · · a 99 and the hash value of its digest made from the restoration X t , and they are the same as Eqs. (3.5) and (3.6), respectively.
We note that the graphs in Figs.3 & 4 say that the concealed data, U 1 t , U 2 t , and U 3 t , are merely analogue data. If a wiretapper becomes aware that the concealed data are for digital ones and knows our A/D transformation in some way, then the wiretapper gets a binary word from the concealed data as follows:   for U 3 t . Here, since the wiretapper does not know that we remove the first bit, every concealed data U i t make the word consisting of 100 characters. In Fig.6 we show the comparison of the original binary pulse X t , its restoration X t , and the concealed data U i t , i = 1, 2, 3.    concealed data U 3 t defined by Eq.(3.3) is in Fig.16. We can restore the pictorial image with Figure 15: The concealed data, U 1 t (left) and U 2 t (right), for the analogue data Xt in Fig.14 the restoration X t as in Fig.17. If a wiretapper becomes aware our method to make a pictorial image from analogue data, then the wiretapper gets pictorial images from the concealed data U i t , i = 1, 2, 3, as in Fig.18.
In Fig.19 we show the comparison of the original binary pulse X t , its restoration X t , and the concealed data U i t , i = 1, 2, 3.

Conclusion and Future Work
We have proposed a mathematical technique for concealing data on the physical layer of the OSI reference model by using random noise disturbance, and moreover, a mathematical technique for restoring the concealed data by using the stochastic process estimation. In this concealing-restoring system, the functionals determining SDEs play a role of secret or common keys, and moreover, the proper noise-filtering theory forms a nucleus to restore the concealed data. In addition, we have showed the simulation result for the data on physical layer and some applications of the two techniques to the pictorial images. We have opened one of examples of the functionals. Then, we have showed how to conceal the data by using the noise-disturbance, and have demonstrated how to restore the data by removing the noises.
Here, the significant point to be emphasized is that any composition of the SES and any form of the individual functional will do so long as a proper noise-filtering method is established for    Figure 19: Xt (Fig.14) and Xt (Fig.17) from the above of the left 2 graphs. U 1 them. Although the state space determined by Eq.(3.1) is linear, Gaussian, and we used the linear Kalman filtering theory in §3 , we can make it more general: nonlinear, non-Gaussian state space. Then, we should employ the nonlinear Kalman filtering theory and particle filtering theory [4]. In fact, we already checked that the particle filtering theory works. We have used the scalar-valued processes, and thus, prepared just one common key for one SDE. We can prepare some common keys for one SDE by using the vector-valued processes.
Although we have employed the message digest to make the check of the coincidence of the binary word and the detection of the falsification at the same time, we are now developing a method with low complexity so that we can make them for data on the physical layer.
According to our several experiments including the concrete examples in §.4, we think that the nonlinearity enhances the noise-disturbance. For instance, the pictorial images in Fig.20 are the case N = 1. Comparing the pictorial images of U 2 t and X 2 t = f −1 (U 2 t ) in Fig.21, we can say that the enhancement of noise-disturbance appears with color. We will study the roles of several parameters.