We consider sequential detection based on quantized data in the presence of eavesdropper. Stochastic encryption is employed as a counter measure that flips the quantization bits at each sensor according to certain probabilities, and the flipping probabilities are only known to the legitimate fusion center (LFC) but not the eavesdropping fusion center (EFC). As a result, the LFC employs the optimal sequential probability ratio test (SPRT) for sequential detection whereas the EFC employs a mismatched SPRT (MSPRT). We characterize the asymptotic performance of the MSPRT in terms of the expected sample size as a function of the vanishing error probabilities. We show that when the detection error probabilities are set to be the same at the LFC and EFC, every symmetric stochastic encryption is ineffective in the sense that it leads to the same expected sample size at the LFC and EFC. Next, in the asymptotic regime of small detection error probabilities, we show that every stochastic encryption degrades the performance of the quantized sequential detection at the LFC by increasing the expected sample size, and the expected sample size required at the EFC is no fewer than that is required at the LFC. Then the optimal stochastic encryption is investigated in the sense of maximizing the difference between the expected sample sizes required at the EFC and LFC. Although this optimization problem is nonconvex, we show that if the acceptable tolerance of the increase in the expected sample size at the LFC induced by the stochastic encryption is small enough, then the globally optimal stochastic encryption can be analytically obtained; and moreover, the optimal scheme only flips one type of quantized bits (i.e., 1 or 0) and keeps the other type unchanged.
Stochastic encryptions were originally proposed in @cite_21 for physical-layer security in the context of fixed-sample-size estimation problems with quantized data. For the fixed-sample-size hypothesis testing in sensor networks, the joint design of the stochastic encryption and the LFC decision rule that minimizes the LFC detection error probabilities subject to a constraint on the EFC error probabilities is studied in @cite_31 . Nonetheless, the design approach in @cite_31 is ad hoc and results in a suboptimal stochastic encryption. This design approach is made more rigorous in @cite_17 , where the optimal stochastic encryption is obtained with respect to the J-divergence which is adopted as the performance metric for both LFC and EFC. However, the approach proposed in @cite_17 cannot be applied to sequential detection.
@cite_31 @cite_21 @cite_17
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The problem of binary hypothesis testing is considered in a bandwidth-constrained densely populated low-power wireless sensor network operating over insecure links. Observations of the sensors are quantized and encrypted before transmission. The encryption method maps the output of the quantizer to one of the possible quantizer output levels randomly according to a probability matrix. The intended (ally) fusion center (AFC) is aware of the encryption keys (probabilities) while the unauthorized (third party) fusion center (TPFC) is not. A constrained optimization problem is formulated from the point of view of AFC in order to design its decision rule along with the encryption probabilities. The objective function to be minimized is the error probability of AFC and the constraint is a lower bound on the error probability of TPFC. In the binary case the optimal solution is found and in the nonbinary case a good suboptimal solution is analytically obtained. Numerical results are presented to show that it is possible to degrade the error probability of TPFC significantly and still achieve very low probability of error for AFC. The proposed method which may be considered a PHY-layer security scheme is highly scalable since it does not increase the packet overhead or transmit power of the sensors and has very low computational complexity. A scheme is described to randomize the keys so as to defeat any key space exploration attack. We consider the problem of secure detection in wireless sensor networks operating over insecure links. It is assumed that an eavesdropping fusion center (EFC) attempts to intercept the transmissions of the sensors and to detect the state of nature. The sensor nodes quantize their observations using a multilevel quantizer. Before transmission to the ally fusion center (AFC), the senor nodes encrypt their data using a probabilistic encryption scheme, which randomly maps the sensor's data to another quantizer output level using a stochastic cipher matrix (key). The communication between the sensors and each fusion center is assumed to be over a parallel access channel with identical and independent branches, and with each branch being a discrete memoryless channel. We employ J-divergence as the performance criterion for both the AFC and EFC. The optimal solution for the cipher matrices is obtained in order to maximize J-divergence for AFC, whereas ensuring that it is zero for the EFC. With the proposed method, as long as the EFC is not aware of the specific cipher matrix employed by each sensor, its detection performance will be very poor. The cost of this method is a small degradation in the detection performance of the AFC. The proposed scheme has no communication overhead and minimal processing requirements making it suitable for sensors with limited resources. Numerical results showing the detection performance of the AFC and EFC verify the efficacy of the proposed method. We consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion in the setting of stochastic search, with application to target localization. Each player yields a noisy response to a binary query governed by a certain error probability. First, we propose a sequential policy for constructing questions that queries each player in sequence and refines the posterior of the target location. Second, we consider a joint policy that asks all players questions in parallel at each time instant and characterize the structure of the optimal policy for constructing the sequence of questions. This generalizes the single player probabilistic bisection method for stochastic search problems. Third, we prove an equivalence between the two schemes showing that, despite the fact that the sequential scheme has access to a more refined filtration, the joint scheme performs just as well on average. Fourth, we establish convergence rates of the mean-square error and derive error exponents. Finally, we obtain an extension to the case of unknown error probabilities. This framework provides a mathematical model for incorporating a human in the loop for active machine learning systems. This paper introduces clockless stochastic decoding for high-throughput low-density parity-check (LDPC) decoders. Stochastic computation provides ultra-low-complexity hardware using simple logic gates. Clockless decoding eliminates global clocking, which eases the worst-case timing restrictions of synchronous stochastic decoders. The lack of synchronization might use outdated bits to update outputs in computation nodes; however, it does not significantly affect output probabilities. A timing model of clockless-computation behaviours under a 90 nm CMOS technology is used to simulate the BER performance of the proposed decoding scheme. Based on our models, the proposed decoding scheme significantly reduces error floors due to the "lock-up" problem and achieves superior BER performance compared with conventional synchronous stochastic decoders. The timing model includes metastability to verify the affect on BER performance.