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

Abstract

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.

1 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 equipment, and even brain–computer/brain–machine interfaces from the outside through wireless communications or the so-called IoT (i.e., Internet of Things). Moreover, it enables the automatic operation of such things with the remote control. They are going to infiltrate society and play several important roles in every area of society. We then have to establish the data security for them (Youm 2017; Román-Castro et al. 2018; Lin et al. 2018; Clausen et al. 2017). 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) (Kain and Agrawala 1992) (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 (Cuomo and Oppenheim 1993; Grassi and Mascolo 1999; Lenug and Lam 1997; Wu and Chua 1993) uses chaos to make the disturbance. The method using cryptographic hash functions for the disturbance has lately been gaining a practical position (Merkle 1979, 1989; Damgård 1989; Schneier 2015). There have been some endeavors for the concealing technique on physical layers: the chaos multiple-input multiple-output (Okamoto and Iwanami 2006; Zheng 2009; Okamoto 2011; Okamoto and Inaba 2015; Ito et al. 2019). Meanwhile, it is noteworthy that the secured telecommunication using noises has been actively studied (Wyner 1975; Hero 2003; Goel and Negi 2008; Swindlehurst 2009; Mukherjee and Swindlehurst 2011). In that technique, we send some noises from interference antennas to the signal on a carrier wave sent from an antenna; we have the signal interfering with the noises and make it an interference wave. There, however, may be a way to remove the noises from the interference wave and to wiretap the original signal (Ohno et al. 2012).

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 a data link layer above the physical layer (see Fig. 1). Although we employ the disturbance by random noises instead of the chaotic one, we can design our concealing-restoring system so that it includes the chaotic disturbance (Fujii and Hirokawa 2020). The idea of the concealing-restoring system was primarily originated in keeping security for the data processed on the physical layer of our developing quantum-sensing equipment over a necessary period. This equipment detects and handles some ultimate personal information. Since we must remove several noises on the physical layer in any case, we make our concealing-restoring system coexist with the denoising system of the equipment. We then consider the information concealing method for qubits (i.e., quantum bits) using the random noises in classical physics. The qubits $|0\rangle$ and $|1\rangle$ are represented by spin states $|\!\!\uparrow \rangle$ and $|\!\!\downarrow \rangle$, namely, $|0\rangle =|\!\!\uparrow \rangle =(1,0)$ and $|1\rangle =|\!\!\downarrow \rangle =(0,1)$. A general qubit $|q\rangle$ can be described with the superposition of the qubits $|0\rangle$ and $|1\rangle$: $|q\rangle =\alpha |0\rangle +\beta |1\rangle$ for some complex numbers $\alpha$ and $\beta$ with $|\alpha |^{2}+|\beta |^{2}=1$. Thus, the qubit can have the representation, $|q\rangle =(\mathfrak {R}\alpha ,\mathfrak {I}\alpha ,\mathfrak {R}\beta ,\mathfrak {I}\beta )$, and an information sequence of qubits, $|q_{1}\rangle , |q_{2}\rangle , \ldots , |q_{\nu }\rangle$, is expressed with a finite sequence,

$$ \mathfrak {R}\alpha _{1}\,\,\,\mathfrak {I}\alpha _{1}\,\,\,\mathfrak {R}\beta _{1}\,\,\,\mathfrak {I}\beta _{1}\,\,\, \mathfrak {R}\alpha _{2}\,\,\,\mathfrak {I}\alpha _{2}\,\,\,\mathfrak {R}\beta _{2}\,\,\,\mathfrak {I}\beta _{2}\,\,\, \ldots \,\,\, \mathfrak {R}\alpha _{\nu }\,\,\,\mathfrak {I}\alpha _{\nu }\,\,\,\mathfrak {R}\beta _{\nu }\,\,\,\mathfrak {I}\beta _{\nu }. $$

We transform it into an electrical signal $X_{t}$, $0\le t\le 4\nu$, using linear interpolation. We process the electrical signal in a microprocessor, made by some semiconductors, of our quantum-sensing equipment. Since the microprocessor is for the conventional computation (i.e., not quantum computation), we need to transport the electrical signal to memory or register according to a microarchitecture. To keep the security for the electric signal $X_{t}$ while processing, storing, and saving it, we employ a mathematical idea to conceal it using the noise disturbance. In this paper, we introduce that mathematical idea for more general signals on the physical layer and more broad applications.

Fig. 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

As some applications derive therefrom, we first 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 of 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 on conventional computers. 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 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. Concretely, we hope to apply the implementation of our techniques to the remote control of drones and devices on them, and to the security of some data sent from those devices. Moreover, we suppose the situation where it is too harsh to make a remote maintenance of the physical layer, for example, in outerspace development or seafloor development.

2 Mathematical Setups

We first explain the outline of how to make our concealing-restoring system for data $X_{t}$, $t\in \mathbb {R}$. The concealing-restoring system 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 $F_{i}$, $i=1, 2, \ldots , N$, 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 the 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 $W_{t}^{j,i}$, $j=1, 2$; $i=1, 2, \ldots , N$, for the SDEs, and a nonlinear bijection f for the NLE. The SDEs for processes $X_{t}^{i}$, $i=1, 2, \ldots , N$, and the NLE for the process $X_{t}^{N+1}$ 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, $U_{t}^{i}$, $i=1, 2, \ldots , N, N+1$, 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 analog 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 $U_{t}^{i}$, and we estimate the process $X_{t}^{i}$. We denote the estimate by $\widehat{X}_{t}^{i}$, and call it estimated data for the process $X_{t}^{i}$. We regard the estimate $\widehat{X}_{t}^{1}$ as the restoration of the original data $X_{t}$. We denote it by $\widehat{X}_{t}$.

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, $a_{0}, a_{1}, \ldots , a_{n} \in \{ 0, 1\}$, we concatenate them and make a word $a_{0}a_{1}\ldots a_{n}$. We employ the following linear interpolation as a simple digital–analog (D/A) transformation. We first define $X_{i}$ by

$$X_{i}= {\left\{ \begin{array}{ll} +1 &{} \text{ if }~a_{i}=1, \\ -1 &{} \text{ if }~a_{i}=0, \end{array}\right. } \qquad i=0, 1, \ldots , n. $$

We connect $X_{i}$ and $X_{i+1}$ with a straight line for each $i=0, 1, \ldots , n-1$, and we have a polygonal line $X_{t}$, $0\le t\le n$. When the data $X_{t}$ are made from the binary word $a_{0}a_{1}\ldots a_{n}$, we call $X_{t}$ a binary pulse for the word $a_{0}a_{1}\ldots a_{n}$. As for the restoration of the word, we use the simple analog–digital (A/D) transformation to seek the character $\widehat{a}_{i}\in \{ 0, 1\}$ for each $i=0, 1, \ldots , n$, and make a word $\widehat{a}_{0}\widehat{a}_{1}\ldots \widehat{a}_{n}$ for the original word $a_{0}a_{1}\ldots a_{n}$ 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 $i=0, 1, \ldots , n$, we define the character $\widehat{a}_{i}$ by

$$ \widehat{a}_{i}= {\left\{ \begin{array}{ll} 1 &{} \text{ if }~\widehat{X}_{i} > \mathrm {threshold}, \\ 0 &{} \text{ if }~\widehat{X}_{i} \le \mathrm {threshold}. \end{array}\right. } $$

We call the word $\widehat{a}_{0}\widehat{a}_{1}\ldots \widehat{a}_{n}$ restored word from $\widehat{X}_{t}$. We note that the mean and the variance play important roles to define a threshold between ‘low’ and ‘high’ of signals, in particular, when we use $\nu$-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

$$\begin{aligned}{} & {} F_{i}(X_{t}^{i}, \dot{X}_{t}^{i}, U_{t}^{i}, W^{1,i}_{t})=0, \qquad i=1, 2, \ldots , N, \end{aligned}$$

(1)

$$\begin{aligned}{} & {} X_{t}^{i+1}=c^{i}X_{t}^{i}+W_{t}^{2,i}, \qquad i=1, 2, \ldots , N, \end{aligned}$$

(2)

$$\begin{aligned}{} & {} U_{t}^{N+1}=f\left( X_{t}^{N+1}\right) . \end{aligned}$$

(3)

In the above system, $\dot{X}_{t}^{i}$ stands for the time derivative $dX_{t}^{i}/dt$ of the process $X_{t}^{i}$, and $c^{i}$ is a constant. The initial data $X_{t}^{1}$ is given by $X_{t}^{1}=X_{t}$. The concealed data $U_{t}^{i}$, $i=1, 2, \ldots , N, N+1$, are directly defined by Eqs. (1) and (3), not Eq. (2). That is, we can hide the linear part of our system because we do not have to make an interference wave. This is the point of our method that is different from that of telecommunication using noises (Wyner 1975; Hero 2003; Goel and Negi 2008; Swindlehurst 2009; Mukherjee and Swindlehurst 2011). Introducing functionals, $G_{i}$, $i=1, 2, \ldots , N$, and using them for Eq. (2), we can introduce the chaotic disturbance in our concealing-restoring system (Fujii and Hirokawa 2020).

Equations (1) and (3) are the mathematical realization of c1. The repetition of Eq. (1) from $i=1$ to $i=N$ with the help of Eq. (2) is for the realization of c2. We can mathematically realize c3 as follows: Take numbers $r_{\ell }$, $\ell =1, 2, \ldots , M$, with $\sum _{\ell =1}^{M}r_{\ell }=0$, and define

$$ U_{t}^{\ell } =\frac{1}{M}\left( U_{t}^{i}+r_{\ell }U_{t}^{j}\right) ,\quad \ell =1, 2, \ldots , M, $$

where $i\ne j$. Then, we can split the data $U_{t}^{i}$ into the data $U_{t}^{\ell }$, $\ell =1, 2, \ldots , M$. In the case $M=2$, for instance, we generate a random number r with $r\ne 0$, and set $r_{1}$ and $r_{2}$ as $r_{1}=r$ and $r_{2}=-r$. From the split data, $U_{t}^{\ell }$, $\ell =1, 2, \ldots , M$, we can restore the data $U_{t}^{\ell }$ to the data $U_{t}^{i}$ and $U_{t}^{j}$ by

$$ U_{t}^{i}=\sum _{\ell =1}^{M}U_{t}^{\ell }\qquad \text{ and }\qquad U_{t}^{j}=r_{\ell }^{-1}\left( MU_{t}^{\ell }-U_{t}^{i}\right) $$

for an $\ell$ satisfying $r_{\ell }\ne 0$. We can also use the sequence, $r_{1}, r_{2}, \ldots , r_{M}$, as a secret or common key.

We note that the last stochastic process appearing in Eq. (3) has the form,

$$\begin{aligned} X_{t}^{N+1} = c^{1}\cdots c^{N}X_{t} +\sum _{i=1}^{N-1}\left( \prod _{j=i+1}^{N} c^{j}\right) W_{t}^{2,i} +W_{t}^{2,N}. \end{aligned}$$

(4)

2.1 How to Conceal Data

We take the original data $X_{t}$ as initial data,

$$ X_{t}^{1}=X_{t}. $$

Inputting it into Eq. (1) with the noise $W^{1,1}_{t}$, we conceal it by the SDE,

$$ F_{1}(X_{t}^{1}, \dot{X}_{t}^{1}, U_{t}^{1}, W^{1,1}_{t})=0. $$

We seek $U_{t}^{1}$ in the above and obtain a concealed data $U_{t}^{1}$. By Eq. (2),

$$ X_{t}^{2}=c^{1}X_{t}^{1}+W_{t}^{2,1}, $$

we have data $X_{t}^{2}$ for the next step. These data $X_{t}^{2}$ consist of the superposition (i.e., linear combination) of $X_{t}^{1}$ and $W_{t}^{2,1}$, and thus, there is a possibility that a wiretapper removes the noise $W_{t}^{2,1}$ and wiretap $X_{t}^{1}$. Thus, to improve the security with another noise-disturbance, we have the same procedure again. We input the data $X_{t}^{2}$ into Eq. (1) with the noise $W_{t}^{1,2}$,

$$ F_{2}(X_{t}^{2}, \dot{X}_{t}^{2}, U_{t}^{2}, W^{1,2}_{t})=0. $$

We then obtain the concealed data $U_{t}^{2}$. Repeating the same procedures, we obtain the concealed data, $U_{t}^{1}, U_{t}^{2}, \ldots , U_{t}^{N}$, and hide the data, $X_{t}^{1}, X_{t}^{2}, \ldots , X_{t}^{N}$.

At last, input the concealed data $X_{t}^{N}$ into Eq. (2) and get the data $X_{t}^{N+1}$. We input this into Eq. (3) and hide it. We then obtain the last concealed data $U_{t}^{N+1}$. In this way, the sequence of the concealed data, $U_{t}^{1}, U_{t}^{2}, \ldots , U_{t}^{N}, U_{t}^{N+1}$, is created.

In the case where the original data are digital, and they give the binary pulse $X_{t}$, the concealed data, $U_{t}^{i}$, $i=1, 2, \ldots , N, N+1$, merely become analog data. So, a wiretapper has to know A/D transformation to obtain the original digital data as getting the concealed data. Therefore, the D/A and A/D transformations play an important role for the concealing-restoring system for some digital data. We can also use them as secret or common keys.

2.2 How to Restore Data

Since the nonlinear function f is bijective, we can restore the concealed data $U_{t}^{N+1}$ to the data $X_{t}^{N+1}$ by

$$ X_{t}^{N+1} =f^{-1}\left( U_{t}^{N+1}\right) . $$

In the light of the stochastic filtering theory, Eqs. (1) and (2) are the state equation and the observation equation, respectively, and they make the system of the noise-filtering. Inputting the above $X_{t}^{N+1}$ into Eq. (2), and the concealed data $U_{t}^{N}$ into Eq. (1), we have simultaneous equations to seek the data $X_{t}^{N}$,

$$\begin{aligned}{} & {} F_{N}(X_{t}^{N}, \dot{X}_{t}^{N}, U_{t}^{N}, W^{1,N}_{t})=0, \\{} & {} X_{t}^{N+1}=c^{N}X_{t}^{N}+W_{t}^{2,N}. \end{aligned}$$

Since we cannot completely restore the noises to the original ones, $W^{1,N}_{t}$ and $W^{2,N}_{t}$, we cannot completely seek the stochastic process $X_{t}^{N}$. Thus, we estimate it with the help of a proper stochastic filtering theory to remove the random noises. We then obtain the estimated data $\widehat{X}_{t}^{N}$.

Inputting the estimated data $\widehat{X}_{t}^{N}$ into the slot of $X_{t}^{N}$ of Eq. (2), and the concealed data $U_{t}^{N-1}$ into Eq. (1), we reach simultaneous equations to seek the data $X_{t}^{N-1}$,

$$\begin{aligned}{} & {} F_{N-1}(X_{t}^{N-1}, \dot{X}_{t}^{N-1}, U_{t}^{N-1}, W^{1,N-1}_{t})=0, \\{} & {} \widehat{X}_{t}^{N}=c^{N-1}X_{t}^{N-1}+W_{t}^{2,N-1}. \end{aligned}$$

In the same way as in the above, the stochastic filtering theory gives us the next estimated data $\widehat{X}_{t}^{N-1}$. We repeat this procedure, and obtain the estimated data, $\widehat{X}_{t}^{N}, \widehat{X}_{t}^{N-1}, \ldots , \widehat{X}_{t}^{2}, \widehat{X}_{t}^{1}$, by turns, and we pick up the last estimate $\widehat{X}_{t}^{1}$. This is the restoration $\widehat{X}_{t}$ of the original data $X_{t}$.

3 Example of Functionals and Simulation

As for how to determine each functional, $F_{i}$, $i=1, 2, \ldots , N$, any definition of it is fine so long as a noise-filtering theory is established for the system with $F_{i}$. To restore the concealed data, $U_{t}^{1}, U_{t}^{2}, \ldots , U_{t}^{N}, U_{t}^{N+1}$, generally speaking, we have to know the concrete forms of the functionals, and the noise-filtering theory. Therefore, we must hide both for securing the original data. In this paper, however, we disclose one of examples of the concrete definition of the functionals and one of examples of the noise-filterings, which should actually be supposed to be in secret. We point out that the example of concealing-restoring system introduced in this section is not valid for other functionals. In particular, it is not tolerant of nonlinearity. See Sect. 5.

3.1 An Example of the Set of Functionals

We release an example of functionals in this section. We determine functions $A^{i}(t), v^{i}(t)$, and non-zero constants $b_{u}^{i}, b^{i}$ in secret. Here $v^{i}(t)$ can be a random noise. For instance, we often make $v^{i}(t)$ by the linear interpolation based on normal random numbers. Namely, we first assign a normal random number with $N(0,\sigma _{v}^{2})$ to $v^{i}(k)$ for each i and k, and then, connect them by linear interpolation. Here, $N(0,\sigma _{v}^{2})$ means the normal distribution whose mean and standard deviation are, respectively, 0 and $\sigma _{v}$. We give each functional $F_{i}$ such that it makes a SDE,

$$\begin{aligned} dX_{t}^{i}=\left( A^{i}\left( t\right) -1\right) X_{t}^{i}dt +b_{u}^{i}U_{t}^{i}dt +b^{i}v^{i}(t)dt -b_{u}^{i}dB_{t}^{i}, \end{aligned}$$

(5)

for $i=1, 2, \ldots , N$. That is,

$$\begin{aligned} \dot{X}_{t}^{i}=\left( A^{i}\left( t\right) -1\right) X_{t}^{i} +b_{u}^{i}U_{t}^{i} +b^{i}v^{i}(t) -b_{u}^{i}W_{t}^{1,i}. \end{aligned}$$

(6)

Here, $W_{t}^{1,i}$ and $W_{t}^{2,i}$ are Gaussian white noises whose mean $m^{j,i}$ and variance $V^{j,i}$ are, respectively, 0 and $(\sigma _{j}^{i})^{2}$. $B_{t}^{i}$ is the Brownian motion given by $W_{t}^{1,i}=dB_{t}^{i}/dt$, $i=1, 2, \ldots , N$. We assume that the noises $W_{t}^{1,i}$ and $W_{t}^{2,i}$ are independent for each $i=1, 2, \ldots , N$, but the noises $W_{t}^{2,i}$, $i=1, 2, \ldots , N$, are not always independent. Thus, in the case where they are not independent, the linear combination of white noises appearing in Eq. (4) is not always white noise.

We regard the functions $A^{i}(t)$, the constants $b_{u}^{i}, b^{i}$, and the mean $m^{j,i}$ and variance $V^{j,i}=(\sigma _{j}^{i})^{2}$ of the white noises as secret keys which are known only by the administrator of our concealing-restoring system. We use functions $v^{i}(t)$ as common keys. Since Eqs. (5) and (2), respectively, play the individual roles of the state equation and observation equation in the stochastic filtering theory, we employ the linear Kalman filtering theory (Kalman 1960; Kallianpur 1980; Bain and Crisan 2009; Grewal and Andrews 2015) to obtain the restoration $\widehat{X}_{t}$.

Using Eq. (6) we give the concealed data $U_{t}^{i}$, $i=1, 2, \ldots , N$, by

$$\begin{aligned} U_{t}^{i}= \frac{1}{b_{u}^{i}} \left\{ dX_{t}^{i}+\left( 1-A^{i}(t)\right) X_{t}^{i}-b^{i}v^{i}(t) \right\} +dB_{t}^{i}. \end{aligned}$$

(7)

In addition to these concealed data, we give the last concealed data $U_{t}^{N+1}$ by Eq. (3). Conversely, since we obtain the data $X_{t}^{N+1}$ by $X_{t}^{N+1}=f^{-1}(U_{t}^{N+1})$, we can estimate the data, $X_{t}^{N}$, $X_{t}^{N-1}$, $\ldots$, $X_{t}^{1}$, from the concealed data, $U_{t}^{N}$, $U_{t}^{N-1}$, $\ldots$, $U_{t}^{1}$, using the linear Kalman filtering theory.

3.2 Simulation of Concealing and Restoring Data on Physical Layer

In our simulation of concealing and restoring data on the physical layer, we employ the message digest (Rivest 1991, 1992a, b; Suhaili and Watanabe 2017; MessageDigest 2020) to check the coincidence of the original word $a_{0}a_{1}\ldots a_{n}$ and its restored word $\widehat{a}_{0}\widehat{a}_{1}\ldots \widehat{a}_{n}$ 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 $a_{0}a_{1}\ldots a_{n}$ as a message, and then, produce its digest. We also produce the digest for the restored word $\widehat{a}_{0}\widehat{a}_{1}\ldots \widehat{a}_{n}$. 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. The check and detection should be performed on a layer out of layers between Layer 3 and Layer 7. In our simulation, we employ SHA-256 to make the hash values (Secure Hash Standard 2015).

To make the estimation in the simulation, we employ the linear Kalman filtering theory under the following conditions. We make Eqs. (1)–(3) for $N=2$ with $A^{i}(t)=0.1$ (constant function), $b^{i}=1$, $b_{u}^{i}=1$, 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 means of white noises are all 0. The standard deviation of the white noise $W_{t}^{j,1}$ is $\sigma _{j}^{1}=0.1$, and that of the white noise $W_{t}^{j,2}$ is $\sigma _{j}^{2}=1$. The length of the word $a_{0}a_{1}\ldots a_{n}$ is 100, and therefore, $n=99$.

Our original word $a_{1}a_{2}\ldots a_{99}$ is given by Eq. (8). We here note that we remove the character $a_{0}$ because we cannot estimate the first bit in our concealing-restoring system.

$$\begin{aligned}{} & {} \mathrm {00001100100111001000100000101110111111111001000110} \nonumber \\{} & {} \mathrm {1010011110111101100101010100010110111100110111001}. \end{aligned}$$

(8)

Then, we get its binary pulse $X_{t}$ as in Fig. 2. The hash value of the digest made from the original word (8) is

$$\begin{aligned} \text{979bca61579e002c9097c78088740e9fdaf21535d6a5c5876bd8623a86185292 }.\nonumber \\ \end{aligned}$$

(9)

Fig. 2

The binary pulse $X_{t}$ transformed from the original word (8)

We make the concealed data, $U_{t}^{1}$ and $U_{t}^{2}$, by Eq. (7) with the help of the linear equation given in Eq. (2). We finally make the concealed data $U_{t}^{3}$ using the nonlinear equation given in Eq. (3) with $f(\xi )=\xi ^{3}$. Their graphs are in Figs. 3 and 4. Following the Kalman filtering theory, we remove the white noises, and estimate the binary pulse $X_{t}$. Then, we obtain the restoration $\widehat{X}_{t}$ as in Fig. 5. The concrete algorithm to seek the restoration $\widehat{X}_{t}$ comes out in Ref. Fujii and Hirokawa (2020). Let us take 0 as the threshold. Then, we obtain the restored word $\widehat{a}_{1}\widehat{a}_{2}\ldots \widehat{a}_{99}$ and the hash value of its digest made from the restoration $\widehat{X}_{t}$. We can achieve positive results that they are the same as Eqs. (8) and (9), respectively.

Fig. 3

The concealed data, $U_{t}^{1}$ (left) and $U_{t}^{2}$ (right), for the binary pulse $X_{t}$ in Fig. 2

Fig. 4

The concealed data $U_{t}^{3}$ for the binary pulse $X_{t}$ in Fig. 2

Fig. 5

The restoration $\widehat{X}_{t}$ for the binary pulse $X_{t}$ in Fig. 2

We note that the graphs in Figs. 3 and 4 say that the concealed data, $U_{t}^{1}$, $U_{t}^{2}$, and $U_{t}^{3}$, are merely analog 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:

$$\begin{aligned}{} & {} \mathrm {00111011000111011000111000001001101011111001101100} \\{} & {} \mathrm {11011111101001111000010111100101101011000111100110} \end{aligned}$$

for $U_{t}^{1}$,

$$\begin{aligned}{} & {} \mathrm {00011011000111011010110000100100111001111011001010} \\{} & {} \mathrm {01011001001001111010010111110101000010001110110110} \end{aligned}$$

for $U_{t}^{2}$, and

$$\begin{aligned}{} & {} \mathrm {10000000000010110101110000010001001100111100100100} \\{} & {} \mathrm {00000101100111110101100010100010000001000111011001} \end{aligned}$$

for $U_{t}^{3}$. Here, since the wiretapper does not know that we removed the first bit, every concealed data $U_{t}^{i}$ makes the word consisting of 100 characters.

In Fig. 6 we show the comparison of the original binary pulse $X_{t}$, its restoration $\widehat{X}_{t}$, and the concealed data $U_{t}^{i}$, $i=1, 2, 3$.

Fig. 6

$X_{t}$ (Fig. 2) and $\widehat{X}_{t}$ (Fig. 5) from the above of the left 2 graphs. $U_{t}^{1}$ (Fig. 3), $U_{t}^{2}$ (Fig. 3), and $U_{t}^{3}$ (Fig. 4) from the above of the right 3 graphs. Here $t\in [0,99]$

4 Application to Data on Physical Layer and Presentation Layer

4.1 Binary Data of Pictorial Image

We now apply the technology of our mathematical method to the binary data of a pictorial image. We use digital data of a pictorial image in the ORL Database of Faces, an archive of AT&T Laboratories Cambridge (The ORL Database of Faces 2020). The data have the grayscale value of 256 gradations (8bit/pixel). We set our parameters as $A=A^{i}=0.1$, $b=b^{i}=1$, $b_{u}=b_{u}^{i}=1$, $c=c^{i}=1$, $\sigma _{1}=\sigma _{1}^{i}=0.1$, and $\sigma _{2}=\sigma _{2}^{i}=1$. We determine the common key $v^{i}(t)$ in the same way as in Sect. 3.2 with $\sigma _{v}=2$. The original pictorial image and its binary pulse $X_{t}$ are obtained as in Fig. 7. Here, the upper bound of t is $92\times 112=10304$ and t runs over [0, 10304]. We obtain the concealed data, $U_{t}^{1}$ and $U_{t}^{2}$, by Eq. (7) as in Fig. 8, and the concealed data $U_{t}^{3}$ by Eq. (3) as in Fig. 9. The restoration $\widehat{X}_{t}$ and the restored pictorial image from it are in Fig. 10.

Fig. 7

The original pictorial image (left) with the digital data, and its binary pulse $X_{t}$ (right) only for $t\in [0,200]$

Fig. 8

The concealed data, $U_{t}^{1}$ (left) and $U_{t}^{2}$ (right), for the binary pulse $X_{t}$ in Fig. 7. Here $t\in [0,200]$ only

Fig. 9

The concealed data $U_{t}^{3}$ for the binary pulse $X_{t}$ in Fig. 7. Here $t\in [0,200]$ only

Fig. 10

The restoration $\widehat{X}_{t}$ for the binary pulse $X_{t}$ in Fig. 7 only for $t\in [0,200]$ (right) and the restored pictorial image (left) of $\widehat{X}_{t}$

If a wiretapper tries to get the original pictorial image from the concealed data $U_{t}^{i}$, $i=1, 2, 3$, since the concealed data are analog as in Figs. 8 and 9, the wiretapper has to know our A/D transformation, and our transformation from the digital data to a pictorial image as well as some keys used in SES. The latter transformation should be done on upper layers. We now assume that the wiretapper can know the transformations. Then, each pictorial image of the concealed data, $U_{t}^{i}$, $i=1, 2, 3$, is in Fig. 11. The format of the pictorial image of Fig. 7 is PGM (i.e., portable gray map). In fact, we cannot restore the PGM header from the concealed data, that is, the header of the PGM is completely broken. Thus, the wiretapper has to realize that the concealed data are for PGM in some way, and he/she has to write the header by himself/herself to restore the pictorial image.

Fig. 11

From the left, pictorial images of the concealed data, $U_{t}^{1}, U_{t}^{2}$ in Fig. 8, and $U_{t}^{3}$ in Fig. 9, for the binary pulse $X_{t}$ in Fig. 7. Here $(\sigma _{v})^{2}=4$

As for the role of the common key $v^{i}(t)$, comparing Fig. 12 with Fig. 11, we can realize the effect of the variance of the common key $v^{i}(t)$ and the nonlinear function $f(\xi )$. The variance of the common key $v^{i}(t)$ is smaller in Fig. 12 than it is in Fig. 11, that is, $(\sigma _{v})^{2}=4$ for Fig. 11 and $(\sigma _{v})^{2}=1$ for Fig. 12, though other parameters for Fig. 12 are the same as for Fig. 11. The contour of the face in the pictorial image of $U_{t}^{1}$ in Fig. 12 stands out more clearly than in Fig. 11. Meanwhile, the nonlinearity conceals the contour as in the pictorial image of $U_{t}^{3}$ in Fig. 12.

Fig. 12

From the left, pictorial images of the concealed data, $U_{t}^{1}, U_{t}^{2}$ in Fig. 8, and $U_{t}^{3}$ in Fig. 9, for the binary pulse $X_{t}$ in Fig. 7. Here $(\sigma _{v})^{2}=1$

In Fig. 13 we show the comparison of the original binary pulse $X_{t}$, its restoration $\widehat{X}_{t}$, and the concealed data $U_{t}^{i}$, $i=1, 2, 3$.

Fig. 13

$X_{t}$ (Fig. 7) and $\widehat{X}_{t}$ (Fig. 10) from the above of the left 2 graphs. $U_{t}^{1}$ (Fig. 8), $U_{t}^{2}$ (Fig. 8), and $U_{t}^{3}$ (Fig. 9) from the above of the right 3 graphs. Here $t\in [0,200]$ only

4.2 Analog Data of Pictorial Image

We use analog data of a pictorial image in the Olivetti faces database (The Olivetti Faces Database 2020), where the data of pictorial images are transformed to analog data from the original ones in the ORL Database of Faces, an archive of AT&T Laboratories Cambridge (The ORL Database of Faces 2020). The data have the grayscale value of 256 gradations (8bit/pixel). Our parameters are $A=A^{i}=0.1$, $b=b^{i}=1$, $b_{u}=b_{u}^{i}=1$, $c=c^{i}=1$, $\sigma _{1}=\sigma _{1}^{i}=0.1$, and $\sigma _{2}=\sigma _{2}^{i}=1$ again. We also use the common key $v^{i}(t)$ in the same way as in Sect. 3.2 with $\sigma _{v}=2$. The original analog data $X_{t}$ and their pictorial image are in Fig. 14. Here, the upper bound of t is $64\times 64=4096$ and t runs over [0, 4096]. The concealed data, $U_{t}^{1}$ and $U_{t}^{2}$, defined by Eq. (7) are in Fig. 15, and the concealed data $U_{t}^{3}$ defined by Eq. (3) are in Fig. 16. We can restore the pictorial image with the restoration $\widehat{X}_{t}$ as in Fig. 17. If a wiretapper becomes aware of our method to make a pictorial image from analog data, then the wiretapper gets pictorial images from the concealed data $U_{t}^{i}$, $i=1, 2, 3$, as in Fig. 18.

Fig. 14

The original pictorial image (left) with the analog data, and the analog data $X_{t}$ only for $t\in [0,200]$ (right)

Fig. 15

The concealed data, $U_{t}^{1}$ (left) and $U_{t}^{2}$ (right), for the analog data $X_{t}$ in Fig. 14. Here, $t\in [0,200]$ only

Fig. 16

The concealed data $U_{t}^{3}$ for the analog data $X_{t}$, $t\in [0,200] \subset [0,4096]$, in Fig. 14

Fig. 17

The restoration $\widehat{X}_{t}$ (right) for the analog data $X_{t}$ in Fig. 14 only for $t\in [0,200]$, and the pictorial image (left) of $\widehat{X}_{t}$

Fig. 18

From the left, pictorial images of the concealed data, $U_{t}^{1}$ (Fig. 15), $U_{t}^{2}$ (Fig. 15), and $U_{t}^{3}$ (Fig. 16)

In Fig. 19 we show the comparison of the original binary pulse $X_{t}$, its restoration $\widehat{X}_{t}$, and the concealed data $U_{t}^{i}$, $i=1, 2, 3$.

Fig. 19

$X_{t}$ (Fig. 14) and $\widehat{X}_{t}$ (Fig. 17) from the above of the left 2 graphs. $U_{t}^{1}$ (Fig. 15), $U_{t}^{2}$ (Fig. 15), and $U_{t}^{3}$ (Fig. 16) from the above of the right 3 graphs. Here $t\in [0,200]$ only

5 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. Then, 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 them. We make briefly some comments about it at the tail end of this section.

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 Sect. 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 $U_{t}^{2}$ and $X_{t}^{2}=f^{-1}(U_{t}^{2})$ in Fig. 21, we can say that the enhancement of noise-disturbance appears with the black color. We will study the roles of several parameters including the nonlinearity. We here introduce the effect coming from the nonlinearity beforehand. The state space determined by Eq. (5) is constructed by the linear Gaussian model, and thus, we used the linear Kalman filtering theory in Sects. 3 and 4. We can make it more general: nonlinear, non-Gaussian state space. Then, we should employ another noise-filtering theory such as the particle filtering theory (Bain and Crisan 2009). In fact, putting a concrete nonlinearity $N_A$ or another nonlinearity $N_B$ in the functional $F_{i}$ of Eq. (1), we have concealed data $U_{t}^{A,i}$ or $U_{t}^{B,i}$, $i=1, 2, 3$, different from those in this paper. Then, the linear Kalman filtering theory is not useful any longer. For instance, we respectively conceal the data in Figs. 7 and 14 using such functionals with the nonlinearity $N_A$ or $N_B$. Then, we cannot estimate the data from the concealed ones by the linear Kalman filter to our satisfaction. See Figs. 22, 23, 24, and 25. The difference between the restorations in Figs. 22 and 23 or between those in Figs. 24 and 25 depends on the degree of nonlinearity. We show the restoring system using the particle filter in Ref. Fujii and Hirokawa (2020).

Fig. 20

From the left, the original pictorial image, the individual pictorial images of the concealed data $U_{t}^{1}$ and $U_{t}^{2}$, and the pictorial image of the restored data. The original pictorial image is a bitmap image, and the parameter t of the original data $X_{t}$ runs over $[0 , 90123\mathrm {byte}]$ 

Fig. 21

Comparison between the pictorial images of $U_{t}^{2}$ with nonlinearity (left) and $X_{t}^{2}=f^{-1}(U_{t}^{2})$ without nonlinearity (right)

Fig. 22

The left graph is restoration $\widehat{X}_{t}$, $0\le t\le 200$, from the concealed data, $U_{t}^{A,i}$, $i=1, 2, 3$, with the nonlinearity $N_A$ using the Kalman filtering. The right picture is the pictorial image restored from such a restoration $\widehat{X}_{t}$

Fig. 23

The left graph is restoration $\widehat{X}_{t}$, $0\le t\le 200$, from the concealed data, $U_{t}^{B,i}$, $i=1, 2, 3$, with the nonlinearity $N_B$ using the linear Kalman filtering. The right picture is the pictorial image restored from such a restoration $\widehat{X}_{t}$

Fig. 24

The left graph is restoration $\widehat{X}_{t}$, $0\le t\le 200$, from the concealed data, $U_{t}^{A,i}$, $i=1, 2, 3$, with the nonlinearity $N_A$ using the linear Kalman filtering. The right picture is the pictorial image restored from such a restoration $\widehat{X}_{t}$

Fig. 25

The left graph is restoration $\widehat{X}_{t}$, $0\le t\le 200$, from the concealed data, $U_{t}^{B,i}$, $i=1, 2, 3$, with the nonlinearity $N_B$ using the Kalman filtering. The right picture is the pictorial image restored from such a restoration $\widehat{X}_{t}$