A Queueing-Stability Criterion for Causal IPD-QIM Network Flow Watermarking
Cryptography and Security
Summary
The authors study how to hide information in delays between packets on encrypted networks like Tor. They focus on a method called causal QIM watermarking, which can only delay packets, not speed them up, and model it using a queue system to understand when this hiding method remains stable without causing too much delay. Their analysis shows that stability depends mainly on the average traffic rate and the delay settings, providing a clear rule to decide when the watermark can be embedded reliably. They confirm their model with simulations and real data, helping guide how to configure these watermarks in practice.
Authors
Jiuxiang Cai, Guang Cheng, Guangjie Liu
Abstract
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 $Y_i=δ_i-r_i$ 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 $μ_d>Δ/4$ (i.e. $Δ<4μ_d$) for i.i.d. backgrounds, and, under stationary-ergodic and finite-state Markov-modulated traffic with instantaneous overload, iff the time-average intensity $\barρ<1$. With the exogenous decoding floor $Δ\ge cσ_ξ$ ($c=4Q^{-1}(ε/2)$), this yields the operating window $Δ\in[cσ_ξ,4\barμ_d)$. 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.