Secure Estimation and Zero-Error Secrecy Capacity

Abstract

We study the problem of securely estimating the states of an unstable dynamical system subject to nonstochastic disturbances. The estimator obtains all its information through an uncertain channel which is subject to nonstochastic disturbances as well, and an eavesdropper obtains a disturbed version of the channel inputs through a second uncertain channel. An encoder observes and block-encodes the states in such a way that, upon sending the generated codeword, the estimator's error is bounded and such that a security criterion is satisfied ensuring that the eavesdropper obtains as little state information as possible. Two security criteria are considered and discussed with the help of a numerical example. A sufficient condition on the uncertain wiretap channel, i.e., the pair formed by the uncertain channel from encoder to estimator and the uncertain channel from encoder to eavesdropper, is derived which ensures that a bounded estimation error and security are achieved. This condition is also shown to be necessary for a subclass of uncertain wiretap channels. To formulate the condition, the zero-error secrecy capacity of uncertain wiretap channels is introduced, i.e., the maximal rate at which data can be transmitted from the encoder to the estimator in such a way that the eavesdropper is unable to reconstruct the transmitted data. Lastly, the zero-error secrecy capacity of uncertain wiretap channels is studied.

Bounded State Estimation over Finite-State Channels: Relating Topological Entropy and Zero-Error Capacity

“…When estimating the state of a linear system via a noisy memoryless channel, it is known that the relevant figure of merit for achieving estimation errors that are almost-surely uniformly bounded is the zero-error capacity, C 0 , of the channel, and not the ordinary capacity [12][13][14]. The zero-error capacity is defined as the maximum block coding rate yielding exactly zero decoding errors at the receiver.…”

Zero-Error Feedback Capacity for Bounded Stabilization and Finite-State Additive Noise Channels

“…Next, we show the upper and lower bounds in (32). According to the Perron-Frobenius Theorem, for an irreducible |S|×|S| matrix A (or, equivalently, the adjacency matrix for a strongly connected graph), the entries of eigenvector v ∈ R |S| corresponding to λ are strictly positive [49,Thm.…”

“…Let α := v min /v max = 1/β > 0, hence αλ n ≤ Y(s 1 , x 1:n ) which combining it with (101) results in (32).…”

“…This suits settings where the channel noise distributions are not available or where the noise is not random. In such situations, worst-case analysis is more relevant [28]- [32], which is equivalent to the presence of an omniscient adversary who knows the transmitted codeword a priori and can input malicious noise to the channel [31]. Here, we use the uncertain variable framework introduced in [22] for studying the zero-error capacity.…”

Transmission rate conditions for distributed filtering in sensor networks against eavesdropper

“…The last one is based on secrecy capacity of transmission, which will be applied in the current paper to defend against the eavesdropper. Zero-error secrecy capacity and the secure estimation with nonstochastic disturbances have been studied in Wiese et al (2019), where an uncertain channel is considered as a mapping and the denotation of secrecy capacity has been proposed. The information theory has been employed to study the channel capacity requirement which guarantees the leakage of information to the eavesdropper is negligible (see, for example, Li et al, 2011;Matveev and Savkin, 2009).…”