2026/07/16 by Jiuxiang Cao, Guang Cheng, Guangjie Liu
#cs.CR
On multi-hop encrypted links such as Tor and cascaded VPNs, tunneling flattens packet lengths and protocol fields, leaving inter-packet delay (IPD) as the main carrier for active flow attribution. Causality lets the embedder delay packets but never advance them, so each quantization-index-modulation (QIM) alignment injects nonnegative dwell into a delay buffer; unbounded dwell breaks lattice alignment and delays the host connection unacceptably. Whether a causal QIM watermark embeds stably on bursty traffic has largely been left to empirical configuration rather than analysis. We model the embedder as a reflected dwell queue under the fixed dual-lattice, equiprobable-bit rule, where injection is state-dependent -- set by the current interval and bit -- rather than exogenous. The substitution Yi=δi-ri gives only an algebraic Lindley-form identity; stability is governed by the busy-state drift at large dwell, where the effective interval collapses to zero and the mean injection becomes Δ/4. Away from the critical boundary, the buffer is stable iff μd>Δ/4 (i.e. Δ<4μd) for i.i.d. backgrounds, and, under stationary-ergodic and finite-state Markov-modulated traffic with instantaneous overload, iff the time-average intensity ρ<1. With the exogenous decoding floor Δ≥ cσξ (c=4Q-1(ε/2)), this yields the operating window Δ∈[cσξ,4μd). Simulations confirm a sharp transition at ρ=1 set only by the mean; on four real IPD traces, with each simulated chain confined to a single flow, the criterion gives the correct stability direction under flow-local correlation and burstiness, while pooled cross-flow means overestimate the margin. These results give a testable stable-embeddability criterion and a quantization-step configuration baseline for causal QIM network flow watermarking.