Tamper-resistant controller using neural network and time-varying quantization

Abstract

In this paper, we consider a tamper-resistant control system aiming at protecting the knowledge of the controller from attackers. In this control system, the controller operates normally only for a limited number of time-varying specific states; otherwise, it outputs an incorrect value. We propose to realize the tamper-resistant controller by employing a neural network and time-varying quantization. Furthermore, we make it possible for only one trained neural network to be used for all quantization based on the local approximation linearity of the trained neural network. Without this approach, the neural network needs to be trained for every possible quantization, which leads to huge computation. We provide simulations to demonstrate the security and feasibility of the proposed method.

A Proof

1.1 A.1 Proof of the security condition of $V_t$

Let A be an arbitrary non-empty open set in $\chi$. To prove the security condition, we need to show that

$$\begin{aligned} P\left( A \cap \bigcup _{t=0}^{\infty } V_t = \emptyset \right) =0. \end{aligned}$$

(16)

If Eq. (16) is satisfied, then by defining A as $A=N_{\epsilon }(q_t)\backslash \{q_t\}$ for some $q_t\in V_t$ and $\epsilon >0$, where $=N_{\epsilon }(q_t)$ is the open ball centered at $q_t$ with radius $\epsilon$, it can be concluded that there are some $q_{t'}\in V_{t'}$ that satisfy $0< \Vert q_t-q_{t'}\Vert < \epsilon$ almost surely.

Proof

As $\bigcup _{t=0}^{\infty } V_t$ is a countable set, we can enumerate the elements of $\bigcup _{t=0}^{\infty } V_t$ as $x_1,\ x_2,\ \cdots$, where $x_i\ (i=1, 2, \ldots )$ can be regarded as independent samples from the uniform distribution in $\chi$. Because of the assumption of independent sampling, we have

$$\begin{aligned}&P\left( A \cap \bigcup _{t=0}^{\infty } V_t = \emptyset \right) \nonumber \\&\quad = P\left( x_1 \notin A, x_2 \not \in A, \ldots \, x_i \not \in A, \ldots \ \right) \nonumber \\&\quad = P\left( x_1 \notin A\right) \ P\left( x_2 \notin A\right) \ \ldots \ P\left( x_i \notin A\right) \ \ldots \nonumber \\&\quad = \prod _{i=1}^{\infty } (1 - P(x_i \in A)) \nonumber \\&\quad = \prod _{i=1}^{\infty } (1 - P(x \in A)) \end{aligned}$$

(17)

Since A is an non-empty open set, $P(x \in A) > 0$. Therefore,

$$\begin{aligned}&P\left( A \cap \bigcup _{t=0}^{\infty } V_t = \emptyset \right) = 0. \end{aligned}$$

(18)

This completes the proof. $\square$

1.2 A.2 Proof that $\{\cos (t) | t\in \mathbb {N} \}$ is dense in $[-1,1]$

To prove that $\{\cos (t) | t\in \mathbb {N} \}$ is dense in $[-1,1]$, it is sufficient to show that for any $\epsilon >0$ and any $\beta \in [0, 2 \pi )$, there exist $n,k\in \mathbb {N}$ that satisfy

$$\begin{aligned} | \beta +2\pi k-n|&<\epsilon \nonumber \\ \Leftrightarrow |kx-n|&<\epsilon , \end{aligned}$$

(19)

where $x=(\beta / k +2\pi )\in \mathbb {R}$.

Proof

From Dirichlet’s approximation theorem: Given any real number $\theta$ and any positive integer N, there exist integers h and k with $0<k\le N$ such that

$$\begin{aligned} | k\theta -h|&< \frac{1}{N}. \end{aligned}$$

(20)

Therefore, for an $N>1 / \epsilon$, there exist natural numbers n and k with $0<k\le N$ such that

$$\begin{aligned} | kx-n|&< \frac{1}{N}<\epsilon , \end{aligned}$$

(21)

which completes the proof. $\square$