Replay attack detection based on smooth watermarking and encoding and decoding

Model

A discrete time-delay system in the form of the following equations is considered:

where x ( k ) ∈ R n x , u ( k ) ∈ R n u , and y ( k ) ∈ R n y indicate the state vector, the control input vector, and the output vector of the system, respectively. A 1 , A 2 , B , and C are the known matrices with appropriate dimensions. The constant time delay is symbolized by τ > 0 . The process noise is W ( k ) ~ N ( 0 , Q ) , and v ( k ) ~ N ( 0 , R ) means the measurement noise. They are Gaussian random variables and independent of each other. Q and R are the corresponding non-negative definite covariance matrices.

The delay-free system

Solving the time-delay problem in the system model is crucial, prompting the revision of the original model to remove delays. Extending the system variables creates a larger state space for achieving the delay-free system, which attenuates the system’s complexity due to time delay, making it easier to handle and analyze. Let z ( k + 1 ) = [ x ( k + 1 ) x ( k ) ⋯ x ( k − τ + 1 ) ] T , Ψ ( k ) = [ y ( k ) y ( k − 1 ) ⋯ y ( k − τ ) ] T ; then equation (1) can be reconstructed as

where A = [ A 1 0 n x ⋯ 0 n x A 2 I n x 0 n x ⋯ 0 n x 0 n x ⋱ ⋱ ⋮ ⋮ ⋱ 0 n x 0 n x I n x 0 n x ] , B = [ B 0 n u ⋯ 0 n u ] T , Γ = diag [ I n w 0 n w ⋯ 0 n w ] , C = diag [ C ⋯ C ] , V ( k ) = [ v ( k ) ⋯ v ( k − τ ) ] T , W ( k ) , and V ( k ) are mutually independent. V ( k ) ~ N ( 0 , R ¯ ) and the symmetric positive definite covariance matrix is R ¯ .

Event-triggered Kalman filter estimation

The data need to be transferred via LANs, WANs, and even the Internet. The Kalman filter (Li et al., 2024) is integrated with the event-triggered communication strategy (Li et al., 2021) to optimize transmission efficiency and improve resource utilization.

According to Liu et al. (2017), there exists a unitary matrix satisfying the following conditions:

where λ 1 ( k ) , λ 2 ( k ) , ⋯ , λ n y ( k ) is the eigenvalue of CP ( k | k − 1 ) C T + R ¯ at time k . Set Ω ( k ) = Δ ℑ ( k ) ( diag [ λ 1 ( k ) , λ 2 ( k ) , ⋯ λ n y ( k ) ] ) − 1 / 2 ∈ R n y × n y . The Ω ( k ) is determined for each moment based on the preceding formula, and define the measurement update as Ψ ~ ( k ) = Ψ ( k ) − Ψ ^ ( k | k − 1 ) . Let ℵ ( k ) = Δ Ω ( k ) T Ψ ~ ( k ) ; the event-triggered mechanism can be set as

where μ > 0 is the triggering parameter and the data set of the actual communication at time k is N ( k ) = Δ { Δ ( 0 ) Ψ ( 0 ) , ⋯ , Δ ( k ) Ψ ( k ) } ∪ { Δ ( 0 ) , ⋯ , Δ ( k ) } . By evaluating the stated conditions, Δ ( k ) is obtained. Once Δ ( k ) = 1 , the latest data is transmitted to the estimator, causing the filter to become a conventional Kalman filter. In the case of Δ ( k ) = 0 , the state estimation z ^ ( k | k ) is replaced by z ^ ( k | k − 1 ) , subsequently influencing the measurement update. The detailed procedure of Kalman estimation based on equation (4) is as follows:

Taking into account the triggering condition, if the posterior estimate and the posterior covariance of the Kalman filter at time k − 1 are z ^ ( k − 1 | k − 1 ) and P ( k − 1 | k − 1 ) , the prior estimation and prior covariance after time update can be expressed as (Huang et al., 2024):

When Δ ( k ) = 1 , N ( k ) = N ( k − 1 ) ∪ { Δ ( k ) = 1 } , the posterior estimate can be characterized as

Ψ ( k ) is transmitted by the sensor to the estimator, resulting in the estimation error being ( z ( k ) − z ^ ( k | k − 1 ) ) − K ( k ) Ψ ~ ( k ) . As indicated in Liu et al. (2017), it is evident that

Let the partial derivative of P ( k | k ) to K ( k ) be zero, and we can deduce

and

When Δ ( k ) = 0 , the posterior estimate is z ^ ( k | k ) = z ^ ( k | k − 1 ) .

Following the calculation, it became clear that

where ℓ ( μ ) = 2 2 π μ e − μ 2 2 ( 1 − 2 θ ( μ ) ) − 1 , θ ( μ ) = Δ ∫ μ + ∞ 1 2 π e − z 2 2 dz .

Combined with the preceding content, the event-triggered Kalman estimation is

The following equation is chosen as the linear quadratic Gaussian (LQG) cost (Su et al., 2020):

where Z and U are the corresponding positive definite weight matrices. Define S = Δ lim k → ∞ S ( k ) ; S ( k ) = A T S ( k + 1 ) A − A T S ( k + 1 ) B ( B T S ( k + 1 ) B + U ) − 1 B T S ( k + 1 ) A + Z satisfies the Riccati difference equation. The optimal control law at steady state can be achieved:

where H = − ( B T SB + U ) − 1 B T SA is the gain matrix of optimal control. The optimal control performance loss is (Mo et al., 2014)

where tr is the trace of the corresponding matrix and K and P are the steady-state values of filter gain and prediction error covariance corresponding to the system at steady state, respectively.

Smoothing of watermark

A replay attack is a well-known threat that is often found on stable systems (Li et al., 2022). During message delivery, the communication link may be threatened by a potential replay attack. The watermark signal is viewed as a successful tool in defending against attacks. The watermark signals, labeled as Δ u ¯ ( k ) ~ N ( 0 , Λ ) , consist of Gaussian random noises with the same probability distribution and being mutually independent (Naha et al., 2023). Λ is the diagonal variance matrix of normal distribution. Using a weighted sliding average method helps keep the watermark signal steady when faced with sudden changes such as system disruptions, abnormal occurrences, or external influences.

where Δ u ( k ) is the watermark signal after smoothing, ϖ ( k ) represents weight and is independent of Δ u ( k ) , ϖ ( k ) ~ U ( 0 , 1 ) . As ϖ ( k ) is uniformly distributed between (0,1), the observed value obtained after weighting calculation is typically lower than the true value. The preceding equation is modified by incorporating a correction factor to address this deviation:

where ϖ ¯ ( k ) = ∑ ρ = 1 L ( ϖ ( k − ρ ) + ϖ ( k + ρ ) ) + ϖ ( k ) 2 L + 1 and L signifies the radius of the sliding window.

Coding and decoding

The watermark signal, through encoding, boosts data security and allows traceability, reducing chances of tampering. Before being transmitted to the Kalman estimator, the data must be encoded. The coding model can be specified as

The model of the system after adding the smoothed watermark signal and the coded signal is

where the initial state is z 0 * and z 0 * = z 0 . Decode the encoded data at the state estimator receiver side to restore the original system model. The decoded signal can be written as

The initial values of the states of the encoded and decoded signals satisfy z 0 c = z 0 d . During stable operation of the system, the signal received by the Kalman estimator receiver is

The final output signal Ψ ke ( k ) can be accurately restored to the output signal Ψ ( k ) of the original system model. The state prior estimation after adding smooth watermarking, encoding, and decoding is recorded as z ^ ke ( k | k − 1 ) . Comparing the priori estimates z ^ ke ( k | k − 1 ) with z ^ ( k | k − 1 ) , it can be seen that

where Ξ = ( A + BH ) ( I − K C ) . The invisibility of a replay attack is discussed in depth by Mo et al. (2014). When Ξ is unstable, the χ 2 detector can accurately identify the attack signal. If Ξ is stable, the detector has trouble spotting anomalies as the attacker inserts legitimate historical messages, enabling the replay attack to evade detection. The following analyzes how the detector identifies a replay attack while Ξ remains stable. After adding the smoothing watermark, coding signal, and decoding signal to the system, the state prior estimation is equal to the initial state prior estimation when the system is running smoothly. Figure 1 presents a comprehensive overview of the proposed methodology.

OPEN IN VIEWER

Figure 1.

The system frame diagram.

Detectability of replay attack

Provided that the system operates stably, the residual signal after Kalman state estimation is

It is apparent from the preceding equation that the residual signal in the current case is consistent with the residual signal when no signal has been added, thus preserving the stability of the system.

The detection function ξ ke ( k ) following Kalman estimation can be outlined as

where ξ ( k ) serves as the primary detection signal, ξ ke ( k ) follows χ 2 distribution, the degree of freedom is m ( τ + 1 ) , and δ is the detection threshold. When the communication link is not subjected to a replay attack, that is, H is satisfied, it stays within the range of the set detection threshold. If H ′ is satisfied, it indicates that the communication link is vulnerable to a replay attack. If the false alarm rate is set to L , then we can get Prob ( r T ( k ) Π − 1 r ( k ) > δ | H ) < L (Guan and Ge, 2018).

Assume that the communication link is exposed to a replay attack in the time domain of [ 0 , k ] and the attacker opts to replay legitimate historical data Ψ * ′ ( k ) . The corresponding state and watermark are z * ′ ( k ) and Δ u ′ ( k ) . r ke ′ ( k ) is the residual signal after decoding, while z ^ ke ′ ( k | k − 1 ) reflects the state prior estimation.

Proof. When the replay attack occurs, it is known that the attacker’s historical legitimate data satisfies z ^ ke ′ ( k | k − 1 ) = z ^ ′ ( k | k − 1 ) , Ψ ke ′ ( k ) = Ψ ′ ( k ) , and r ke ′ ( k ) = r ′ ( k ) . The actual residual is

where z ^ ′ ( k | k − 1 ) − z ^ ( k | k − 1 ) = Ξ k ( z ^ ′ ( 0 | − 1 ) − z ^ ( 0 | − 1 ) ) + ∑ ρ = 0 k − 1 Ξ k − 1 − ρ ( A + B H ) K ( Ψ ′ ( ρ ) − Ψ ( ρ ) ) . The attacker conducted a replay attack, replacing real data with historical data, leading to Ψ ( ρ ) = Ψ ′ ( ρ ) , ρ = 0 , 1 , . . . , k . The system’s consistent stability over time suggests that z ^ ′ ( k | k − 1 ) − z ^ ( k | k − 1 ) = 0 .

where u ′ ( k − 1 ) − u ( k − 1 ) = 0 . Let Δ u ′ ( k ) − Δ u ( k ) = Ω ( k ) ; the injected data from the attacker contains the encoding model. The decoding model’s independence prevents it from being susceptible to replay attacks, as it belongs to an external system. From z 0 d ′ = z 0 d , we can get

The residual of the system subjected to a replay attack can be described as

where r ′ ( k ) is the historically legitimate residual. ξ ke ¯ ( k ) can be reformulated as

where ξ ke ′ ( k ) is a legitimate history detection value. Based on Δ u ′ ( k ) ≠ Δ u ( k ) , when a replay attack is implemented on the communication link, it holds that ξ ke ¯ ( k ) > ξ ke ′ ( k ) , indicating the detector’s capability to capture the hacker’s replay attack.

Suppose that after adding the smooth watermarking, the state variables z * ( k ) of the system satisfy z * ( k ) = z ( k ) + z ~ ( k ) and z ~ i ( k ) is the difference between the state with and without smooth watermarking. Let ε ( k ) = [ u ( k ) z ( k ) z * ( k ) − z ( k ) ] T ; the LQG cost is restated as

where Θ = [ U 0 UH 0 Z Z H T U Z Z ] and the additional performance loss is Δ J = J − J ¯ . The additional performance loss that injects the unsmoothed watermark signal is Δ J ~ = tr ( ( B T SB + U ) Λ ) . J and Δ J after adding the smoothed watermark signal are J = tr ( Cov { ε ( k ) Θ } + 2 Λ U 2 + 3 L ) and Δ J = 2 Δ J ~ 2 + 3 L .

It can be concluded that as L increases, the variance of the smooth watermarking decreases, ultimately reducing Δ J . In recognizing the replay attack, aim for a high variance in the smooth watermark signal. The larger the variance, the better the detector’s indicator and the easier it is to recognize anomalies. Obviously, L cannot be set to infinity in real production. This means that L needs to be enlarged to minimize Δ J while ensuring that the watermark signal is large enough for the system to sense replay attacks in time. The proof is complete.

The system performance

The best LQG performance can be calculated by thoroughly assessing the sliding window radius, watermark signal variance, residual covariance, detection rate, and false alarm rate in the event of a replay attack. The aim is to strike a balance between system stability and the capacity to detect anomalies, guaranteeing efficient and orderly system operation during an attack.

where ϑ is the upper bound of Δ J caused by adding the smooth watermark signal and ∂ is a lower bound of the mathematical expectation of Δ ξ . Let ς = ∑ ρ = 0 ∞ ( ( A ρ B ) Λ ( A ρ B ) T ) = A ς A T + B Λ B T .

The asymptotic expectation of the detection function under attack is

Let the parameter Δ ξ conform to Δ ξ = 4 2 + 3 L tr ( C ς C T ) , so if Δ ξ > δ − m ( τ + 1 ) , that is, the value of the detection function exceeds the preset threshold, the detector is triggered, indicating that the communication link is under a replay attack.

Given that the false alarm rate is under L , it is acknowledged:

In the case where H holds, r k e T ( k ) Π − 1 r ke ( k ) obeys the χ 2 distribution, and the preceding equation is equivalent to δ ≥ χ L . Let’s use ι to symbolize the detection rate of the attack. When it is certain that the detection rate is not lower than ι , it is recognized that

As a result, the inequality below is valid:

where the smallest eigenvalues of Π ¯ and Π − 1 are denoted as λ min , Π ¯ and λ min , Π − 1 , respectively, while λ max , Π represents the largest eigenvalue of Π . The calculation gives r k e T ( k ) Π − 1 r ke ( k ) ≥ λ min , Π ¯ λ max , Π r k e T ( k ) Π ¯ − 1 r ke ( k ) .

Furthermore, the inequality δ λ min , Π ¯ / λ max , Π ≤ χ ι holds.