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We consider sequential detection based on quantized data in the presence ofeavesdropper. Stochastic encryption is employed as a counter measure that flipsthe quantization bits at each sensor according to certain probabilities, andthe flipping probabilities are only known to the legitimate fusion center (LFC)but not the eavesdropping fusion center (EFC). As a result, the LFC employs theoptimal sequential probability ratio test (SPRT) for sequential detectionwhereas the EFC employs a mismatched SPRT (MSPRT). We characterize theasymptotic performance of the MSPRT in terms of the expected sample size as afunction of the vanishing error probabilities. We show that when the detectionerror probabilities are set to be the same at the LFC and EFC, every symmetricstochastic encryption is ineffective in the sense that it leads to the sameexpected sample size at the LFC and EFC. Next, in the asymptotic regime ofsmall detection error probabilities, we show that every stochastic encryptiondegrades the performance of the quantized sequential detection at the LFC byincreasing the expected sample size, and the expected sample size required atthe EFC is no fewer than that is required at the LFC. Then the optimalstochastic encryption is investigated in the sense of maximizing the differencebetween the expected sample sizes required at the EFC and LFC. Although thisoptimization problem is nonconvex, we show that if the acceptable tolerance ofthe increase in the expected sample size at the LFC induced by the stochasticencryption is small enough, then the globally optimal stochastic encryption canbe analytically obtained; and moreover, the optimal scheme only flips one typeof quantized bits (i.e., 1 or 0) and keeps the other type unchanged.

Asymptotically Optimal Stochastic Encryption for Quantized Sequential Detection in the Presence of Eavesdroppers