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.

中文翻译：

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存在窃听者的量化序列检测的渐近最优随机加密

我们考虑在窃听者存在的情况下基于量化数据的顺序检测。采用随机加密作为对策，根据一定的概率翻转每个传感器的量化位，翻转概率只有合法融合中心（LFC）知道，而不是窃听融合中心（EFC）知道。因此，LFC 使用最佳顺序概率比测试 (SPRT) 进行顺序检测，而 EFC 使用不匹配的 SPRT (MSPRT)。我们根据作为消失误差概率的函数的预期样本大小来表征 MSPRT 的渐近性能。我们表明，当 LFC 和 EFC 的检测错误概率设置为相同时，每个对称随机加密都是无效的，因为它导致 LFC 和 EFC 的预期样本大小相同。接下来，在小检测错误概率的渐近机制中，我们表明每个随机加密都会通过增加预期样本大小来降低 LFC 量化顺序检测的性能，并且 EFC 所需的预期样本大小不低于在 LFC 需要。然后在最大化 EFC 和 LFC 所需的预期样本大小之间的差异的意义上研究最佳随机加密。虽然这个优化问题是非凸的，但我们表明，如果随机加密引起的 LFC 预期样本大小增加的可接受容差足够小，则可以解析得到全局最优随机加密；而且，最优方案只翻转一种类型的量化位（即1或0），而保持另一种类型不变。