Stability Analysis and Dynamic Quantizer for Controller Encryption

Shimada

IET Control Theory & Appl

2020

Self Cite

“…In this case, we can interpret that

normalΦ = F

ξ false(t false) = x false(t false)

, and

ψ false(t false) = u false(t false)

Section: Encrypted Control System With Dynamic Elgamal Cryptosystemmentioning

confidence: 97%

“…Definition 3 Let

C_{Φ}

C_{ξ} false(t false)

, and

C_{Ψ} false(t false)

be ciphertexts of

normalΦ

ξ false(t false)

, and

normalΨ false(t false)

at time

t \in Z^{+}

, respectively. Suppose a controller f is given as

f = f^{+} \circ f^{\times}

[5], and

ε

is modified to

ε^{⋆} = false(G e n, E n c, {sans-serifD sans-serife sans-serifc}^{+}, {sans-serifE sans-serifc sans-serifd}_{γ}, {sans-serifD sans-serifc sans-serifd}_{γ} false)

, where

{sans-serifE sans-serifc sans-serifd}_{γ}

and

{sans-serifD sans-serifc sans-serifd}_{γ}

are an encoder and decoder, respectively [22]. Then,

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ {E c d}_{γ} & : R ∋ x \mapsto \overset{ˇ}{x} = ⌈ γ x + α (γ x) ⌋ \in M, \\ α (γ x) & : = \{\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em... \end{matrix} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

\overset{false¯}{F} = {sans-serifD sans-serifc sans-serifd}_{γ_{c}} false({sans-serifE sans-serifc sans-serifd}_{γ_{c}} false(F false) false)

be a quantised state‐feedback gain. Assume that

A + B F

is a Schur matrix [22], and

u false(t false) = \overset{false¯}{F} x false(t false)

. If

γ_{c}

satisfies

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ | γ_{c} | > \frac{1}{Ω (P, Q)} d_{\max}, \\ Ω (P, Q) : = \frac{2}{\sqrt{m n} ∥ B^{normal⊤} P B ∥} (- ∥ (A + B F)^{normal⊤} P B ∥ \\ + \sqrt{∥ false(A + B F false)^{⊤} P B ∥^{2} + λ_{\min} false(Q false) ∥ B^{⊤} P B ∥}), \end{matrix}

for a given matrix

Q = Q^{⊤} > 0

and the corresponding matrix

P = P^{⊤} > 0

satisfying

false(A + B F false)^{⊤} P false(A + B<...>

…”

Section: Encrypted Control System With Dynamic Elgamal Cryptosystemmentioning

confidence: 99%

“…□ Remark 4 In encrypted state‐feedback control systems, the following homomorphism holds:

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ sans-serifD sans-serife sans-serifc (sans-serifE sans-serifn sans-serifc ({E c d}_{γ_{c}} (F), sans-serifp sans-serifk (t)) * sans-serifE sans-serifn sans-serifc ({E c d}_{γ_{p} (t)} (x (t)), sans-serifp sans-serifk (t)) \mod p, sans-serifs sans-serifk (t)) \\ = f^{\times} ({E c d}_{γ_{c}} (F), {E c d}_{γ_{p} (t)} (x (t))) . \end{matrix}

Remark 5

scriptM

is a finite set. Therefore,

μ_{c}

and

μ_{p}

should be chosen such that encoded values do not overflow or underflow [22]. Remark 6 In general, not all elements of

x false(t false)

are directly observable.…”

Section: Encrypted Control System With Dynamic Elgamal Cryptosystemmentioning

confidence: 99%

See 2 more Smart Citations

Stability‐guaranteed dynamic ElGamal cryptosystem for encrypted control systems

Shimada

IET Control Theory & Appl

2020

Self Cite

“…The quantization approach for average consensus control, event-triggered control, and control of nonlinear scalar systems were also considered [15,16]. The authors of [17] proposed a dynamic quantizer for encrypted control systems with ElGamal encryption. The dynamic quantizer guarantees that a closed-loop system with an encrypted state-feedback controller inherits the asymptotic stability of an unencrypted closed-loop system.…”

Section: Introductionmentioning

confidence: 99%

ElGamal-type encryption for optimal dynamic quantizer in encrypted control systems

SICE Journal of Control, Measurement, and System Integration

2021

Self Cite

This study considers a quantizer design problem with controller encryption for minimizing performance degradation caused by encryption. It is difficult to design an optimal dynamic quantizer that converts real numbers to plaintexts for encrypted control systems with ElGamal encryption because the plaintext space of ElGamal encryption is intermittent and does not include zero and negative numbers. A variant of ElGamal encryption is proposed to apply a conventional optimal dynamic quantizer for encrypted control systems. The proposed multiplicative homomorphic cryptosystem, wherein the plaintext space is consecutive integers within a certain range, can handle zero and negative integers properly. Numerical simulations demonstrate that the optimal dynamic quantizer with the proposed cryptosystem improves the control performance of an encrypted regulator.

show abstract

Control-theoretic Approach to Malleability Cancellation by Attacked Signal Normalization

2019

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