  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.
