Square-Root Law for Covert Communication with Warden-Favorable Side Information
Covert communication enables Alice to transmit to Bob while making the transmission difficult for Willie to detect. We study a scalar Gaussian covert-overlay model in which Alice's low-power covert signal is superimposed on an aggregate public component generated by Alice or other trackable sources. Willie is given all physically obtainable side information, including protocol details, timing, pilots, channel estimates, and calibration information, and subtracts his best estimate of the public component before testing. Covertness is imposed on the resulting residual through a relative-entropy constraint with budget $δ$ conditioned on Willie's side information. In the stationary case, the residual under no covert transmission has variance $σ_0^2=σ_W^2+σ_e^2$, where $σ_W^2$ is Willie's receiver-noise variance and $σ_e^2$ is the irreducible cancellation error. Over $n$ channel uses, the maximal reliably transmissible covert payload is $R_C^\star\sqrt{n}(1+o(1))$ bits, where $R_C^\star=\frac{σ_0^2}{σ_B^2\ln 2}\sqrtδ$, and $σ_B^2$ is Bob's receiver-noise variance. Thus, the square-root-law (SRL) constant is governed by the variance at Willie's actual detector input, not by receiver noise alone. Low-power Gaussian signaling achieves this constant, and a matching converse establishes first-order optimality within the conditioned additive Gaussian innovation model. For known time-varying conditioned residual variances, we also derive the first-order allocation, which assigns more covert power to larger residual variances. The results require a Gaussian post-cancellation null residual with known conditioned variance; non-Gaussian residuals and fixed non-vanishing variance uncertainty are outside the scope of this paper.