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# Towards Provably Secure Encrypted Control Using
Homomorphic Encryption∗

Kaoru Teranishi1,2 and Kiminao Kogiso1 ∗This work was supported by JSPS Grant-
in-Aid for JSPS Fellows Grant Number JP21J22442 and for Scientific Research
(B) Grant Number JP22H01509.1Department of Mechanical and Intelligent Systems
Engineering, The University of Electro-Communications, 1-5-1 Chofugaoka,
Chofu, Tokyo 1828585, Japan {teranishi<EMAIL_ADDRESS>Fellow of
Japan Society for the Promotion of Science

###### Abstract

Encrypted control is a promising method for the secure outsourcing of
controller computation to a public cloud. However, a feasible method for
security proofs of control has not yet been developed in the field of
encrypted control systems. Additionally, cryptography does not consider
certain types of attacks on encrypted control systems; therefore, the security
of such a system cannot be guaranteed using a secure cryptosystem. This study
proposes a novel security definition for encrypted control under attack for
control systems using cryptography. It applies the concept of provable
security, which is the security of cryptosystems based on mathematical proofs,
to encrypted control systems. Furthermore, this study analyzes the relation
between the proposed security and the conventional security of cryptosystems.
The results of the analysis demonstrated that the security of an encrypted
control system can be enhanced by employing secure homomorphic encryption.

##  I Introduction

The encrypted control method is based on the homomorphism of cryptosystems
that enables implementation of controller computation in an encrypted manner
[1, 2, 3, 4]. The controller parameters and inputs/outputs of an encrypted
controller are encrypted against attackers originating outside the closed-loop
system. This control method can be implemented for the secure outsourcing
computation of controllers to a public cloud because the control inputs are
computed without using a decryption key.

Although encrypted control is a promising method for secure control, research
on the security of encrypted control systems remains limited. In most of the
previous studies, an encrypted control system was assumed to be secure if the
underlying cryptosystem was secure. However, certain attacks against encrypted
control systems are not considered in cryptography. Fig. 1(a) depicts a
typical situation in standard signal encryption. Alice encrypts a message and
transmits the encrypted message to Bob, who decrypts it. Eve, an attacker,
intercepts the encrypted data and tries to recover it. Eve eavesdrops on the
input and output of the encrypted controller while attacking the encrypted
control system, as shown in Fig. 1(b). This is because control systems are
typically closed-loop systems. In this case, Eve’s objective is not necessary
to obtain the original controller input and output. For example, Eve may
attempt to recover an encrypted controller parameter from encrypted messages.
This objective may be achievable even if Eve fails to decrypt the encrypted
input and output. Although this type of attack on encrypted control systems is
fairly common, it is not considered in standard cryptosystems, including the
ElGamal encryption [5] and Paillier encryption [6], which are widely used for
encrypted control. Therefore, using a secure cryptosystem does not necessarily
mean that an encrypted control system is secure.

(a) Eavesdropping in encrypted network.

(b) Eavesdropping in encrypted control.

Figure 1: Eavesdropping in encrypted network and encrypted control.

Furthermore, the definition of security for encrypted control systems is
ambiguous, and the types of attacks against the systems are not clearly
defined. Meanwhile, the security of cryptosystems has been mathematically
defined and analyzed in the field of cryptography [7]. One of the security
approaches, called provable security, is formulated based on the objectives
and capabilities of the attackers. Additionally, it is proved under some
assumptions of computational complexity. More precisely, the security is
usually demonstrated using proof by contradiction, namely if there exists an
algorithm to break a cryptosystem, then a practically difficult computational
problem can be efficiently solved by using the algorithm. This proof
construction is called the reductionist approach and is fundamental in proving
the security of modern cryptosystems. The security of encrypted control
systems can be clarified and formally analyzed by applying a similar security
definition and proof construction. Consequently, the standard security
definition of cryptosystems must be expanded to encompass the specific attacks
for encrypted control systems because the attack scenarios considered in
cryptography and control systems vary significantly. Some studies have already
applied cryptographic security to encrypted control protocols. The privacy of
cloud-based optimization, model predictive control, and state estimation is
achieved through the computational indistinguishability of two random
ensembles [8, 9, 10], and that of distributed and cooperative controls is
achieved based on a cryptographic game [11, 12]. However, the guaranteed
privacy of the aforementioned studies is based on the input/output privacy of
the systems rather than a controller parameter and is provided for specific
protocols.

This study proposes a novel security definition of encrypted control systems
under a parameter estimation attack via a game based on a cryptographic
security notion. In a parameter estimation attack, an attacker attempts to
disclose the original controller parameter of an encrypted controller from
multiple data of encrypted input and output. The proposed security is defined
as an attacker obtaining no information about a controller parameter by the
attack. Additionally, we analyze the relation of the proposed security to the
standard security of cryptosystems. Contrary to the existing studies [8, 9,
10, 11, 12], this study aims to expand the application of security notion in
cryptography to encrypted control systems. Moreover, it contributes to the
establishment of a methodology of security proofs for general encrypted
control. The proposed security definition allows for the analysis of the
security of a broader encrypted control system under a parameter estimation
attack. The analysis of this study clarifies the strength of the proposed
security. It is shown that the proposed security is weaker than or equal to
the standard security of cryptosystems. This implies that a secure encrypted
control system under the attack can be realized using a secure cryptosystem
that satisfies the standard cryptographic security. In addition, this study
discusses a condition in which the proposed security is as secure as the
standard cryptosystem security. The condition suggests that the proposed
security for most controls may be achievable by a simple and lightweight
scheme rather than a cryptosystem that satisfies the standard security.

The remainder of this paper is organized as follows. Section II introduces
homomorphic encryption and encrypted control employed in this study. In
addition, the concept of provable security in cryptography is provided.
Section III proposes a novel security definition for encrypted control and
presents the analysis results. Section IV presents the scope for further
development of the proposed security definition. Section V presents the
conclusions and future work.

##  II Preliminaries

###  II-A Notation

The set of real numbers is denoted by $\mathbb{R}$. The sets of
$n$-dimensional column-vectors and $m$-by-$n$ matrices of which elements
belong to a set $X$ are denoted by $X^{n}$ and $X^{m\times n}$, respectively.
An algorithm $\mathcal{A}$ is polynomial-time algorithm if there exists a
polynomial $p$ such that, for every $k$-bit length input $x$, $\mathcal{A}(x)$
terminates within at most $p(k)$ steps [7]. We say a function
$\epsilon:\\{1,2,3,\dots\\}\to\mathbb{R}$ is negligible if, for every positive
integer $c>0$, there exists an integer $N$ such that $|\epsilon(n)|<n^{-c}$
holds for all $n>N$ [7].

###  II-B Homomorphic encryption

This section summarizes the basics of homomorphic encryption. Homomorphic
encryption is a cryptosystem that enables the application of arithmetic over a
ciphertext space. The definition of a cryptosystem is as follows.

######  Definition 1 (Cryptosystem [13])

A cryptosystem is defined as a tuple
$\Pi=(\mathsf{KeyGen},\mathsf{Enc},\mathsf{Dec})$, which satisfies the
following properties:

  * •

$\mathsf{KeyGen}:\mathcal{S}\to\mathcal{K}$ is a key-generation function that
determines keys $(\mathsf{pk},\mathsf{sk})\in\mathcal{K}$ from a security
parameter $\lambda\in\mathcal{S}$, where $\mathcal{K}$ is a key space, and
$\mathcal{S}$ is a set of security parameters.

  * •

$\mathsf{Enc}_{\mathsf{pk}}:\mathcal{M}\to\mathcal{C}$ is an encryption
function that outputs a ciphertext $c\in\mathcal{C}$ from a plaintext
$m\in\mathcal{M}$, where $\mathcal{M}$ is a plaintext space, and $\mathcal{C}$
is a ciphertext space.

  * •

$\mathsf{Dec}_{\mathsf{sk}}:\mathcal{C}\to\mathcal{M}$ is a decryption
function that outputs a plaintext $m\in\mathcal{M}$ from a ciphertext
$c\in\mathcal{C}$.

  * •

For any $\mathsf{pk}$, there exists $\mathsf{sk}$ such that
$\mathsf{Dec}_{\mathsf{sk}}(\mathsf{Enc}_{\mathsf{pk}}(m))=m$ for all
$m\in\mathcal{M}$.

Further, we often say $\Pi$ is symmetric-key encryption if
$\mathsf{pk}=\mathsf{sk}$; otherwise it is public-key encryption.

Homomorphic encryption is defined as follows.

######  Definition 2 (Homomorphic encryption)

A cryptosystem $\Pi$ in Definition 1 is called homomorphic encryption if there
exist binary operations $\bullet:\mathcal{M}\times\mathcal{M}\to\mathcal{M}$
and $\circ:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ such that
$\mathsf{Dec}_{\mathsf{sk}}(\mathsf{Enc}_{\mathsf{pk}}(m)\circ\mathsf{Enc}_{\mathsf{pk}}(m^{\prime}))=m\bullet
m^{\prime}$ for all $m,m^{\prime}\in\mathcal{M}$.

Homomorphic encryption is classified based on the allowed arithmetic, into
additive, multiplicative, somewhat, and fully homomorphic encryption. The
additive and multiplicative homomorphic encryption satisfy
$\mathsf{Dec}_{\mathsf{sk}}(\mathsf{Enc}_{\mathsf{pk}}(m)\oplus\mathsf{Enc}_{\mathsf{pk}}(m^{\prime}))=m+m^{\prime}$
and
$\mathsf{Dec}_{\mathsf{sk}}(\mathsf{Enc}_{\mathsf{pk}}(m)\otimes\mathsf{Enc}_{\mathsf{pk}}(m^{\prime}))=mm^{\prime}$,
respectively. Somewhat homomorphic encryption can perform the arithmetic
$\oplus$ and $\otimes$ only a limited number of times. Fully homomorphic
encryption allows the computation of any number of arbitrary arithmetic.

###  II-C Encrypted control

Various encrypted control frameworks have been proposed in conventional works
according to the controller type, such as static output feedback [2],
cooperative control [14, 12], and model predictive control [15, 16]. However,
this study does not assume a specific type of controller to be encrypted. A
controller is considered as a map from a plant output to the control input to
analyze the cryptographic security for as broad a class of encrypted
controllers as possible.

######  Definition 3 (Encrypted control)

Given a control law

$\bm{u}=f(\bm{y};\bm{K}),$ (1)

where $\bm{u}\in\mathcal{M}^{q}$ is a control input,
$\bm{y}\in\mathcal{M}^{\ell}$ is a plant output, and
$\bm{K}\in\mathcal{M}^{r}$ is a controller parameter. Let $\Pi$ be homomorphic
encryption in Definition 2. We say a map $f_{\Pi}$ is an encrypted control law
of $f$ with $\Pi$ if

$\mathsf{Dec}_{\mathsf{sk}}(f_{\Pi}(\mathsf{Enc}_{\mathsf{pk}}(\bm{y});\mathsf{Enc}_{\mathsf{pk}}(\bm{K}))=f(\bm{y};\bm{K})$

holds, where for a plaintext vector $\bm{m}=[m_{1}\ \cdots\
m_{k}]^{\top}\in\mathcal{M}^{k}$ and ciphertext vector $\bm{c}=[c_{1}\ \cdots\
c_{n}]^{\top}\in\mathcal{C}^{n}$ the encryption and decryption functions
perform each element of the vectors, namely
$\mathsf{Enc}_{\mathsf{pk}}(\bm{m})=[\mathsf{Enc}_{\mathsf{pk}}(m_{1})\
\cdots\ \mathsf{Enc}_{\mathsf{pk}}(m_{k})]^{\top}$ and
$\mathsf{Dec}_{\mathsf{sk}}(\bm{c})=[\mathsf{Dec}_{\mathsf{sk}}(c_{1})\
\cdots\ \mathsf{Dec}_{\mathsf{sk}}(c_{n})]^{\top}$.

###### Remark 1

In practice, $\bm{y}$ and $\bm{K}$ are given by real-valued vectors, and
$\mathcal{M}$ is not necessarily the same as $\mathbb{R}$. Therefore, the
elements of $\bm{y}$ and $\bm{K}$ must be encoded into plaintext before
encryption [2, 17, 18]. This encoding process is omitted in this study because
it is focused on the cryptographic properties of the encrypted controls.

###  II-D Provable Security

The security of modern cryptosystems is demonstrated through mathematical
proofs using the model of an attacker, and the security proved by such formal
procedures is called provable security. A popular method to define the
security is a game between an attacker and a challenger. The game simulates a
situation in which an attacker attempts to break a cryptosystem using an
oracle, which is a black box that outputs an ideal response for an input. The
oracle represents the capabilities of the attacker. It should be noted that
specific attack methodologies for cryptosystems, such as brute-force and side-
channel attacks, are not considered in provable security. In the framework of
provable security, attackers are considered as probabilistic polynomial-time
algorithms.

This section introduces indistinguishability under chosen plaintext attack
(IND-CPA), which is a traditional security concept for public-key encryption,
as an example of provable security.

######  Definition 4 (IND-CPA [7])

Let $\Pi\\!=\\!(\mathsf{KeyGen},\mathsf{Enc},\mathsf{Dec})$ be public-key
encryption. Define a game between an attacker
$\mathcal{A}=(\mathcal{A}_{1},\mathcal{A}_{2})$ and a challenger as follows.

$\mathsf{Game}^{\mathsf{IND-CPA}}_{\Pi,\mathcal{A}}$: 1.
$(\mathsf{pk},\mathsf{sk})\leftarrow\mathsf{KeyGen}(\lambda)$ 2.
$(m_{0},m_{1},\sigma)\leftarrow\mathcal{A}_{1}(\mathsf{pk})$ 3.
$b\overset{\$}{\leftarrow}\\{0,1\\}$ 4.
$c\leftarrow\mathsf{Enc}_{\mathsf{pk}}(m_{b})$ 5.
$\hat{b}\leftarrow\mathcal{A}_{2}(c,\sigma)$

  * •

Setup: The challenger computes keys $\mathsf{pk}$ and $\mathsf{sk}$ from a
security parameter $\lambda$ using the key-generation function
$\mathsf{KeyGen}$ and gives $\mathsf{pk}$ to the attacker (Line $1$).

  * •

Challenge: The attacker computes two different plaintexts
$m_{0},m_{1}\in\mathcal{M}$, which are of identical size, and an intermediate
state $\sigma$ using the polynomial-time algorithm $\mathcal{A}_{1}$ (Line
$2$). The attacker gives $m_{0}$ and $m_{1}$ to the challenger. The challenger
randomly selects a bit $b\in\\{0,1\\}$ and computes a ciphertext $c$ of
$m_{b}$ using the encryption function $\mathsf{Enc}$ with $\mathsf{pk}$ (Lines
$3$–$4$). The challenger gives $c$ to the attacker.

  * •

Guess: The attacker guesses the plaintext that has been encrypted from the
given ciphertext $c$ using the polynomial-time algorithm $\mathcal{A}_{2}$ and
outputs a bit $\hat{b}\in\\{0,1\\}$ (Line $5$). If $\hat{b}=b$, the attacker
wins the game; otherwise, the challenger wins.

The game is called the IND-CPA game.

Define the advantage of the attacker in the IND-CPA game as
$\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}\coloneqq\left|\Pr\left[\hat{b}=b\mid\mathsf{Game}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}\right]-1/2\right|$. We say $\Pi$ is IND-CPA secure, or
$\Pi$ satisfies the IND-CPA security if $\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}$ is negligible, that is, there exists a negligible
function $\epsilon$ such that $\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}\leq\epsilon$.

###### Remark 2

Public-key encryption must at least satisfy the IND-CPA security to be used
for secure communication. If the advantage of the attacker is not negligible,
he/she can guess the encrypted message with a significant probability.

##  III Provable Security for Encrypted Control

This section proposes a novel security notion of encrypted control based on
the traditional provable-security notion in cryptography and reveals the
relationship between the notions.

###  III-A Indistinguishability against parameter estimation attack

This study considers an attack scenario in which an attacker attempts to
estimate the controller parameters from the ciphertexts of the controller
inputs and outputs. The indistinguishability for encrypted control against the
attack can be defined as follows.

######  Definition 5 (IND-PEA)

Let $\Pi=(\mathsf{KeyGen},\mathsf{Enc},\mathsf{Dec})$ be homomorphic
encryption, and let $f_{\Pi}$ be an encrypted control law of (1) with $\Pi$.
Define a game between an attacker
$\mathcal{B}=(\mathcal{B}_{1},\mathcal{B}_{2})$ and a challenger as follows.

$\mathsf{Game}^{\mathsf{IND-PEA}}_{\Pi,f,\mathcal{B}}$: 1.
$(\mathsf{pk},\mathsf{sk})\leftarrow\mathsf{KeyGen}(\lambda)$ 2.
$(\bm{K}_{0},\bm{K}_{1},\sigma)\leftarrow\mathcal{B}_{1}(\mathsf{pk})$ 3.
$b\overset{\$}{\leftarrow}\\{0,1\\}$ 4.
$\bm{c_{K}}\leftarrow\mathsf{Enc}_{\mathsf{pk}}(\bm{K}_{b})$ 5.
$\hat{b}\leftarrow\mathcal{B}_{2}^{\mathcal{O}}(\sigma)$

  * •

Setup: The challenger computes keys $\mathsf{pk}$ and $\mathsf{sk}$ from a
security parameter $\lambda$ using the key-generation function
$\mathsf{KeyGen}$ and gives $\mathsf{pk}$ to the attacker (Line $1$).

  * •

Challenge: The attacker computes two different parameters
$\bm{K}_{0},\bm{K}_{1}\in\mathcal{M}^{r}$, which are of identical size, and an
intermediate state $\sigma$ using the polynomial-time algorithm
$\mathcal{B}_{1}$ (Line $2$). The attacker gives $\bm{K}_{0}$ and $\bm{K}_{1}$
to the challenger. The challenger randomly selects a bit $b\in\\{0,1\\}$ and
computes a ciphertext $\bm{c_{K}}$ of $\bm{K}_{b}$ using the encryption
function $\mathsf{Enc}$ with $\mathsf{pk}$ (Lines $3$–$4$). The challenger
sets $\bm{c_{K}}$ to the encrypted controller.

  * •

Guess: The attacker guesses the selected parameter using the polynomial-time
algorithm $\mathcal{B}_{2}$ and by querying an encrypted control oracle
$\mathcal{O}$ polynomial times of $\lambda$ and outputs a bit
$\hat{b}\in\\{0,1\\}$ (Line $5$). The oracle receives a query
$\bm{c_{y}}\in\mathcal{C}^{\ell}$ and returns an output
$\bm{c_{u}}=f_{\Pi}(\bm{c_{y}};\bm{c_{K}})$. If $\hat{b}=b$, the attacker wins
the game; otherwise, the challenger wins.

The game is called the indistinguishability under parameter estimation attack
(IND-PEA) game.

Define the advantage of the attacker in the IND-PEA game as
$\mathsf{Adv}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}\coloneqq\left|\Pr\left[\hat{b}=b\mid\mathsf{Game}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}\right]-1/2\right|$. We say $f_{\Pi}$ is IND-PEA
secure, or $f_{\Pi}$ satisfies the IND-PEA security if
$\mathsf{Adv}^{\mathsf{IND-PEA}}_{\Pi,f,\mathcal{B}}$ is negligible.

Fig. 2 presents the schematic diagrams of the IND-CPA and IND-PEA games. In
the IND-CPA game of Fig. 2(a), an attacker guesses whether the plaintext,
$m_{0}$ or $m_{1}$, has been encrypted from a given ciphertext $c$ of $m_{b}$.
Note that an attacker in this game can obtain the ciphertext of the target
plaintext. Conversely, an attacker in the IND-PEA game of Fig. 2(b) cannot
obtain a ciphertext of the target parameter, $\bm{K}_{b}$. An attacker can use
the encrypted control oracle $\mathcal{O}$ with the encrypted parameter
$\bm{c_{K}}=\mathsf{Enc}_{\mathsf{pk}}(\bm{K}_{b})$ to collect the encrypted
controller outputs $\bm{c_{y}}$ for any input $\bm{c_{u}}$. Therefore, the
IND-PEA game simulates different situations within the IND-CPA game.

It should be noted that the oracle to be used by an attacker indicates that
encrypted control is provided as a cloud service. Most studies conducted on
encrypted control consider cloud-based control systems [19, 15, 16, 8, 20,
21], and in such cases, an attacker can freely use an encrypted control
algorithm. Therefore, the use of the oracle in the IND-PEA game is reasonable
when considering the security for the encrypted control.

(a) IND-CPA game.

(b) IND-PEA game.

Figure 2: Schematic pictures of IND-CPA and IND-PEA games.

###  III-B Relationship between the security notions

The security of encrypted control can be demonstrated using mathematical
proofs based on the definition of IND-PEA security. Furthermore, it gives the
other benefit of discussing how much security strength is required to achieve
secure encrypted control. The following theorems disclose the relation between
the IND-PEA security and IND-CPA security and clarify the security strength of
the IND-PEA security.

###### Theorem 1

Let $\Pi$ be homomorphic encryption and let $f_{\Pi}$ be an encrypted control
law of (1) with $\Pi$. If $\Pi$ is IND-CPA secure, $f_{\Pi}$ is IND-PEA
secure.

###### Proof:

It is proved that if $f_{\Pi}$ is not IND-PEA secure, $\Pi$ is not IND-CPA
secure. Define polynomial-time algorithms $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ as follows.

$\mathcal{A}_{1}(\mathsf{pk})$: 1.
$(m_{0},m_{1},\sigma)\leftarrow\mathcal{B}_{1}(\mathsf{pk})$ 2. return
$(m_{0},m_{1},\sigma)$

$\mathcal{A}_{2}(c,\sigma)$: 1. return $\mathcal{B}_{2}^{\mathcal{O}}(\sigma)$

Substituting the algorithms into $\mathsf{Game}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}$ in Definition 4, we obtain the following game.

$\mathsf{Game}_{\Pi,f,\mathcal{B}}$: 1.
$(\mathsf{pk},\mathsf{sk})\leftarrow\mathsf{KeyGen}(\lambda)$ 2.
$(m_{0},m_{1},\sigma)\leftarrow\mathcal{B}_{1}(\mathsf{pk})$ 3.
$b\overset{\$}{\leftarrow}\\{0,1\\}$ 4.
$c\leftarrow\mathsf{Enc}_{\mathsf{pk}}(m_{b})$ 5.
$\hat{b}\leftarrow\mathcal{B}_{2}^{\mathcal{O}}(\sigma)$

  * •

Setup: This phase is identical to the setup phase in Definition 4.

  * •

Challenge: The attacker computes two different parameters
$m_{0},m_{1}\in\mathcal{M}$, which are of identical size, and an intermediate
state $\sigma$ using the polynomial-time algorithm $\mathcal{B}_{1}$ (Line
$2$). The attacker gives $m_{0}$ and $m_{1}$ to the challenger. The challenger
randomly selects a bit $b\in\\{0,1\\}$ and computes a ciphertext $c$ of
$m_{b}$ using the encryption function $\mathsf{Enc}$ with $\mathsf{pk}$ (Lines
$3$–$4$). The challenger sets $c$ to the encrypted controller.

  * •

Guess: This phase is identical to the guess phase in Definition 5.

Define the advantage of the attacker in the game
$\mathsf{Game}_{\Pi,f,\mathcal{B}}$ as
$\mathsf{Adv}_{\Pi,f,\mathcal{B}}\coloneqq\left|\Pr\left[\hat{b}=b\mid\mathsf{Game}_{\Pi,f,\mathcal{B}}\right]-1/2\right|$.
Here, we consider the following lemmas to prove the theorem.

###### Lemma 1

$\left|\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}-\mathsf{Adv}_{\Pi,f,\mathcal{B}}\right|$ is
negligible.

###### Proof:

The substitution to obtain the game $\mathsf{Game}_{\Pi,f,\mathcal{B}}$ does
not change the probability of the game between the challenger and the
attacker. ∎

###### Lemma 2

$\left|\mathsf{Adv}_{\Pi,f,\mathcal{B}}-\mathsf{Adv}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}\right|$ is negligible.

###### Proof:

The game $\mathsf{Game}_{\Pi,f,\mathcal{B}}$ corresponds to the IND-PEA game
$\mathsf{Game}^{\mathsf{IND-PEA}}_{\Pi,f,\mathcal{B}}$ with $r=1$, which is
the dimension of the controller parameter. The advantage of the attacker does
not change between these games. ∎

From Lemma 1 and Lemma 2, it follows that the difference of advantages,
$\left|\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}-\mathsf{Adv}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}\right|$, is negligible. Furthermore,
$\mathsf{Adv}^{\mathsf{IND-PEA}}_{\Pi,f,\mathcal{B}}$ is not negligible
because $f_{\Pi}$ is not IND-PEA secure, based on the assumption of this
proof. Therefore, $\mathsf{Adv}^{\mathsf{IND-CPA}}_{\Pi,\mathcal{A}}$ is not
negligible, and so $\Pi$ is not IND-CPA secure. ∎

The theorem demonstrates that the IND-PEA security is weaker than or equal to
the IND-CPA security. This implies that secure encrypted control against the
parameter estimation attack is achievable by using secure homomorphic
encryption. The following theorem presents one of the conditions so that the
IND-PEA security is equivalent to the IND-CPA security.

###### Theorem 2

Let $\Pi$ be homomorphic encryption and let $f_{\Pi}$ be an encrypted control
law of (1) with $\Pi$. Suppose $f$ in (1) is bijective for a fixed
$\bm{y}\in\mathcal{M}^{\ell}$. If $f_{\Pi}$ is IND-PEA secure and if the
attacker can compute an encrypted control law of $f^{-1}$ with $\Pi$, then
$\Pi$ is IND-CPA secure.

###### Proof:

Under the assumptions, we prove that if $\Pi$ is not IND-CPA secure, $f_{\Pi}$
is not IND-PEA secure. Let $f^{-1}_{\Pi}$ be an encrypted control law of
$f^{-1}$ with $\Pi$. Since $f$ in (1) is bijective for some fixed
$\bm{y}\in\mathcal{M}^{\ell}$, it follows that

$f^{-1}(\bm{y};f(\bm{y};\bm{K}))=\bm{K}$ (2)

for all $\bm{K}\in\mathcal{M}^{r}$. Moreover,

$f^{-1}_{\Pi}(\bm{c_{y}};f_{\Pi}(\bm{c_{y}};\bm{c_{K}}))=\bm{c_{K}}$ (3)

holds for $\bm{c_{y}}=\mathsf{Enc}_{\mathsf{pk}}(\bm{y})$ and
$\bm{c_{K}}=\mathsf{Enc}_{\mathsf{pk}}(\bm{K})$.

Define polynomial-time algorithms $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ as
follows.

$\mathcal{B}_{1}(\mathsf{pk})$: 1.
$(K_{0},K_{1},\sigma)\leftarrow\mathcal{A}_{1}(\mathsf{pk})$ 2.
$\bm{K}_{0}\leftarrow[K_{0}\ \cdots\ K_{0}]^{\top}$,
$\bm{K}_{1}\leftarrow[K_{1}\ \cdots\ K_{1}]^{\top}$ 3. Fix
$\bm{y}\in\mathcal{M}^{\ell}$ to satisfy (2) 4.
$\sigma\leftarrow(\sigma,\bm{y})$ 5. return $(\bm{K}_{0},\bm{K}_{1},\sigma)$

$\mathcal{B}_{2}^{\mathcal{O}}(\sigma)$: 1. $(\sigma,\bm{y})\leftarrow\sigma$
2. $\bm{c_{y}}\leftarrow\mathsf{Enc}_{\mathsf{pk}}(\bm{y})$ 3.
$\bm{c_{u}}\leftarrow\mathcal{O}(\bm{c_{y}})$ 4.
$\hat{\bm{c}}_{\bm{K}}\leftarrow f^{-1}_{\Pi}(\bm{c_{y}};\bm{c_{u}})$,
$c_{K}\leftarrow\hat{\bm{c}}_{\bm{K},1}$ 5. return
$\mathcal{A}_{2}(c_{K},\sigma)$

Substituting the algorithms into $\mathsf{Game}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}$ in Definition 5, we obtain the following game.

$\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}$: 1.
$(\mathsf{pk},\mathsf{sk})\leftarrow\mathsf{KeyGen}(\lambda)$ 2.
$(K_{0},K_{1},\sigma)\leftarrow\mathcal{A}_{1}(\mathsf{pk})$ 3.
$\bm{K}_{0}\leftarrow[K_{0}\ \cdots\ K_{0}]^{\top}$,
$\bm{K}_{1}\leftarrow[K_{1}\ \cdots\ K_{1}]^{\top}$ 4. Fix
$\bm{y}\in\mathcal{M}^{\ell}$ to satisfy (2) 5.
$\sigma\leftarrow(\sigma,\bm{y})$ 6. $b\overset{\$}{\leftarrow}\\{0,1\\}$ 7.
$\bm{c_{K}}\leftarrow\mathsf{Enc}_{\mathsf{pk}}(\bm{K}_{b})$ 8.
$(\sigma,\bm{y})\leftarrow\sigma$ 9.
$\bm{c_{y}}\leftarrow\mathsf{Enc}_{\mathsf{pk}}(\bm{y})$ 10.
$\bm{c_{u}}\leftarrow\mathcal{O}(\bm{c_{y}})$ 11.
$\hat{\bm{c}}_{\bm{K}}\leftarrow f^{-1}_{\Pi}(\bm{c_{y}};\bm{c_{u}})$,
$c_{K}\leftarrow\hat{\bm{c}}_{\bm{K},1}$ 12.
$\hat{b}\leftarrow\mathcal{A}_{2}(c_{K},\sigma)$

  * •

Setup: This phase is identical to the setup phase in Definition 5.

  * •

Challenge: The attacker computes two different plaintexts
$K_{0},K_{1}\in\mathcal{M}$, which are of identical size, and an intermediate
state $\sigma$ using the polynomial-time algorithm $\mathcal{A}_{1}$ (Line
$2$). Subsequently, the attacker constructs vectors
$\bm{K}_{0},\bm{K}_{1}\in\mathcal{M}^{r}$ from $K_{0},K_{1}$ and gives them to
the challenger (Line $3$). Additionally, the attacker fixes
$\bm{y}\in\mathcal{M}^{\ell}$ to satisfy (2) and updates $\sigma$ to
$(\sigma,\bm{y})$ (Lines $4$–$5$). The challenger randomly selects a bit
$b\in\\{0,1\\}$ and computes a ciphertext $\bm{c_{K}}$ of $\bm{K}_{b}$ using
the encryption function $\mathsf{Enc}$ with $\mathsf{pk}$ (Lines $6$–$7$). The
challenger sets $\bm{c_{K}}$ to the encrypted controller.

  * •

Guess: The attacker takes out the intermediate state and $\bm{y}$ from
$\sigma$ and computes the ciphertext $\bm{c_{y}}$ of $\bm{y}$ (Lines $8$–$9$).
Subsequently, the attacker queries the encrypted control oracle $\mathcal{O}$
to obtain a controller output $\bm{c_{u}}$ and computes a ciphertext
$\hat{\bm{c}}_{K}$ by using the encrypted controller of $f^{-1}$ with $\Pi$
(Line $10$–$11$). The attacker guesses the parameter that has been encrypted
from the first element $c_{K}$ of the ciphertext vector $\hat{\bm{c}}_{K}$
using the polynomial-time algorithm $\mathcal{A}_{2}$ and outputs a bit
$\hat{b}\in\\{0,1\\}$ (Line $11$–$12$). If $\hat{b}=b$, the attacker wins the
game; otherwise, the challenger wins.

Define the advantage of the attacker in the game
$\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}$ as
$\mathsf{Adv}^{1}_{\Pi,f,\mathcal{A}}\coloneqq\left|\Pr\left[\hat{b}=b\mid\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}\right]-1/2\right|$.
Furthermore, we modify lines $3$–$5$ and $7$–$11$ of
$\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}$ to obtain the following game.

$\mathsf{Game}^{2}_{\Pi,\mathcal{A}}$: 1.
$(\mathsf{pk},\mathsf{sk})\leftarrow\mathsf{KeyGen}(\lambda)$ 2.
$(K_{0},K_{1},\sigma)\leftarrow\mathcal{A}_{1}(\mathsf{pk})$ 3.
$b\overset{\$}{\leftarrow}\\{0,1\\}$ 4.
$c_{K}\leftarrow\mathsf{Enc}_{\mathsf{pk}}(K_{b})$ 5.
$\hat{b}\leftarrow\mathcal{A}_{2}(c_{K},\sigma)$

  * •

Setup: This phase is identical to the setup phase in Definition 5.

  * •

Challenge: The attacker computes two different plaintexts
$K_{0},K_{1}\in\mathcal{M}$, which are of identical size, and an intermediate
state $\sigma$ using the polynomial-time algorithm $\mathcal{A}_{1}$ (Line
$2$). The attacker gives $K_{0}$ and $K_{1}$ to the challenger. The challenger
randomly selects a bit $b\in\\{0,1\\}$ and computes a ciphertext $c_{K}$ of
$K_{b}$ using the encryption function $\mathsf{Enc}$ with $\mathsf{pk}$ (Lines
$3$–$4$). The challenger gives $c_{K}$ to the attacker.

  * •

Guess: This phase is identical to the guess phase in Definition 4.

Define the advantage of the attacker in the game
$\mathsf{Game}^{2}_{\Pi,\mathcal{A}}$ as
$\mathsf{Adv}^{2}_{\Pi,\mathcal{A}}\coloneqq\left|\Pr\left[\hat{b}=b\mid\mathsf{Game}^{2}_{\Pi,\mathcal{A}}\right]-1/2\right|$.
Here, we consider the following lemmas to prove the theorem.

###### Lemma 3

$\left|\mathsf{Adv}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}-\mathsf{Adv}^{1}_{\Pi,f,A}\right|$ is negligible.

###### Proof:

The substitution for obtaining $\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}$ does
not change the probability of the game between the challenger and the
attacker. ∎

###### Lemma 4

$\left|\mathsf{Adv}^{1}_{\Pi,f,A}-\mathsf{Adv}^{2}_{\Pi,A}\right|$ is
negligible.

###### Proof:

It follows from (3) that
$\hat{\bm{c}}_{\bm{K}}=\bm{c_{K}}=\mathsf{Enc}_{\mathsf{pk}}(\bm{K}_{b})=[\mathsf{Enc}_{\mathsf{pk}}(K_{b})\
\cdots\ \mathsf{Enc}_{\mathsf{pk}}(K_{b})]^{\top}$. Therefore, we can rewrite
line $11$ in $\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}$ as line $4$ in
$\mathsf{Game}^{2}_{\Pi,\mathcal{A}}$ without changing the probability of the
games. The lines $3$–$5$ and $7$–$10$ of
$\mathsf{Game}^{1}_{\Pi,f,\mathcal{A}}$ do not affect the advantage of the
attacker in the game, and so we can remove the lines. ∎

###### Lemma 5

$\left|\mathsf{Adv}^{2}_{\Pi,A}-\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}\right|$ is negligible.

###### Proof:

By definition, the game $\mathsf{Game}^{2}_{\Pi,A}$ is the same as the game
$\mathsf{Game}^{\mathsf{IND-CPA}}_{\Pi,A}$. ∎

It follows from Lemma 3 to Lemma 5 that the difference of advantages,
$\left|\mathsf{Adv}^{\mathsf{IND-
PEA}}_{\Pi,f,\mathcal{B}}-\mathsf{Adv}^{\mathsf{IND-
CPA}}_{\Pi,\mathcal{A}}\right|$, is negligible. Furthermore,
$\mathsf{Adv}^{\mathsf{IND-CPA}}_{\Pi,\mathcal{A}}$ is not negligible since
$\Pi$ is not IND-CPA secure based on the assumption of this proof. Therefore,
$\mathsf{Adv}^{\mathsf{IND-PEA}}_{\Pi,f,\mathcal{B}}$ is not negligible, and
so $f_{\Pi}$ is not IND-PEA secure. ∎

Note that few controllers can satisfy the assumption in Theorem 2 in practice.
For example, in a static output feedback controller $\bm{u}=\bm{F}\bm{y}$,
where $\bm{F}\in\mathcal{M}^{q\times\ell}$ is a feedback gain, the existence
of inverse mapping $f^{-1}$ in Theorem 2 is equivalent to the condition that
$q=r$, namely $\ell=1$. Such a controller is a particular case of this
control, and hence, it seems that the IND-PEA security is weaker than the IND-
CPA security for most of the controllers formulated by (1). Although one may
think that this is a negative result, it suggests that homomorphic encryption
satisfying IND-CPA security may not necessarily be required to prevent the
parameter estimation attack. In other words, the IND-PEA security might be
achieved using more lightweight homomorphic encryption schemes than the
conventional schemes to reduce the computation costs. This implication is
significant because encrypted control generally increases computational costs,
and control systems require real-time computation.

##  IV Further Development

The security definition and analysis presented in this study were based on the
assumption that the control law (1) was employed. Although the control law
represents all static controllers with a controller parameter $\bm{K}$, there
is scope for further expansion. The results of this study can be expanded upon
using a dynamical controller $\bm{x}_{t+1}=f(\bm{x}_{t},\bm{y}_{t};\bm{K})$,
$\bm{u}_{t}=g(\bm{x}_{t},\bm{y}_{t};\bm{K})$, a time-varying controller
$\bm{u}_{t}=f(\bm{y}_{t};\bm{K}_{t})$ instead of (1), where $\bm{x}$ is a
controller state, and $t$ is a time step. According to this change, the
encrypted control oracle must be modified.

For example, the modified oracle for the dynamical controller must include a
state updated by each query of an attacker and return an output based on the
current state and input. Note that the oracle state is hidden against the
attacker. The security of dynamical controllers under a parameter estimation
attack requires to be analyzed while considering the state trajectory of the
controller because the controller output depends on both the controller state
and the parameter. Such analysis is challenging because it needs to consider
the effect of fundamental properties of dynamical systems, such as stability,
controllability, and observability, on the provable security for encrypted
control. Moreover, an attacker may aim to disclose the (initial) state of an
encrypted dynamical controller. In this case, the security is defined through
a game that differs from the IND-PEA game.

Furthermore, the security of controllers formulated by stochastic processes
such as a Markov decision process can also be considered. An example of this
type of controller is reinforcement learning, which is an application of
encrypted control [21]. In this case, the state of the environment randomly
moves in each time step, and an action, which is an input for the environment,
is determined by a policy that maximizes the expected cumulative reward. An
attacker for an agent of reinforcement learning may be interested in the
policy rather than the controller parameter. Further research on provably
secure encrypted control involves considering such stochastic controllers.

##  V Conclusions

This study defined a novel security for encrypted control under a parameter
estimation attack in terms of provable security in cryptography and analyzed
its security strength. The definition enables us to prove the security for
encrypted control by mathematical procedures. The analysis revealed that
encrypted control is secure if homomorphic encryption used for the control is
secure. This result means that most existing encrypted controls are secure
under a parameter estimation attack. The proposed security can be extended in
the future to encrypted control using dynamical controllers.

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